{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2026-04-16T00:12:35.000Z","description":"Problems submitted by members of the MATLAB Central community.","is_default":true,"created_by":161519,"badge_id":null,"featured":false,"trending":false,"solution_count_in_trending_period":0,"trending_last_calculated":"2026-04-16T00:00:00.000Z","image_id":null,"published":true,"community_created":false,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":null,"description_html":null,"published_at":null},"problems":[{"id":61080,"title":"Calculate Reynolds Number","description":"Write a MATLAB function that calculates the Reynolds number for flow over a flat plate:\r\n​\r\nρ = fluid density (kg/m³)\r\nV = velocity (m/s)\r\nL = characteristic length (m)\r\nμ = dynamic viscosity (Pa·s)","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.440001px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 142.75px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 71.375px; transform-origin: 408px 71.375px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eWrite a MATLAB function that calculates the Reynolds number for flow over a flat plate:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"81\" height=\"19\" style=\"width: 81px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 81.75px; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 40.875px; transform-origin: 392px 40.875px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4375px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.21875px; text-align: left; transform-origin: 364px 10.21875px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eρ \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e= fluid density (kg/m³)\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4375px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.21875px; text-align: left; transform-origin: 364px 10.21875px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eV \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e= velocity (m/s)\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4375px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.21875px; text-align: left; transform-origin: 364px 10.21875px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eL\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e= characteristic length (m)\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4375px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.21875px; text-align: left; transform-origin: 364px 10.21875px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eμ\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e= dynamic viscosity (Pa·s)\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function Re = calculateReynolds(rho, V, L, mu)\r\n% CALCULATEREYNOLDS computes the Reynolds number\r\n%\r\n% Inputs:\r\n%   rho - fluid density in kg/m^3\r\n%   V   - velocity in m/s\r\n%   L   - characteristic length in m\r\n%   mu  - dynamic viscosity in Pa*s\r\n%\r\n% Output:\r\n%   Re  - Reynolds number (dimensionless)\r\n\r\n% Your code here\r\n\r\nend\r\n","test_suite":"%% Test 1\r\nRe1 = calculateReynolds(1.225, 10, 0.5, 1.8e-5);\r\nassert(abs(Re1 - 340277.7778) \u003c 1e-1, 'Test 1 failed');\r\n\r\n%% Test 2\r\nRe2 = calculateReynolds(1.0, 15, 1, 2e-5);\r\nassert(abs(Re2 - 750000) \u003c 1e-1, 'Test 2 failed');\r\n\r\n%% Test 3\r\nRe3 = calculateReynolds(1.2, 5, 0.2, 1.5e-5);\r\nassert(abs(Re3 - 80000) \u003c 1e-1, 'Test 3 failed');\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":2,"created_by":4707073,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":34,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2025-11-18T11:13:52.000Z","updated_at":"2026-03-04T21:48:28.000Z","published_at":"2025-11-18T11:13:52.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a MATLAB function that calculates the Reynolds number for flow over a flat plate:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eRe=\\\\rho VL/\\\\mu\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e​\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eρ \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e= fluid density (kg/m³)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eV \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e= velocity (m/s)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eL\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e= characteristic length (m)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eμ\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e= dynamic viscosity (Pa·s)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60778,"title":"Complete hydraulic geometry relations","description":"Hydraulic geometry relations express the velocity , width , and depth  of a river as a function of the discharge (or flow) , which is the volume of water that passes a cross section in a unit time. These relations have the form (e.g., Leopold and Mattuck 1953)\r\n\r\nwhere the coefficients have the appropriate dimensions. \r\nWrite a function that takes two each of the coefficients and exponents and determines the third such that the relation between flow and velocity (i.e., ) is satisfied. The coefficients and exponents will be given as 1x3 vectors in the order velocity, width, and depth, and the unknown values will be given as NaN. ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 200px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 100px; transform-origin: 407px 100px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eHydraulic geometry relations express the velocity \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eV\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, width \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eB\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and depth \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eH\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e of a river as a function of the discharge (or flow) \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eQ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, which is the volume of water that passes a cross section in a unit time. These relations have the form (e.g., Leopold and Mattuck 1953)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 26px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 13px; text-align: left; transform-origin: 384px 13px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZcAAAA0CAYAAABGmXX5AAAAAXNSR0IArs4c6QAAFmBJREFUeF7t3QW0bV1VB/CJ3ditYKMOFbsDFQsLsbu7MLAVwcbuxMDu7m5BEMXuLhRsxT4/WOuNddfd++w4+5x7z31zjfHG9713dqw119wz/3OuO0SOpEBSICmQFEgKbEyBO2z8vHxcUiApkBRICiQFIpVLMkFSICmQFEgKbE6BVC6bkzQfmBRICiQFkgKpXJIHkgJJgaRAUmBzCqRy2Zyk+cCkQFIgKZAUSOWSPJAUSAokBZICm1MglcvmJM0HJgWSAkmBpEAql+SBpEBSICmQFNicAqlcNidpPjApkBRICiQFUrmcPw88S0S8W0TcKSJeLCI+JiJ+4PyXlSvYgAJPFRHvHBEvGBF3jYiviogvjoj/2+DZ+YibSQE64WUi4m0igmzBQ+8SEX++dLmpXJZS7Ppc//gR8T4RcZ+IeMeI+KUdU3x/RDxJRLxWRDz6+kw1Z3JiCviu7xERXxERHxYRD4qIr4mIV4+IV4yIPzzxfPJ150GBpy3GxzNFxFvs+Oc5IuKnIuKzIuKjli4hlctSil2P6+3bh0bEJ0fEu0fEAyPilXZ/fjQiPjciPiIi/vt6TDVncQUUuFdEfHNEfFJE3K94tT+9UzgPiYi33Cmcf7+COeUrrzcF7hgR3xYRzxcRrxcRv1l4h/H6hhHxQ0unn8plKcWux/UvubNCCYuH7v68QUQ8fxEm31ssjH/qpvnsEXH/Eg4RJslxcynA2vzxnWB4vIh4jbJMikZY4/2b8MYzF56giJ4mIv4kIj59Fwr5kh0P/dfNJU+ubIQCPNxPiYiPjIgHlKiI///AIlv+J+Kx7cLeJCI+MSJeICL+NyK+dRcl+YCI+Ov+ualczo/X7BlPRSjsZ3ZeymMi4vcj4st3FsbDuni6eCkGETN9+hIaeYfzW3LOeAEFqpB4RFEkwqNfGRE/scvHERAGY4Ml+hc7S/XrihJ6u/Kb+wmXzMssIPqZX/psEfFzhS++JyKeuuRthVP/qlnbW0fEvSPig4oy+fiIeKuI+MaIwD8XoiWpXM6PKygJcdBnjYhXi4hf3bOEJypC4nUi4rtTuZzfZi+csXyb0Iac25tGxHeN3H/fwj9ydtVLEWP/hh0A4E9LXobiyXF7UOA1I+KHd/v+I8Uz+deBZcvDMDo+eOfVPLL87t9+MiKeLCJerlNE2RX5DHkHguMXd/F0iqNNzkrwv9Aupv7bA2ENobNULme42Qun/BQRITSKL+4WET9b7mdECmP8WQltCH19XPF46yuq0XLnXVjtVUrIdeHr8/IzpUCVDwwTObnqgYh8CJ/+bvnvExYeqsus/Maoee2dV/OP7frTczk/brCRLMw3LvmVTy2QQVYFy+Orm/BHXV0ql/Pb57Uz/owStgA7fu9dfJxA+ISI+Ltief7HLrdCKPxLF/qqgoISSkTZWuqf5332XA73yYvX+wvFOJHM9+fXRpYl+Q9EJExGMV0YqVzOkxnEzL8+Il65TF/uBXrswSOx8lQu57nPa2YN9fOFO2tTfNz4jWKE8GhqzmXoudVz4d3cMxFla0h/1ve8bjFMn6FEPr60JO7bnEu7QLBlyXwy56OH0KljyoVr5IYX7sgl1v/thXnb5M0TRMT7FYTB85R7IAk+vzA2KynHMAWernghCiFfoliarvzbsnngxosLmLpXHVO52HsWMmTS6xeU0tBKxfbBG7+oeF49oi354zIFeB3CWxB+8ihQXQZaQoSBGjMstki+v3xJ8gOK+Ma3GmQJRNobNbxdn02eKOqUEDZ4T2L6bzYgeyhJ9RZqd3LspwDEoET72xY6Qg4aEIEiG7zbtd8f78b3Tj8ImxmUzLsuCYs9V2FcSALjC4oC2cfIz7mLvXGpJJuhB742uWCUAjYGpO99y0ZzLW2SuKUNBBsVzmBt2jjIjbVC5JjKpV0gZJo1Gf+5i/lLFBJ+hKQwHmsIEoV1DDv/8OSPQQrIn715Ebz4hAAW/iQcCArFkIQsISL8hYeGkrBzyWt/7A2l1Sb5594/57qWNxie1ncplFIexOAS4n3x8ndrVM+VEOn9lBbRUOfG8/yDEtJSWC0UiqbKEd6rhEgBOBgoawcP2R5+dknocy44E7fGvrAYi5RFweowWElTVZpQSRbzfeXFWaw1vHWEwzcVeDCUBivjUd2ltVCSUJn6GKcY5CqUyy8PdApgSVGSBquVp5Ne7cXdk0BllPEE5UkoYQZbP7ToACemrL/sQKXgG+cdCaVdSMpOMdaC31vlYt77Qm9QSD+/Qzc+dzGoFPX94IJ33W6XkhWMeeUIDAWADZDhXv76jZMgSvIPJQkvrHXIqN80uY+PKLLHjqmcS8sQYvweNGY9P2kptiE4E20yvl2KkChtG/1jZUPGPuiKP9c3TC0L2rL6l45TKJfeGPm84n21c9X7jCfzlLvwxz+XnNE+KPXSdZ779faZYSYcPfXxt/VOvknQ4zXhLErqM0svKZ7RMUbPG4xUxurY0G1CXY77FOclwGCcVvhA1EMBrDFlaNQCbPDhLQw83hL0Kvl0wVicUi5VKJk0WCMLglAYGncv1gWm+dgDQjjHYO7r8sxWsUiUUcI2ZWz0H+V7Fqtk6XpOoVxaRWh+Yuzgz+2ojFhDraxyRVs5Htd4tCoW9JgTVtZcsIaeL1mOM4jqnYooCadDLdh9r2t5gyKUP4IyGhu1ENTva9Y1Y+k34pJesWjvIzH/93tWJ28nWvJSRUZP7cUUoWpphDCm8P2tMaVcWi2nuOpSoUx5kvgbIQFBwLqWjM5xkQLqB3gqXH1D7FMsfWpIwL19uUjzwTUV9qdQLrUQC0+NWZutRcplf9XS72qKBjf995rzkEw3ILvEs6fCyq3xt+/7HKIfRS+MoveYEFQd5qKztt88c4vR7jvviCcyVqTZG1QfXvJNW8zjpj3jZUvehBcidC4tQcjvGxVy7tszDqWvva0GwIV3TymXqpUk6v9tT7iLRS2WB+WxxjW/aZver8cHy0JEH2OOheG6/kNb68bymCRP1yqnOfsjkS+MagxZm3gN2kcfIqNWA2fO5XHh5pqLaoEQU3RvPZd932f/nNqkEOxUN+12aM8v9Mqg2ar56RJPpPVyvL8tBp2ix+30uwS9ozVeuiz6W0pocwr00Houbp0rE4S0pUbUvCh7gGDlTAjxy6EJrV5ImUwpl17LDYUxnrcU4AibYfapxV1nBoDMoSglN1W6Wz9FYEMIAEnFtoJ17lpqyLBCAud6LXITBDXrwFiqXGy+OKhCqBrHByFkGW8ZX5+yhqCfWOUQSWiAtnjp9+YS8BpcZw0sQ5YeYwH98Mt3FFQfQWiflrazf8ZifVbY/1yvBUnanOhc5SI3ymOu3nBPWgJiywR6bUnjmcaUpdx6wKDrLGzAhnMdDPQPKUcgyC9C0tpjuS6ep+LnKaDU0NrJKbB+Y67X4to+ND1XuQihUmCUmfeBhgPtAA8M1tdNKRe/i+nWgiwYdBqqjmqNUir6XHnh2tGGf9Y+Y+4H1j/fOoQhCD/KhCCkmf27xDR4prEm51GBDoS8IRyg+PGPZixSCE1DOQgiYy4jzHj0ppewdhkXlBnotOQyi5jSESYFU6zKrRZnLcHZXyVvIJSu074D8HzfgnwBIVzRkfhkbW6gFRLeVRsBTm1Q79UuDYtNPX+r31senuOJtB4wCDJDbw0Evw3pH7KWtd8cA0RHYfBfyFDfAJ4nDxgk2qXMyT8Nzb03SHxrl9qvjCy6DVG6ZErZr6bdlHLx4NY66hFAhKQPTSM87ZrXMEGd/FUJEBYpyx4QYQi9VV16UGEb81sLqd0KXrcu+WBaK869vXJfOJWjXV7Dbl7AqpFzqwrRv+ELH6lzZsYqfvdN7qp4w5xeoSTa/b8PuE18twJsjeHRW/VT+YiWRrWiXj85A+8CURxS73IMBuGhjjXQnHrfGprWZ16lcqFAyErdyIfQW5Wf13pmvVxYoiDeo8v18iiPcnLtHOXSCo7WOqsWOUz6FEJhiomu6ve2lkQ4Q23BH3eTUZTE6phCy42toY2LuwYWfG5eqrXixOJ5h0M1D1dFv/reSiN/V9xXUSN98aSQKXooFj3EEDnVemstiRwFj4IF2o76kcP2rwEn9J7pFNy/fXdvgWp/zsi7bqPlDT3xGEhjQ48rCEPh4LVRiKtef1tLAnYv9NWWGrQGxZL9btfVyoUlwJje210SRVlM1znKRYxNrJ8yaQvj5CCcE0HzTSEUFk/sRDdUz0uH4SHop0p5VhelM1S3MWea7ce1xPvpLdMlru+ceW11TZ8gxBfAC+1goFAo+O1chEabMB1LloLwaqWxlQW6xFJv+eq6Gh59znCIN1o+aQ2xoSLcrXj2mM+p4IwxYEZbILpkv+uce293Ce/1UZTBc1i2Is4c5dImgGg6cGSDQlE8c65tGdok9Bh6q92MoboNdJDUBd0UntBSoUc/tSGdJYzQxvO9Z07dw1Z8seQ5bfhhTHn2Cqj1bpa865TX1nDomJBoP/I+1Om7olC1tZdrkodSW0AZaV9eRwslXhJ/r+do3KU8aC5S6JT08672+5kyrNqiUPeuNeZOvcb2ffUUUCCnsT2p3i6Poy82r6eDKjfgAf1l4RkniVZPvwfPLMn1aROErsYSVOIqms5RLq31gSCggZLT4olTRYCrJnWim1rhfakvTplDFTBDdRs8OW0UCAxIijEkV6tc5qK9ehDAdYbttjHcsfX11tZcOpyIFS69phXeY61KWqXaW+RCP1B5tUGgv4PzQ8dBDlakXqtclnh0LQhAJT/QxK9cFbH2vLf1RKb2vPfUx4y5a7jMW1OqwnsMcScsBalHdvaeGU9Z2FUojbGihkV5B+O1DYf3ymUu4KAHAZgHubcV3PzSvsxRLj1iDPoBESX65xQBzmWGUyZtWyvJxyk89uvdRFsrZMg6YFl4zosUKDbvZ6hXVhu+mPrA6hRa6PJUG5C59D3GdXNavnhvG17096XJ51Pyhvm1Seghw6MVEr1FDlEmp0bp1uR/LTLGZ23ytU3MzlUuvZC4rscSL/VEtm75cuqEfiv0Qe0phL/pPrp2Tr1nptMwHsDr1Uupyqrlmd5Qm6tcWoOkN3KOIRsme4vVl7aIMf82lKg6dIKnFCBtmGYottsX/O1r2lkZZky5qIuRyDTmxJHbWP+hDSsP3ZOp++e0fPGM3ipdGvI4JW+Yb02YjkFnW+Xf7ynFhCck11vQQvXwWmHQdh6fA9PFlyrqeUXGVB+pqf075u9LPZG20HKp8TG0jlMrlxac0Z/oaH59NKL1diFWhU8hUds6werZ9vB0vCXkagy9q6dHrUVUc3MyY3WO52KiLWLs6LG6Y3J8eXbbeWDIm4DoUs+j4ImVCrQAcq3CXpik7d0zpVxaATLmJdUlt0gTikVNRRtvPRZpMDelZm/Nce6Y0/LFs9oK9CUFX3PnsfV1VZkNeRNCoMAJzi1ipNT8EbSYfFut9brVHbZMrgqK1lDBWyrzGSDGWHi2rg9f4geFqN7r+qkWMYfSZi1vLPFEemt8Tgf2Q9e19f2tMuu9iYpKVS8nIoJPeLGQqfhCfuYx3YTco7zDc+17izhz9o6z6wGRxryk+rjajQEoybd9j67dz9Z0uPW8ucqlRYx9TmnnfA5Q0jHCtZ5LW1tQiym5oxBPKmCFPQgOIS+JWd1H27VPKRdzaF3SMfqxbLRHYeGqSD70vIW5TNMqNPcIdTpLZk6nhRYSORby0yAPjp4la7C8e6t+7lxPdV27rjb2zwLEFzwGNTusVXv7iFIwJwc31uHavuKnvvqdxUpQoA/FNNSbD18qTPXu2vvLUQxz9ugQmh3CG0tavrRW/xJgwyFr2/redg3tt1Dr6JxNQxEA/wD21G4PzlTpz7Wpcggf4r++OL2PrIx1/FDULGqiOaV3cxKW1umtptNc5VIRY7TrUC3I6glc0Y19eEF7bx8uge43yCyIDZYDa57wAAVWcdt/0HOUi4/UxyZfZTgEzLMVvPmNNUFpsYbr6Z1LKtgPIWPrxXnO3Nh/j1hSIS4kpMWFD0qjTpYaRW2N6KbNhcOFji0UD6GHe1uIOjCH00ApFlamZCwF4sxxDQMfWpKieGespU71UNB6KC/Xnu8j5Iz3dHDAixBX3o+2tXsE4XSKsZY3+r5X++o5rFFOobbgX1K3cQoazH1H3z8QzJfRwOAgP3wHjp/Wdofx+MjiffY9uRhjaCH0Wr9H3UN0zG6HPCZZoZuIaAAjBxDAt+U33rBnM1qFUYWij+3lXpjgXOUCMaYfDovtppwuieiOBoDkwRisg08rLqpNIEhYFGLhrA2J+SGhOEe5IHpN/hMUrBahDccXSAQSzOiqT8+a81rmfgBD11k7j6m2uHHNvlb4QhhopgGlxof7xqN3z3pYsZ6+c6IV+CFr2Ppee0Xh2w+hUR0H7A8DQIcBxpaP/UULv/D0KKGxAflDwIqdj7W2J5B1igDtdwAYHnzi8l/fHv5j4PCeTzWW8kY97pwRBh3Xjv6I9HqksRNK79pdS/DydoX+KPFzGTwFQryGORU8+971JLRv6OJbsz4hzbGQN+NMaQMDlKc7FvpynagKD6eWiJAp5DV5pg6RwtnXgv9otJ2rXI42gRvw4LnKpS6VBUzQElhCHf0JlGMKgJV8rFMC6zsJTci4dyoW+Q3Ynitfgpg3xfLAPcf61knW0ChLlYfTtsIfW4hvmDLi6R5T8SRvnJ6VqgfIm9l37oocjM4GCr7n5uEYiYyBo3UlT+VyOMMsUS6VCTTFlOQWRpnaXNYQC1pIRAfVY+a6hHZ4LVzt6x66Onznjv8Elj9LVmuhqb2r8XEQVkAOIRWGx779ZrkKtxA8xzyiGKWSN47PL0NvqEjdoWhCm4vjSYsoKLzswST9c4WsAUmEqPt8z2arTOVyOCnnKBdCQBxdspzAaQchzvVXwc1rcDIl5mDx6tkmTwOpNjfJvnZFpzjudu3czvG+ihCS6GVN7lPWwmsK6GrFfbte4Y3KG2DPED94CN8BFRAUBM+WRyj09E7euDoOlKuVq5QHbI8E5+U6V0VPuX7I5ZIZQosAA1Bp+E+eVLQEqEZe59IZLFsuM5XL4dScUi4sUkoFUkNcXvLNv0m6TQ1xdygj8f5jeiy6Lijiunc5amBqXvn7fgr4ruSwJOOd5dEmUuVf/JFDYXS4Tq5PWItRIVwxlcuqb5eTkPs6Zifk5I2r4/ZapCz31p5MyqAQahUuq7k+rWN6w3Vo5jwbgCVK56gjlcvh5K0wbYnaISSQ2hjhDd5Ja72yIiBCJDS1dYfMMepBPGotAChOhRo7nBL5hEoBdQmSqTzRdv8ICWhLPCHZS/k48ledQ985V50IUIk2/7yfKjgIB0eKSwz/TpL8RlCAEgEtByLhiUj01wJqxglvo3qmrnXGDXkDhVbzbGS5Giz8gr94w9VIIXcAaxg0zpI5CWoslct63uSWQn0JebDuMAP0mcSaTc9xe1IAegdgQ5J9aIz1Krs9qZWrRgFymAfKg2VEQJM9vEHMnaWBmcolmTspkBRICiQFNqdAKpfNSZoPTAokBZICSYFULskDSYGkQFIgKbA5BVK5bE7SfGBSICmQFEgKpHJJHkgKJAWSAkmBzSmQymVzkuYDkwJJgaRAUiCVS/JAUiApkBRICmxOgf8HuvR8gO3VzGcAAAAASUVORK5CYII=\" width=\"203.5\" height=\"26\" alt=\"V = a1 Q^e1, B = a2 Q^e2, H = a3 Q^e3\" style=\"width: 203.5px; height: 26px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere the coefficients have the appropriate dimensions. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function that takes two each of the coefficients and exponents and determines the third such that the relation between flow and velocity (i.e., \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"63\" height=\"18\" alt=\"Q = VBH\" style=\"width: 63px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e) is satisfied. The coefficients and exponents will be given as 1x3 vectors in the order velocity, width, and depth, and the unknown values will be given as NaN. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [anew,enew] = hydraulicGeometry(a,e)\r\n  anew = 1.1*a;\r\n  enew = 1.1*e;\r\nend","test_suite":"%\r\na = [NaN 7.2 0.27];\r\ne = [NaN 0.5 0.3];\r\n[anew,enew] = hydraulicGeometry(a,e);\r\nanew_correct = [0.5144 7.2 0.27];\r\nenew_correct = [0.2 0.5 0.3];\r\nassert(all(abs(anew-anew_correct)\u003c1e-4))\r\nassert(all(abs(enew-enew_correct)\u003c1e-4))\r\n\r\n%%\r\na = [0.5 NaN 0.3];\r\ne = [0.25 NaN 0.3];\r\n[anew,enew] = hydraulicGeometry(a,e);\r\nanew_correct = [0.5 6.6667 0.3];\r\nenew_correct = [0.25 0.45 0.3];\r\nassert(all(abs(anew-anew_correct)\u003c1e-4))\r\nassert(all(abs(enew-enew_correct)\u003c1e-4))\r\n\r\n%%\r\na = [0.45 7 NaN];\r\ne = [0.23 0.48 NaN];\r\n[anew,enew] = hydraulicGeometry(a,e);\r\nanew_correct = [0.45 7 0.3175];\r\nenew_correct = [0.23 0.48 0.29];\r\nassert(all(abs(anew-anew_correct)\u003c1e-4))\r\nassert(all(abs(enew-enew_correct)\u003c1e-4))\r\n\r\n%%\r\na = [0.8 5 NaN];\r\ne = [NaN 0.3 0.4];\r\n[anew,enew] = hydraulicGeometry(a,e);\r\nanew_correct = [0.8 5 0.25];\r\nenew_correct = [0.3 0.3 0.4];\r\nassert(all(abs(anew-anew_correct)\u003c1e-4))\r\nassert(all(abs(enew-enew_correct)\u003c1e-4))\r\n\r\n%%\r\na = [0.6 4 NaN];\r\ne = [0.1 NaN 0.4];\r\n[anew,enew] = hydraulicGeometry(a,e);\r\nanew_correct = [0.6 4 0.4167];\r\nenew_correct = [0.1 0.5 0.4];\r\nassert(all(abs(anew-anew_correct)\u003c1e-4))\r\nassert(all(abs(enew-enew_correct)\u003c1e-4))\r\n\r\n%%\r\na = [NaN 4.6 0.8];\r\ne = [0.15 NaN 0.37];\r\n[anew,enew] = hydraulicGeometry(a,e);\r\nanew_correct = [0.2717 4.6 0.8];\r\nenew_correct = [0.15 0.48 0.37];\r\nassert(all(abs(anew-anew_correct)\u003c1e-4))\r\nassert(all(abs(enew-enew_correct)\u003c1e-4))","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2024-12-14T15:47:02.000Z","deleted_by":null,"deleted_at":null,"solvers_count":8,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2024-12-14T15:46:55.000Z","updated_at":"2026-03-11T11:38:03.000Z","published_at":"2024-12-14T15:47:02.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eHydraulic geometry relations express the velocity \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"V\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eV\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, width \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"B\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eB\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and depth \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"H\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eH\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e of a river as a function of the discharge (or flow) \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Q\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eQ\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, which is the volume of water that passes a cross section in a unit time. These relations have the form (e.g., Leopold and Mattuck 1953)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"V = a1 Q^e1, B = a2 Q^e2, H = a3 Q^e3\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eV = a_1 Q^{e_1}, B = a_2 Q^{e_2}, H = a_3 Q^{e_3}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere the coefficients have the appropriate dimensions. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function that takes two each of the coefficients and exponents and determines the third such that the relation between flow and velocity (i.e., \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Q = VBH\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eQ = VBH\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e) is satisfied. The coefficients and exponents will be given as 1x3 vectors in the order velocity, width, and depth, and the unknown values will be given as NaN. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":51322,"title":"Solve an ODE: diffusion problem 1","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 264.25px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 132.125px; transform-origin: 407px 132.125px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28px 7.91667px; transform-origin: 28px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 295.017px 7.91667px; transform-origin: 295.017px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn the solution of a problem involving diffusion, the following ordinary differential equation arises\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f''+a eta f' = 0\" style=\"width: 91px; height: 18px;\" width=\"91\" height=\"18\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42.25px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.125px; text-align: left; transform-origin: 384px 21.125px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 7.91667px; transform-origin: 21.0083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ea\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 298.233px 7.91667px; transform-origin: 298.233px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is a positive constant and primes denote differentiation with respect to the independent variable \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 45.1167px 7.91667px; transform-origin: 45.1167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The problem has the conditions \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f(0) = f0\" style=\"width: 60px; height: 20px;\" width=\"60\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f(infinity) = 0\" style=\"width: 65px; height: 19px;\" width=\"65\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 194.342px 7.91667px; transform-origin: 194.342px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to solve this problem—that is, return values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ef\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 69.2333px 7.91667px; transform-origin: 69.2333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at specified values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.8333px 7.91667px; transform-origin: 40.8333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eBackground\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 372.133px 7.91667px; transform-origin: 372.133px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe physical problem involves diffusion of a quantity from one point where the concentration is maintained at a constant value into a medium in which the concentration is zero far away. The ODE results from transforming the diffusion equation—a partial differential equation in time and a spatial coordinate—with a similarity solution. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function f = diffusion1ODE(eta,a,f0)\r\n%  eta = independent variable\r\n%  a   = constant\r\n%  f0  = value of f at eta = 0\r\n\r\n   f = g(eta,a,f0)\r\nend","test_suite":"%%\r\na = 1/2; \r\nf0 = 1;\r\neta = 0:0.2:1;\r\nf_correct = [1 0.887537083981715 0.777297410789522 0.671373240540873 0.571607644953331 0.479500122186953];\r\nassert(all(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 2; \r\nf0 = 1;\r\neta = 0.5;\r\nf_correct = 0.479500122186953;\r\nassert(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10)\r\n\r\n%%\r\na = 1;\r\nf0 = 0.5;\r\neta = [0.12 0.23 0.456 0.789 1.011];\r\nf_correct = [0.452241573979416 0.409045884857994 0.324194989107724 0.215056003266342 0.156008214896009];\r\nassert(all(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 2;\r\nf0 = 1;\r\neta = 1:4;\r\nf_correct = [0.157299207050285 0.004677734981047 2.209049699858544e-05 1.541725790028002e-08];\r\nassert(all(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 3/4;\r\nf0 = 4/3;\r\neta = 3*logspace(-2,0,3);\r\nf_correct = [1.305696910508165 1.060016229285651 0.012499691279247];\r\nassert(all(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 0.01;\r\nf0 = 1;\r\neta = rand/120;\r\nf_correct = polyval(flip([1 -0.0797885 0 0.000132981 0 -1.99471e-7]),eta);\r\nf = diffusion1ODE(eta,a,f0);\r\nassert(all(abs(f-f_correct)\u003c1e-7))","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":6,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-04-08T01:02:26.000Z","updated_at":"2025-05-04T20:55:44.000Z","published_at":"2021-04-08T01:09:52.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn the solution of a problem involving diffusion, the following ordinary differential equation arises\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f''+a eta f' = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\\\\prime\\\\prime + a\\\\eta f\\\\prime = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is a positive constant and primes denote differentiation with respect to the independent variable \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The problem has the conditions \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f(0) = f0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef(0) = f_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f(infinity) = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef(\\\\infty ) = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to solve this problem—that is, return values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at specified values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eBackground\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe physical problem involves diffusion of a quantity from one point where the concentration is maintained at a constant value into a medium in which the concentration is zero far away. The ODE results from transforming the diffusion equation—a partial differential equation in time and a spatial coordinate—with a similarity solution. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":59696,"title":"Solve an ODE: Ekman spiral on a solid surface","description":"Problem \r\nWrite a function to solve for  and  as a function of  in this system of ordinary differential equations:\r\n\r\n\r\nwhere , , , and  are constants. The boundary conditions are that  at  and  and  as .\r\nBackground\r\nThis set of equations results from simplifying the Navier-Stokes equations (i.e., conservation of momentum for a fluid with a linear stress-rate of strain relation) for large-scale flow subjected to rotation. The horizontal velocity components are  and . The Coriolis parameter  is related to Earth’s rotation rate and the latitude, and  is the kinematic viscosity, which can be interpreted as an eddy viscosity to account for the effects of turbulence. \r\nThe velocities far above the surface,  and , result from the pressure gradient:  and , where  is pressure and  is density. Notice that far above the surface, the flow is along the isobars, or lines of constant pressure. As the surface is approached, the velocity vector rotates—or spirals. ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 388.2px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 194.1px; transform-origin: 407px 194.1px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 29.9417px 8px; transform-origin: 29.9417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 86.6px 8px; transform-origin: 86.6px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to solve for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eu\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ev\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 51.3333px 8px; transform-origin: 51.3333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e as a function of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ez\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 147.683px 8px; transform-origin: 147.683px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in this system of ordinary differential equations:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 36.6px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.3px; text-align: left; transform-origin: 384px 18.3px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-16px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"121\" height=\"36.5\" alt=\"-fv = -fV + eta d^2u/dz^2\" style=\"width: 121px; height: 36.5px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 36.6px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.3px; text-align: left; transform-origin: 384px 18.3px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-16px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"102.5\" height=\"36.5\" alt=\"fu = fU + eta d^2v/dz^2\" style=\"width: 102.5px; height: 36.5px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 8px; transform-origin: 21.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ef\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eU\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eV\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17.5px 8px; transform-origin: 17.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 152.083px 8px; transform-origin: 152.083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e are constants. The boundary conditions are that \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"62\" height=\"18\" alt=\"u = v = 0\" style=\"width: 62px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 9.71667px 8px; transform-origin: 9.71667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"35\" height=\"18\" alt=\"z = 0\" style=\"width: 35px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"40\" height=\"18\" alt=\"u = U\" style=\"width: 40px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"39\" height=\"18\" alt=\"v = V\" style=\"width: 39px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.275px 8px; transform-origin: 11.275px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e as \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"45\" height=\"18\" alt=\"z --\u003e infinity\" style=\"width: 45px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.8333px 8px; transform-origin: 40.8333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eBackground\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 382.342px 8px; transform-origin: 382.342px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThis set of equations results from simplifying the Navier-Stokes equations (i.e., conservation of momentum for a fluid with a linear stress-rate of strain relation) for large-scale flow subjected to rotation. The horizontal velocity components are \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eu\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ev\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 77.0167px 8px; transform-origin: 77.0167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The Coriolis parameter \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ef\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 168.283px 8px; transform-origin: 168.283px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is related to Earth’s rotation rate and the latitude, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 115px 8px; transform-origin: 115px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the kinematic viscosity, which can be interpreted as an eddy viscosity to account for the effects of turbulence. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 114.35px 8px; transform-origin: 114.35px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe velocities far above the surface, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eU\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eV\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 108.9px 8px; transform-origin: 108.9px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, result from the pressure gradient: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"123.5\" height=\"18.5\" alt=\"U = -(1/rho f) dp/dy\" style=\"width: 123.5px; height: 18.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"112.5\" height=\"18.5\" alt=\"V = (1/rho f) dp/dx\" style=\"width: 112.5px; height: 18.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 24.8917px 8px; transform-origin: 24.8917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, where \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ep\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 51.7333px 8px; transform-origin: 51.7333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is pressure and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eρ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 237.9px 8px; transform-origin: 237.9px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is density. Notice that far above the surface, the flow is along the isobars, or lines of constant pressure. As the surface is approached, the velocity vector rotates—or spirals. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [u,v] = EkmanSolid(z,eta,f,U,V)\r\n%  u,v  =   velocities in the x- and y-directions\r\n%  z    =   distance above the solid surface\r\n%  eta  =   kinematic viscosity (can be interpreted as an eddy viscosity for turbulent flow)\r\n%  f    =   Coriolis parameter\r\n%  U,V  =   velocities in the x- and y-directions far above the surface\r\n\r\n   [u,v] = deal(U,V);","test_suite":"%%\r\neta = 1e-6;             %  Kinematic viscosity (m2/s)\r\nf  = 1e-4;              %  Coriolis parameter (1/s)\r\nU  = 1;                 %  x-velocity far above the surface (m/s)\r\nV  = 0;                 %  y-velocity far above the surface (m/s)\r\nz = [0 0.05 0.1 0.15 0.2:0.1:0.5 0.7];    %  Distance above surface (m)\r\n[u,v] = EkmanSolid(z,eta,f,U,V);\r\nu_correct = [0 0.34124 0.62515 0.83094 0.96209 1.0627 1.0562 1.0269 0.99833];\r\nv_correct = [0 0.24312 0.32032 0.30214 0.24014 0.10216 0.018209 -0.011186 -0.0068865];\r\nassert(isequal([u(1) v(1)],[0 0]))\r\nassert(all(abs(u(2:end)-u_correct(2:end))./u_correct(2:end) \u003c 1e-4))\r\nassert(all(abs(v(2:end)-v_correct(2:end))./v_correct(2:end) \u003c 1e-4))\r\n\r\n%%\r\neta = 1e-6;             %  Kinematic viscosity (m2/s)\r\nf  = 1e-4;              %  Coriolis parameter (1/s)\r\nU  = 0;                 %  x-velocity far above the surface (m/s)\r\nV  = 1;                 %  y-velocity far above the surface (m/s)\r\nz = [0 0.05 0.1 0.15 0.2:0.1:0.5 0.7];    %  Distance above surface (m)\r\n[u,v] = EkmanSolid(z,eta,f,U,V);\r\nu_correct = [0 -0.24312 -0.32032 -0.30214 -0.24014 -0.10216 -0.018209 0.011186 0.0068865];\r\nv_correct = [0 0.34124 0.62515 0.83094 0.96209 1.0627 1.0562 1.0269 0.99833];\r\nassert(isequal([u(1) v(1)],[0 0]))\r\nassert(all(abs(u(2:end)-u_correct(2:end))./u_correct(2:end) \u003c 1e-4))\r\nassert(all(abs(v(2:end)-v_correct(2:end))./v_correct(2:end) \u003c 1e-4))\r\n\r\n%%\r\neta = 3;                %  Eddy viscosity (m2/s)\r\nf  = 1e-4;              %  Coriolis parameter (1/s)\r\nU  = 0.5;               %  x-velocity far above the surface (m/s)\r\nV  = 0.8;               %  y-velocity far above the surface (m/s)\r\nz = [10 50 80 130 190 240 300 400 500 1000 2000];    %  Distance above surface (m)\r\n[u,v] = EkmanSolid(z,eta,f,U,V);\r\nu_correct = [-0.010943 -0.031442 -0.026767 0.0081958 0.077758 0.14605 0.22904 0.3501 0.43683 0.51587 0.49983];\r\nv_correct = [0.052233 0.24389 0.36912 0.54304 0.69822 0.78853 0.85848 0.9072 0.90497 0.80113 0.80021];\r\nassert(all(abs(u-u_correct)./u_correct \u003c 1e-4))\r\nassert(all(abs(v-v_correct)./v_correct \u003c 1e-4))\r\n\r\n%%\r\neta = 0.5;              %  Eddy viscosity (m2/s)\r\nf  = 1e-4;              %  Coriolis parameter (1/s)\r\nU  = 2;                 %  x-velocity far above the surface (m/s)\r\nV  = 0.4;               %  y-velocity far above the surface (m/s)\r\nz = [10 50 80 130 190 240 300 400 500 1000 2000];    %  Distance above surface (m)\r\n[u,v] = EkmanSolid(z,eta,f,U,V);\r\nu_correct = [0.16323 0.81912 1.245 1.7492 2.0401 2.1093 2.0958 2.0295 1.9988 2.0001 2];\r\nv_correct = [0.22054 0.76866 0.91944 0.89604 0.70242 0.54931 0.43377 0.37707 0.38631 0.39997 0.4];\r\nassert(all(abs(u-u_correct)./u_correct \u003c 1e-4))\r\nassert(all(abs(v-v_correct)./v_correct \u003c 1e-4))\r\n\r\n%%\r\neta = 10;               %  Eddy viscosity (m2/s)\r\nf  = 8e-5;              %  Coriolis parameter (1/s)\r\nU  = 8;                 %  x-velocity far above the surface (m/s)\r\nV  = 6;                 %  y-velocity far above the surface (m/s)\r\nz = 200:200:2000;       %  Distance above surface (m)\r\n[u,v] = EkmanSolid(z,eta,f,U,V);\r\nu_correct = [1.4945 3.5616 5.4425 6.8363 7.7122 8.1675 8.3361 8.3398 8.2686 8.1789];\r\nv_correct = [4.3838 6.7003 7.591 7.6499 7.3224 6.8916 6.5067 6.2251 6.0503 5.9609];\r\nassert(all(abs(u-u_correct)./u_correct \u003c 1e-4))\r\nassert(all(abs(v-v_correct)./v_correct \u003c 1e-4))\r\n\r\n%%\r\neta = 12;               %  Eddy viscosity (m2/s)\r\nf  = 8e-5;              %  Coriolis parameter (1/s)\r\nU  = 10;                %  x-velocity far above the surface (m/s)\r\nV  = 5;                 %  y-velocity far above the surface (m/s)\r\nz = 300:300:3000;       %  Distance above surface (m)\r\n[u,v] = EkmanSolid(z,eta,f,U,V);\r\nu_correct = [3.5576 6.9831 9.1755 10.1948 10.468 10.397 10.2354 10.0997 10.0197 9.9861];\r\nv_correct = [5.5429 7.208 6.9985 6.2349 5.551 5.1311 4.9452 4.902 4.9216 4.9554];\r\nassert(all(abs(u-u_correct)./u_correct \u003c 1e-4))\r\nassert(all(abs(v-v_correct)./v_correct \u003c 1e-4))\r\n\r\n%%\r\nfiletext = fileread('EkmanSolid.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'assert') || contains(filetext,'regexp'); \r\nassert(~illegal)\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":3,"created_by":46909,"edited_by":46909,"edited_at":"2024-03-11T02:38:47.000Z","deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":"2024-03-11T02:38:47.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2024-03-10T16:24:15.000Z","updated_at":"2025-09-16T00:37:21.000Z","published_at":"2024-03-10T16:24:26.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to solve for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"u\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eu\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"v\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ev\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e as a function of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"z\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ez\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e in this system of ordinary differential equations:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"-fv = -fV + eta d^2u/dz^2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e-fv = -fV+\\\\eta\\\\frac{d^2u}{dz^2}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"fu = fU + eta d^2v/dz^2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003efu = fU+\\\\eta\\\\frac{d^2v}{dz^2}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eU\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eV\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e are constants. The boundary conditions are that \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"u = v = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eu = v = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"z = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ez = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"u = U\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eu = U\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"v = V\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ev = V\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e as \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"z --\u0026gt; infinity\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ez\\\\to\\\\infty\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eBackground\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis set of equations results from simplifying the Navier-Stokes equations (i.e., conservation of momentum for a fluid with a linear stress-rate of strain relation) for large-scale flow subjected to rotation. The horizontal velocity components are \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"u\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eu\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"v\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ev\\n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The Coriolis parameter \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is related to Earth’s rotation rate and the latitude, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the kinematic viscosity, which can be interpreted as an eddy viscosity to account for the effects of turbulence. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe velocities far above the surface, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eU\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eV\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, result from the pressure gradient: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"U = -(1/rho f) dp/dy\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eU = -(1/\\\\rho f)\\\\partial p/\\\\partial y\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"V = (1/rho f) dp/dx\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eV = (1/\\\\rho f)\\\\partial p/\\\\partial x\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, where \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is pressure and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"rho\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\rho\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is density. Notice that far above the surface, the flow is along the isobars, or lines of constant pressure. As the surface is approached, the velocity vector rotates—or spirals. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":51087,"title":"Solve an ODE: equation for a 2D laminar jet","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 503px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 251.5px; transform-origin: 407px 251.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28px 7.91667px; transform-origin: 28px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 330.55px 7.91667px; transform-origin: 330.55px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn the solution for a two-dimensional laminar jet, the following nonlinear ordinary differential equation arises\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f’’’ + 2(f’2+ff”) = 0\" style=\"width: 147.5px; height: 20px;\" width=\"147.5\" height=\"20\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 236.783px 7.91667px; transform-origin: 236.783px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere primes denote differentiation with respect to the independent variable \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 103.467px 7.91667px; transform-origin: 103.467px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The problem has the conditions \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f(0) = f''(0) = 0\" style=\"width: 115px; height: 19px;\" width=\"115\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f'(oo) = 0\" style=\"width: 69px; height: 19px;\" width=\"69\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 75.0417px 7.91667px; transform-origin: 75.0417px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and the constraint         \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 44px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22px; text-align: left; transform-origin: 384px 22px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"integral of f'^2 from -infinity to infinity = a\" style=\"width: 119.5px; height: 44px;\" width=\"119.5\" height=\"44\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 71.8417px 7.91667px; transform-origin: 71.8417px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e                                     \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 7.91667px; transform-origin: 21.0083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ea\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 43.1667px 7.91667px; transform-origin: 43.1667px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is a constant.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 194.342px 7.91667px; transform-origin: 194.342px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to solve this problem—that is, return values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ef\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 69.2333px 7.91667px; transform-origin: 69.2333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at specified values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 103.575px 7.91667px; transform-origin: 103.575px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The test suite allows MATLAB’s functions for numerical solution of ODEs, but the equation has a relatively simple analytical solution. If you would like a hint for the analytical solution, execute this command:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 296.25px 7.91667px; transform-origin: 296.25px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003echar('Zxj%ymj%uwtizhy%wzqj%tk%inkkjwjsynfynts%y|t%ynrjx%fsi%nsyjlwfyj%ymwjj%ynrjx3'-5)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.8333px 7.91667px; transform-origin: 40.8333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eBackground\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 147px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 73.5px; text-align: left; transform-origin: 384px 73.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 129.925px 7.91667px; transform-origin: 129.925px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe physical problem involves a jet in the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"x, y\" style=\"width: 24.5px; height: 18px;\" width=\"24.5\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 108.133px 7.91667px; transform-origin: 108.133px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e plane emanating from a source at \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"(x,y) = (0,0)\" style=\"width: 87.5px; height: 19px;\" width=\"87.5\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 64.1667px 7.91667px; transform-origin: 64.1667px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. It can be solved by expressing the velocities \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eu\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ev\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 278.117px 7.91667px; transform-origin: 278.117px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in terms of a streamfunction and employing a similarity solution that combines the spatial coordinates into a single variable \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 73.5083px 7.91667px; transform-origin: 73.5083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. In the problem above, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f(eta)\" style=\"width: 32px; height: 19px;\" width=\"32\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 172.308px 7.91667px; transform-origin: 172.308px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is proportional to the streamfunction. The conditions at \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"eta = 0\" style=\"width: 36px; height: 18px;\" width=\"36\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 47.4583px 7.91667px; transform-origin: 47.4583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e define the line \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y = 0\" style=\"width: 37px; height: 18px;\" width=\"37\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 143.508px 7.91667px; transform-origin: 143.508px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e as a line of symmetry; the transverse velocity \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ev\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 128.733px 7.91667px; transform-origin: 128.733px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and the shear stress are zero there. The condition \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f'(oo) = 0\" style=\"width: 69px; height: 19px;\" width=\"69\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 110.45px 7.91667px; transform-origin: 110.45px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e states that the streamwise velocity \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eu\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 203.767px 7.91667px; transform-origin: 203.767px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is zero far from the source. The integral constraint states that the momentum flux of the jet is constant. The value \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"a = 4/3\" style=\"width: 51.5px; height: 19px;\" width=\"51.5\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 200.7px 7.91667px; transform-origin: 200.7px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e corresponds to the physical problem of the jet. Other values are included for variety.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function f = lamjetODE(eta,a)\r\n  f = g(eta,a);\r\nend","test_suite":"%%\r\na = 4/3; \r\neta = 0:0.2:1;\r\nf_correct = [0 0.197375320224904 0.379948962255225 0.537049566998035 0.664036770267849 0.761594155955765];\r\nassert(all(abs(lamjetODE(eta,a)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 4/3; \r\neta = log(2);\r\nf_correct = 0.6;\r\nassert(abs(lamjetODE(eta,a)-f_correct)\u003c1e-10)\r\n\r\n%%\r\na = 1;\r\neta = pi;\r\nf_correct = 0.902552583791843;\r\nassert(abs(lamjetODE(eta,a)-f_correct)\u003c1e-10)\r\n\r\n%%\r\na = 0.5;\r\neta = 1.5;\r\nf_correct = 0.572446107496431;\r\nassert(abs(lamjetODE(eta,a)-f_correct)\u003c1e-10)\r\n\r\n%%\r\na = 0.75;\r\neta = -4:0;\r\nf_correct = [-0.823247562011528 -0.813902901498664 -0.766864130068021 -0.559711742572462 0];\r\nassert(all(abs(lamjetODE(eta,a)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = rand;\r\neta = 4*rand;\r\nassert(abs(lamjetODE(eta,a)+lamjetODE(-eta,a))\u003c1e-10)","published":true,"deleted":false,"likes_count":2,"comments_count":2,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":"2021-03-21T03:15:00.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2021-03-21T02:07:31.000Z","updated_at":"2021-03-22T00:42:37.000Z","published_at":"2021-03-21T02:24:49.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn the solution for a two-dimensional laminar jet, the following nonlinear ordinary differential equation arises\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f’’’ + 2(f’2+ff”) = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\\\\prime\\\\prime\\\\prime + 2(f\\\\prime^2+ff\\\\prime\\\\prime) = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere primes denote differentiation with respect to the independent variable \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The problem has the conditions \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f(0) = f''(0) = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef(0) = f\\\\prime\\\\prime(0) = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f'(oo) = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\\\\prime(\\\\infty ) = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and the constraint         \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"integral of f'^2 from -infinity to infinity = a\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\int_{-\\\\infty}^\\\\infty\\\\left[f\\\\prime(\\\\eta)\\\\right]^2d\\\\eta=a\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e                                     \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is a constant.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to solve this problem—that is, return values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at specified values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The test suite allows MATLAB’s functions for numerical solution of ODEs, but the equation has a relatively simple analytical solution. If you would like a hint for the analytical solution, execute this command:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003echar('Zxj%ymj%uwtizhy%wzqj%tk%inkkjwjsynfynts%y|t%ynrjx%fsi%nsyjlwfyj%ymwjj%ynrjx3'-5)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eBackground\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe physical problem involves a jet in the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x, y\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex, y\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e plane emanating from a source at \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"(x,y) = (0,0)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e(x,y) = (0,0)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. It can be solved by expressing the velocities \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"u\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eu\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"v\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ev\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e in terms of a streamfunction and employing a similarity solution that combines the spatial coordinates into a single variable \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. In the problem above, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f(eta)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef(\\\\eta)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is proportional to the streamfunction. The conditions at \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e define the line \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e as a line of symmetry; the transverse velocity \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"v\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ev\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and the shear stress are zero there. The condition \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f'(oo) = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\\\\prime(\\\\infty ) = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e states that the streamwise velocity \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"u\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eu\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is zero far from the source. The integral constraint states that the momentum flux of the jet is constant. The value \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a = 4/3\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea = 4/3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e corresponds to the physical problem of the jet. Other values are included for variety.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"}],"problem_search":{"errors":[],"problems":[{"id":61080,"title":"Calculate Reynolds Number","description":"Write a MATLAB function that calculates the Reynolds number for flow over a flat plate:\r\n​\r\nρ = fluid density (kg/m³)\r\nV = velocity (m/s)\r\nL = characteristic length (m)\r\nμ = dynamic viscosity (Pa·s)","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.440001px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 142.75px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 71.375px; transform-origin: 408px 71.375px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eWrite a MATLAB function that calculates the Reynolds number for flow over a flat plate:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"81\" height=\"19\" style=\"width: 81px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e​\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 81.75px; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 40.875px; transform-origin: 392px 40.875px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4375px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.21875px; text-align: left; transform-origin: 364px 10.21875px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eρ \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e= fluid density (kg/m³)\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4375px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.21875px; text-align: left; transform-origin: 364px 10.21875px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eV \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e= velocity (m/s)\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4375px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.21875px; text-align: left; transform-origin: 364px 10.21875px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eL\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e= characteristic length (m)\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4375px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.21875px; text-align: left; transform-origin: 364px 10.21875px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eμ\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e= dynamic viscosity (Pa·s)\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function Re = calculateReynolds(rho, V, L, mu)\r\n% CALCULATEREYNOLDS computes the Reynolds number\r\n%\r\n% Inputs:\r\n%   rho - fluid density in kg/m^3\r\n%   V   - velocity in m/s\r\n%   L   - characteristic length in m\r\n%   mu  - dynamic viscosity in Pa*s\r\n%\r\n% Output:\r\n%   Re  - Reynolds number (dimensionless)\r\n\r\n% Your code here\r\n\r\nend\r\n","test_suite":"%% Test 1\r\nRe1 = calculateReynolds(1.225, 10, 0.5, 1.8e-5);\r\nassert(abs(Re1 - 340277.7778) \u003c 1e-1, 'Test 1 failed');\r\n\r\n%% Test 2\r\nRe2 = calculateReynolds(1.0, 15, 1, 2e-5);\r\nassert(abs(Re2 - 750000) \u003c 1e-1, 'Test 2 failed');\r\n\r\n%% Test 3\r\nRe3 = calculateReynolds(1.2, 5, 0.2, 1.5e-5);\r\nassert(abs(Re3 - 80000) \u003c 1e-1, 'Test 3 failed');\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":2,"created_by":4707073,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":34,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2025-11-18T11:13:52.000Z","updated_at":"2026-03-04T21:48:28.000Z","published_at":"2025-11-18T11:13:52.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a MATLAB function that calculates the Reynolds number for flow over a flat plate:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eRe=\\\\rho VL/\\\\mu\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e​\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eρ \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e= fluid density (kg/m³)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eV \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e= velocity (m/s)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eL\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e= characteristic length (m)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eμ\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e= dynamic viscosity (Pa·s)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60778,"title":"Complete hydraulic geometry relations","description":"Hydraulic geometry relations express the velocity , width , and depth  of a river as a function of the discharge (or flow) , which is the volume of water that passes a cross section in a unit time. These relations have the form (e.g., Leopold and Mattuck 1953)\r\n\r\nwhere the coefficients have the appropriate dimensions. \r\nWrite a function that takes two each of the coefficients and exponents and determines the third such that the relation between flow and velocity (i.e., ) is satisfied. The coefficients and exponents will be given as 1x3 vectors in the order velocity, width, and depth, and the unknown values will be given as NaN. ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 200px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 100px; transform-origin: 407px 100px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eHydraulic geometry relations express the velocity \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eV\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, width \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eB\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and depth \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eH\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e of a river as a function of the discharge (or flow) \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eQ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, which is the volume of water that passes a cross section in a unit time. These relations have the form (e.g., Leopold and Mattuck 1953)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 26px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 13px; text-align: left; transform-origin: 384px 13px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"203.5\" height=\"26\" alt=\"V = a1 Q^e1, B = a2 Q^e2, H = a3 Q^e3\" style=\"width: 203.5px; height: 26px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere the coefficients have the appropriate dimensions. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function that takes two each of the coefficients and exponents and determines the third such that the relation between flow and velocity (i.e., \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"63\" height=\"18\" alt=\"Q = VBH\" style=\"width: 63px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e) is satisfied. The coefficients and exponents will be given as 1x3 vectors in the order velocity, width, and depth, and the unknown values will be given as NaN. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [anew,enew] = hydraulicGeometry(a,e)\r\n  anew = 1.1*a;\r\n  enew = 1.1*e;\r\nend","test_suite":"%\r\na = [NaN 7.2 0.27];\r\ne = [NaN 0.5 0.3];\r\n[anew,enew] = hydraulicGeometry(a,e);\r\nanew_correct = [0.5144 7.2 0.27];\r\nenew_correct = [0.2 0.5 0.3];\r\nassert(all(abs(anew-anew_correct)\u003c1e-4))\r\nassert(all(abs(enew-enew_correct)\u003c1e-4))\r\n\r\n%%\r\na = [0.5 NaN 0.3];\r\ne = [0.25 NaN 0.3];\r\n[anew,enew] = hydraulicGeometry(a,e);\r\nanew_correct = [0.5 6.6667 0.3];\r\nenew_correct = [0.25 0.45 0.3];\r\nassert(all(abs(anew-anew_correct)\u003c1e-4))\r\nassert(all(abs(enew-enew_correct)\u003c1e-4))\r\n\r\n%%\r\na = [0.45 7 NaN];\r\ne = [0.23 0.48 NaN];\r\n[anew,enew] = hydraulicGeometry(a,e);\r\nanew_correct = [0.45 7 0.3175];\r\nenew_correct = [0.23 0.48 0.29];\r\nassert(all(abs(anew-anew_correct)\u003c1e-4))\r\nassert(all(abs(enew-enew_correct)\u003c1e-4))\r\n\r\n%%\r\na = [0.8 5 NaN];\r\ne = [NaN 0.3 0.4];\r\n[anew,enew] = hydraulicGeometry(a,e);\r\nanew_correct = [0.8 5 0.25];\r\nenew_correct = [0.3 0.3 0.4];\r\nassert(all(abs(anew-anew_correct)\u003c1e-4))\r\nassert(all(abs(enew-enew_correct)\u003c1e-4))\r\n\r\n%%\r\na = [0.6 4 NaN];\r\ne = [0.1 NaN 0.4];\r\n[anew,enew] = hydraulicGeometry(a,e);\r\nanew_correct = [0.6 4 0.4167];\r\nenew_correct = [0.1 0.5 0.4];\r\nassert(all(abs(anew-anew_correct)\u003c1e-4))\r\nassert(all(abs(enew-enew_correct)\u003c1e-4))\r\n\r\n%%\r\na = [NaN 4.6 0.8];\r\ne = [0.15 NaN 0.37];\r\n[anew,enew] = hydraulicGeometry(a,e);\r\nanew_correct = [0.2717 4.6 0.8];\r\nenew_correct = [0.15 0.48 0.37];\r\nassert(all(abs(anew-anew_correct)\u003c1e-4))\r\nassert(all(abs(enew-enew_correct)\u003c1e-4))","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2024-12-14T15:47:02.000Z","deleted_by":null,"deleted_at":null,"solvers_count":8,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2024-12-14T15:46:55.000Z","updated_at":"2026-03-11T11:38:03.000Z","published_at":"2024-12-14T15:47:02.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eHydraulic geometry relations express the velocity \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"V\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eV\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, width \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"B\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eB\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and depth \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"H\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eH\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e of a river as a function of the discharge (or flow) \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Q\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eQ\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, which is the volume of water that passes a cross section in a unit time. These relations have the form (e.g., Leopold and Mattuck 1953)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"V = a1 Q^e1, B = a2 Q^e2, H = a3 Q^e3\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eV = a_1 Q^{e_1}, B = a_2 Q^{e_2}, H = a_3 Q^{e_3}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere the coefficients have the appropriate dimensions. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function that takes two each of the coefficients and exponents and determines the third such that the relation between flow and velocity (i.e., \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Q = VBH\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eQ = VBH\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e) is satisfied. The coefficients and exponents will be given as 1x3 vectors in the order velocity, width, and depth, and the unknown values will be given as NaN. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":51322,"title":"Solve an ODE: diffusion problem 1","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 264.25px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 132.125px; transform-origin: 407px 132.125px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28px 7.91667px; transform-origin: 28px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 295.017px 7.91667px; transform-origin: 295.017px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn the solution of a problem involving diffusion, the following ordinary differential equation arises\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAALYAAAAkCAYAAAAkXsBMAAAEhklEQVR4nO1bwZGjMBDsHMjACZCAI3AEzoAMNgNXXQT+XAKE4BxclwExOIW9B+pirJNGYiU4Caur/LAlg2ia1sxIAA0NDQ0NDQ0NDQ0NDQ0NO+IE4BzZ7xTo0wPokkf0OejNJ4QY7hsMegATgAeAO4AXgEHp+w3gqRxvMH1uGcd4VFwx833HzP8Ev7nEcF8qTpiv9ct8Yh7iJJwxkzWa7xTlC27HvZn2u3LMp+lzzTfMQ+ILM080kdF8f3j6x3BfGjrM4+V1nrE8zCM2EnhnTvDCMr1doJM7mXbfgE6mfUILRTTQUKT7Urhfnv+EuC8NHfwmx+ufsMH10DHGUEcDToVafzq+L5T5CRj713JDY/DAOp5iuC8NfFB9oROdPPs1vbAuZKBoNYGNyO/Wax/A0kGRfiM+EYzhviRw5tZmoIvoc8l14ivWk/uELq4O+d0aOJ6w6VTTiv+EuC8NdOtv+JPhTvTJljcwUYkll4PQnPiCbTL2owmbM2XszYzhvjQwHwiNm/1eKScbsJRbSO4kfhuUQVwRLt/dkXFKEdhD2B3msUseWKKyZyD2dV1rj4XPTvTnb9LJHuJ3LRyM4b408BpDgn2Ivj8Os0gip0KKJYZcIByydNjGVbYWNh/0EfO0ecH8wNNNpLP2WEzBnmYlrzJpOmHhWN7Im/g9tDCWsigjz5/yiVm8A5aKx1phJ5eHKRQt/ikJWwlblqNsV5YckfDe9B9EG137bsYnj+mahnkjfesEW0AKLeUTmzvJ84VC3RHrjx882J7kpmALYWuilueU4uSCwtVqo6gJitd1U+n2vnWCLZDLsWNDBSns0D2TPCcL+3+QqyFEPIXyDPRbM/tQ1D4O7ko72+je9s3zZfqyzOcrgR0BUtihYoIM35KELeuLpSQke0+V0iV8LsT42iVAaQx2zV6K104ur0rbkSA52C3G3qQonohYx54C/WKmSm4j0KZJ+aDZs4B90+x2GX/bYd5NaTsadq2KAHWSmzPGlsLzhS5agifd3jUe/tcVwjD82Xt33t5VEWC51lAeF9svCBmv1oKcwg5VJUKJT8hhtBj6f4WAe4d6wLuBak7MPsn5ni+xKRk5ha0lNXapzhanXAJ2jUULYWTb3tt5966K8JyhRDlbWCzjw5r2Su8hbIra3uPAlzC42qjxx/+64koZAn3KGzCxu/uS3bpWcnMKWy4I0CXOWEp39uKLfKNFCtcVxtDtXeNcuzfnCND2Y9Nk5bsAPwafkNrIzSnsM96XxMkHiZfCtqseLAG6HEaGKS43dy3PfwI6vO89l2/QTMhksHx6SqlfxyL3yuMJy4YwO7brTNsV7oqIrzqgbQOWoi+lxLo3eiycD8i4r1ySW8tmdaKGN2i02ZCxedLWzAY36Cg1lflqghZfU/S1zZTFgRtzZN2RyUsNu/lqBGfD3/h3XzpfmK5lQaxY2DvTmIF+WuKyF2SN+g/eZ0YmozWVV4sFN8Mz25/QRL0lpLB/YamcDJi5z/0O6EeDmf+AumrWtUKuzvH1MFdlpaGhoaGhoaGhoXz8BQdnXetyYvEFAAAAAElFTkSuQmCC\" alt=\"f''+a eta f' = 0\" style=\"width: 91px; height: 18px;\" width=\"91\" height=\"18\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42.25px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.125px; text-align: left; transform-origin: 384px 21.125px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 7.91667px; transform-origin: 21.0083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ea\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 298.233px 7.91667px; transform-origin: 298.233px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is a positive constant and primes denote differentiation with respect to the independent variable \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 45.1167px 7.91667px; transform-origin: 45.1167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The problem has the conditions \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f(0) = f0\" style=\"width: 60px; height: 20px;\" width=\"60\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f(infinity) = 0\" style=\"width: 65px; height: 19px;\" width=\"65\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 194.342px 7.91667px; transform-origin: 194.342px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to solve this problem—that is, return values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ef\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 69.2333px 7.91667px; transform-origin: 69.2333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at specified values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.8333px 7.91667px; transform-origin: 40.8333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eBackground\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 372.133px 7.91667px; transform-origin: 372.133px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe physical problem involves diffusion of a quantity from one point where the concentration is maintained at a constant value into a medium in which the concentration is zero far away. The ODE results from transforming the diffusion equation—a partial differential equation in time and a spatial coordinate—with a similarity solution. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function f = diffusion1ODE(eta,a,f0)\r\n%  eta = independent variable\r\n%  a   = constant\r\n%  f0  = value of f at eta = 0\r\n\r\n   f = g(eta,a,f0)\r\nend","test_suite":"%%\r\na = 1/2; \r\nf0 = 1;\r\neta = 0:0.2:1;\r\nf_correct = [1 0.887537083981715 0.777297410789522 0.671373240540873 0.571607644953331 0.479500122186953];\r\nassert(all(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 2; \r\nf0 = 1;\r\neta = 0.5;\r\nf_correct = 0.479500122186953;\r\nassert(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10)\r\n\r\n%%\r\na = 1;\r\nf0 = 0.5;\r\neta = [0.12 0.23 0.456 0.789 1.011];\r\nf_correct = [0.452241573979416 0.409045884857994 0.324194989107724 0.215056003266342 0.156008214896009];\r\nassert(all(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 2;\r\nf0 = 1;\r\neta = 1:4;\r\nf_correct = [0.157299207050285 0.004677734981047 2.209049699858544e-05 1.541725790028002e-08];\r\nassert(all(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 3/4;\r\nf0 = 4/3;\r\neta = 3*logspace(-2,0,3);\r\nf_correct = [1.305696910508165 1.060016229285651 0.012499691279247];\r\nassert(all(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 0.01;\r\nf0 = 1;\r\neta = rand/120;\r\nf_correct = polyval(flip([1 -0.0797885 0 0.000132981 0 -1.99471e-7]),eta);\r\nf = diffusion1ODE(eta,a,f0);\r\nassert(all(abs(f-f_correct)\u003c1e-7))","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":6,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-04-08T01:02:26.000Z","updated_at":"2025-05-04T20:55:44.000Z","published_at":"2021-04-08T01:09:52.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn the solution of a problem involving diffusion, the following ordinary differential equation arises\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f''+a eta f' = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\\\\prime\\\\prime + a\\\\eta f\\\\prime = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is a positive constant and primes denote differentiation with respect to the independent variable \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The problem has the conditions \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f(0) = f0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef(0) = f_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f(infinity) = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef(\\\\infty ) = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to solve this problem—that is, return values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at specified values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eBackground\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe physical problem involves diffusion of a quantity from one point where the concentration is maintained at a constant value into a medium in which the concentration is zero far away. The ODE results from transforming the diffusion equation—a partial differential equation in time and a spatial coordinate—with a similarity solution. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":59696,"title":"Solve an ODE: Ekman spiral on a solid surface","description":"Problem \r\nWrite a function to solve for  and  as a function of  in this system of ordinary differential equations:\r\n\r\n\r\nwhere , , , and  are constants. The boundary conditions are that  at  and  and  as .\r\nBackground\r\nThis set of equations results from simplifying the Navier-Stokes equations (i.e., conservation of momentum for a fluid with a linear stress-rate of strain relation) for large-scale flow subjected to rotation. The horizontal velocity components are  and . The Coriolis parameter  is related to Earth’s rotation rate and the latitude, and  is the kinematic viscosity, which can be interpreted as an eddy viscosity to account for the effects of turbulence. \r\nThe velocities far above the surface,  and , result from the pressure gradient:  and , where  is pressure and  is density. Notice that far above the surface, the flow is along the isobars, or lines of constant pressure. As the surface is approached, the velocity vector rotates—or spirals. ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 388.2px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 194.1px; transform-origin: 407px 194.1px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 29.9417px 8px; transform-origin: 29.9417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 86.6px 8px; transform-origin: 86.6px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to solve for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eu\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ev\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 51.3333px 8px; transform-origin: 51.3333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e as a function of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ez\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 147.683px 8px; transform-origin: 147.683px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in this system of ordinary differential equations:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 36.6px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.3px; text-align: left; transform-origin: 384px 18.3px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-16px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"121\" height=\"36.5\" alt=\"-fv = -fV + eta d^2u/dz^2\" style=\"width: 121px; height: 36.5px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 36.6px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.3px; text-align: left; transform-origin: 384px 18.3px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-16px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"102.5\" height=\"36.5\" alt=\"fu = fU + eta d^2v/dz^2\" style=\"width: 102.5px; height: 36.5px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 8px; transform-origin: 21.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ef\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eU\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eV\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17.5px 8px; transform-origin: 17.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 152.083px 8px; transform-origin: 152.083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e are constants. The boundary conditions are that \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHwAAAAkCAYAAABLw14kAAADzElEQVR4Xu1aS6hOURS+d05eIxIDBkQx8CoxMKCkDJTHzMhjKBEGBgYUJkYepWTiMZNSTKW8BpJigGLAiIg533fbq9bdZ+3zn+PPOfu016mvc/9z9tl7rfXt9dh738kJv4qywGRR2rqyE054YZPACXfCC7NAYeq6hzvhhVmgMHXdw4dL+FaIfhSYFVRYhPtj4DDwPaWWEz5Mwu9A7N3ANeBgUGEu7k+AZcBegG0qlxM+PMLPQ+RjwDtgeST+Wvx+Hp6tw/1FrJ4TPizCl0Dc90Fkhu4rhvgP8Yzh/hmwwQkfFsGxtBLKf+PFFsuD8ewkcDZ8uA33R7oT9/BhTYBfEHcGQMJnJkSnd9PLeekcP/VgHMJZJKwJHb/EPVkZDsum2UqriaTX0ntT15/wopLnLcJJJCu/OcBmgOX+/fBMD3AVPw6EB/MyJZy6nAAWB6zAneHuXGQp0WWUIfucDYcw+OUgwCg5hfCKU6c8nIa6BXBW8arkAjz7AswHzOIgYRk9S8cx3imDtLr+WOy8AhgOLWPpyZtc0kQDSLU8jh781qymjU71eG0In9Z/XUiXao/5gh6iQ/Ye/L4dhLqA+/GGWvdFOMWTgucr/l5gyCv6LsW7Dw306ZPwUTZ/C/m5Hq9MqDrC6zxYjMcOLe9vYK/Om+iQaKUgevkqYHs0uTsXNDGgnmCjItxT9LG+DeF6vRd3rt9Z3p+LgWI5dHSxwjYn+JEQCXLU4b+GdO0NcY7R3t0mf/dtRNYl34IQ8aYF1647M/Zuiq3TaNMcXnHIVEjX+W6lCnHauynEqNDSN8nx+FK96vUpdeKhAwmvbEVmpIDeNm1KeKVeSREu+ftumFmiN4uBn6n80MA4fRZtFE82LjThzHc3AGubsk6lros2LX+q8GQbPTFi/syNl9R+LRVktT4bIHEcVHt/A76nvpNdoCbtU23+NbJIMSOGkBMlhsu2Vx+E63Sa2vvQ6bhSq1gerj+QJQqJugRsBD4Bej3L9h+BaXu2ba3XUXshnDtQN4HVQO35cUdyNR2mzeGJdZpmerhsQsgHDBH3AOY4rsX1aQ2JPgPkupSJDamXK5yg+4JOTQ2eQ7u641GZECzWdllOaHm4NgpDH8mUj3VIZoXObddNQJONihyMpT18x4Dkjm1X9w8QC9GY5+VmTZIK6dzJ4fUAuAhI9cqlzRtAtlRPW7MoB2YTMtBQPwAuw4Z+2MNUuj843Wfc+a9Or4HrdZyMc1qWMa8uWsoCTnhhc8MJd8ILs0Bh6rqHO+GFWaAwdd3DnfDCLFCYuu7hhRH+F3iP1yX2h/VEAAAAAElFTkSuQmCC\" width=\"62\" height=\"18\" alt=\"u = v = 0\" style=\"width: 62px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 9.71667px 8px; transform-origin: 9.71667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"35\" height=\"18\" alt=\"z = 0\" style=\"width: 35px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"40\" height=\"18\" alt=\"u = U\" style=\"width: 40px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"39\" height=\"18\" alt=\"v = V\" style=\"width: 39px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.275px 8px; transform-origin: 11.275px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e as \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"45\" height=\"18\" alt=\"z --\u003e infinity\" style=\"width: 45px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.8333px 8px; transform-origin: 40.8333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eBackground\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 382.342px 8px; transform-origin: 382.342px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThis set of equations results from simplifying the Navier-Stokes equations (i.e., conservation of momentum for a fluid with a linear stress-rate of strain relation) for large-scale flow subjected to rotation. The horizontal velocity components are \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eu\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ev\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 77.0167px 8px; transform-origin: 77.0167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The Coriolis parameter \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ef\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 168.283px 8px; transform-origin: 168.283px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is related to Earth’s rotation rate and the latitude, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 115px 8px; transform-origin: 115px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the kinematic viscosity, which can be interpreted as an eddy viscosity to account for the effects of turbulence. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 114.35px 8px; transform-origin: 114.35px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe velocities far above the surface, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eU\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eV\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 108.9px 8px; transform-origin: 108.9px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, result from the pressure gradient: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"123.5\" height=\"18.5\" alt=\"U = -(1/rho f) dp/dy\" style=\"width: 123.5px; height: 18.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"112.5\" height=\"18.5\" alt=\"V = (1/rho f) dp/dx\" style=\"width: 112.5px; height: 18.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 24.8917px 8px; transform-origin: 24.8917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, where \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ep\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 51.7333px 8px; transform-origin: 51.7333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is pressure and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eρ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 237.9px 8px; transform-origin: 237.9px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is density. Notice that far above the surface, the flow is along the isobars, or lines of constant pressure. As the surface is approached, the velocity vector rotates—or spirals. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [u,v] = EkmanSolid(z,eta,f,U,V)\r\n%  u,v  =   velocities in the x- and y-directions\r\n%  z    =   distance above the solid surface\r\n%  eta  =   kinematic viscosity (can be interpreted as an eddy viscosity for turbulent flow)\r\n%  f    =   Coriolis parameter\r\n%  U,V  =   velocities in the x- and y-directions far above the surface\r\n\r\n   [u,v] = deal(U,V);","test_suite":"%%\r\neta = 1e-6;             %  Kinematic viscosity (m2/s)\r\nf  = 1e-4;              %  Coriolis parameter (1/s)\r\nU  = 1;                 %  x-velocity far above the surface (m/s)\r\nV  = 0;                 %  y-velocity far above the surface (m/s)\r\nz = [0 0.05 0.1 0.15 0.2:0.1:0.5 0.7];    %  Distance above surface (m)\r\n[u,v] = EkmanSolid(z,eta,f,U,V);\r\nu_correct = [0 0.34124 0.62515 0.83094 0.96209 1.0627 1.0562 1.0269 0.99833];\r\nv_correct = [0 0.24312 0.32032 0.30214 0.24014 0.10216 0.018209 -0.011186 -0.0068865];\r\nassert(isequal([u(1) v(1)],[0 0]))\r\nassert(all(abs(u(2:end)-u_correct(2:end))./u_correct(2:end) \u003c 1e-4))\r\nassert(all(abs(v(2:end)-v_correct(2:end))./v_correct(2:end) \u003c 1e-4))\r\n\r\n%%\r\neta = 1e-6;             %  Kinematic viscosity (m2/s)\r\nf  = 1e-4;              %  Coriolis parameter (1/s)\r\nU  = 0;                 %  x-velocity far above the surface (m/s)\r\nV  = 1;                 %  y-velocity far above the surface (m/s)\r\nz = [0 0.05 0.1 0.15 0.2:0.1:0.5 0.7];    %  Distance above surface (m)\r\n[u,v] = EkmanSolid(z,eta,f,U,V);\r\nu_correct = [0 -0.24312 -0.32032 -0.30214 -0.24014 -0.10216 -0.018209 0.011186 0.0068865];\r\nv_correct = [0 0.34124 0.62515 0.83094 0.96209 1.0627 1.0562 1.0269 0.99833];\r\nassert(isequal([u(1) v(1)],[0 0]))\r\nassert(all(abs(u(2:end)-u_correct(2:end))./u_correct(2:end) \u003c 1e-4))\r\nassert(all(abs(v(2:end)-v_correct(2:end))./v_correct(2:end) \u003c 1e-4))\r\n\r\n%%\r\neta = 3;                %  Eddy viscosity (m2/s)\r\nf  = 1e-4;              %  Coriolis parameter (1/s)\r\nU  = 0.5;               %  x-velocity far above the surface (m/s)\r\nV  = 0.8;               %  y-velocity far above the surface (m/s)\r\nz = [10 50 80 130 190 240 300 400 500 1000 2000];    %  Distance above surface (m)\r\n[u,v] = EkmanSolid(z,eta,f,U,V);\r\nu_correct = [-0.010943 -0.031442 -0.026767 0.0081958 0.077758 0.14605 0.22904 0.3501 0.43683 0.51587 0.49983];\r\nv_correct = [0.052233 0.24389 0.36912 0.54304 0.69822 0.78853 0.85848 0.9072 0.90497 0.80113 0.80021];\r\nassert(all(abs(u-u_correct)./u_correct \u003c 1e-4))\r\nassert(all(abs(v-v_correct)./v_correct \u003c 1e-4))\r\n\r\n%%\r\neta = 0.5;              %  Eddy viscosity (m2/s)\r\nf  = 1e-4;              %  Coriolis parameter (1/s)\r\nU  = 2;                 %  x-velocity far above the surface (m/s)\r\nV  = 0.4;               %  y-velocity far above the surface (m/s)\r\nz = [10 50 80 130 190 240 300 400 500 1000 2000];    %  Distance above surface (m)\r\n[u,v] = EkmanSolid(z,eta,f,U,V);\r\nu_correct = [0.16323 0.81912 1.245 1.7492 2.0401 2.1093 2.0958 2.0295 1.9988 2.0001 2];\r\nv_correct = [0.22054 0.76866 0.91944 0.89604 0.70242 0.54931 0.43377 0.37707 0.38631 0.39997 0.4];\r\nassert(all(abs(u-u_correct)./u_correct \u003c 1e-4))\r\nassert(all(abs(v-v_correct)./v_correct \u003c 1e-4))\r\n\r\n%%\r\neta = 10;               %  Eddy viscosity (m2/s)\r\nf  = 8e-5;              %  Coriolis parameter (1/s)\r\nU  = 8;                 %  x-velocity far above the surface (m/s)\r\nV  = 6;                 %  y-velocity far above the surface (m/s)\r\nz = 200:200:2000;       %  Distance above surface (m)\r\n[u,v] = EkmanSolid(z,eta,f,U,V);\r\nu_correct = [1.4945 3.5616 5.4425 6.8363 7.7122 8.1675 8.3361 8.3398 8.2686 8.1789];\r\nv_correct = [4.3838 6.7003 7.591 7.6499 7.3224 6.8916 6.5067 6.2251 6.0503 5.9609];\r\nassert(all(abs(u-u_correct)./u_correct \u003c 1e-4))\r\nassert(all(abs(v-v_correct)./v_correct \u003c 1e-4))\r\n\r\n%%\r\neta = 12;               %  Eddy viscosity (m2/s)\r\nf  = 8e-5;              %  Coriolis parameter (1/s)\r\nU  = 10;                %  x-velocity far above the surface (m/s)\r\nV  = 5;                 %  y-velocity far above the surface (m/s)\r\nz = 300:300:3000;       %  Distance above surface (m)\r\n[u,v] = EkmanSolid(z,eta,f,U,V);\r\nu_correct = [3.5576 6.9831 9.1755 10.1948 10.468 10.397 10.2354 10.0997 10.0197 9.9861];\r\nv_correct = [5.5429 7.208 6.9985 6.2349 5.551 5.1311 4.9452 4.902 4.9216 4.9554];\r\nassert(all(abs(u-u_correct)./u_correct \u003c 1e-4))\r\nassert(all(abs(v-v_correct)./v_correct \u003c 1e-4))\r\n\r\n%%\r\nfiletext = fileread('EkmanSolid.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'assert') || contains(filetext,'regexp'); \r\nassert(~illegal)\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":3,"created_by":46909,"edited_by":46909,"edited_at":"2024-03-11T02:38:47.000Z","deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":"2024-03-11T02:38:47.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2024-03-10T16:24:15.000Z","updated_at":"2025-09-16T00:37:21.000Z","published_at":"2024-03-10T16:24:26.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to solve for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"u\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eu\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"v\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ev\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e as a function of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"z\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ez\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e in this system of ordinary differential equations:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"-fv = -fV + eta d^2u/dz^2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e-fv = -fV+\\\\eta\\\\frac{d^2u}{dz^2}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"fu = fU + eta d^2v/dz^2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003efu = fU+\\\\eta\\\\frac{d^2v}{dz^2}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eU\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eV\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e are constants. The boundary conditions are that \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"u = v = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eu = v = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"z = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ez = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"u = U\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eu = U\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"v = V\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ev = V\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e as \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"z --\u0026gt; infinity\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ez\\\\to\\\\infty\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eBackground\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis set of equations results from simplifying the Navier-Stokes equations (i.e., conservation of momentum for a fluid with a linear stress-rate of strain relation) for large-scale flow subjected to rotation. The horizontal velocity components are \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"u\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eu\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"v\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ev\\n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The Coriolis parameter \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is related to Earth’s rotation rate and the latitude, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the kinematic viscosity, which can be interpreted as an eddy viscosity to account for the effects of turbulence. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe velocities far above the surface, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eU\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eV\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, result from the pressure gradient: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"U = -(1/rho f) dp/dy\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eU = -(1/\\\\rho f)\\\\partial p/\\\\partial y\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"V = (1/rho f) dp/dx\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eV = (1/\\\\rho f)\\\\partial p/\\\\partial x\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, where \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is pressure and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"rho\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\rho\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is density. Notice that far above the surface, the flow is along the isobars, or lines of constant pressure. As the surface is approached, the velocity vector rotates—or spirals. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":51087,"title":"Solve an ODE: equation for a 2D laminar jet","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 503px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 251.5px; transform-origin: 407px 251.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28px 7.91667px; transform-origin: 28px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 330.55px 7.91667px; transform-origin: 330.55px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn the solution for a two-dimensional laminar jet, the following nonlinear ordinary differential equation arises\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f’’’ + 2(f’2+ff”) = 0\" style=\"width: 147.5px; height: 20px;\" width=\"147.5\" height=\"20\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 236.783px 7.91667px; transform-origin: 236.783px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere primes denote differentiation with respect to the independent variable \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 103.467px 7.91667px; transform-origin: 103.467px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The problem has the conditions \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f(0) = f''(0) = 0\" style=\"width: 115px; height: 19px;\" width=\"115\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f'(oo) = 0\" style=\"width: 69px; height: 19px;\" width=\"69\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 75.0417px 7.91667px; transform-origin: 75.0417px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and the constraint         \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 44px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22px; text-align: left; transform-origin: 384px 22px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"integral of f'^2 from -infinity to infinity = a\" style=\"width: 119.5px; height: 44px;\" width=\"119.5\" height=\"44\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 71.8417px 7.91667px; transform-origin: 71.8417px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e                                     \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 7.91667px; transform-origin: 21.0083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ea\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 43.1667px 7.91667px; transform-origin: 43.1667px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is a constant.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 194.342px 7.91667px; transform-origin: 194.342px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to solve this problem—that is, return values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ef\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 69.2333px 7.91667px; transform-origin: 69.2333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at specified values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 103.575px 7.91667px; transform-origin: 103.575px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The test suite allows MATLAB’s functions for numerical solution of ODEs, but the equation has a relatively simple analytical solution. If you would like a hint for the analytical solution, execute this command:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 296.25px 7.91667px; transform-origin: 296.25px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003echar('Zxj%ymj%uwtizhy%wzqj%tk%inkkjwjsynfynts%y|t%ynrjx%fsi%nsyjlwfyj%ymwjj%ynrjx3'-5)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.8333px 7.91667px; transform-origin: 40.8333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eBackground\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 147px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 73.5px; text-align: left; transform-origin: 384px 73.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 129.925px 7.91667px; transform-origin: 129.925px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe physical problem involves a jet in the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADEAAAAkCAYAAAAgh9I0AAABi0lEQVRYhe2WUZGEMAyGPw84wAAGUHAK6gAHOMBCNSABD1hAQy1wD22GwtGDtrD70m9mZ3ebIW3S5CdQKBQKhUKhUCgcqAEF9O730dY7ey4N0AJdwF/l9upinP4AIzABq/uMnl156yt/A4yhBwbAOF/mxF/v7RUViCCBGGxGWmDBBqrZB5dDy3bQ/mCrgDlgu0XnOZcA2tSTXiAHXQJ2OUM0DVsQhmd6IISfsLMeNDnOpV5DGXqKmnDtS+8kMzrHc46Tmyxur8lbq9x6soD4TZWrRHfQbLdeubXsW9Dsg3izJ2Av3w02aVm3oNgyIn2hs4/5P35fyPsqSVbBZmFx37D1hTS3lFnt/dfYWh7YSiEF6YuZSDGp3EFHrDIYbCYEX/6U20AF7Ct5NSwJi34v+G/MM4mrL+zHUSRH02XMSCpdGfZC0bcXdnl+Ik+Oe/bq9BUG0stJeu2t0eYWIgipWZxInFSfQgJI1XTN+xJ+ieL+DchAZ7A98ORY/zH86ThXkr+KqN3b81ihcIdfNuyHpTGjL0oAAAAASUVORK5CYII=\" alt=\"x, y\" style=\"width: 24.5px; height: 18px;\" width=\"24.5\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 108.133px 7.91667px; transform-origin: 108.133px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e plane emanating from a source at \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"(x,y) = (0,0)\" style=\"width: 87.5px; height: 19px;\" width=\"87.5\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 64.1667px 7.91667px; transform-origin: 64.1667px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. It can be solved by expressing the velocities \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eu\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ev\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 278.117px 7.91667px; transform-origin: 278.117px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in terms of a streamfunction and employing a similarity solution that combines the spatial coordinates into a single variable \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 73.5083px 7.91667px; transform-origin: 73.5083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. In the problem above, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f(eta)\" style=\"width: 32px; height: 19px;\" width=\"32\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 172.308px 7.91667px; transform-origin: 172.308px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is proportional to the streamfunction. The conditions at \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"eta = 0\" style=\"width: 36px; height: 18px;\" width=\"36\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 47.4583px 7.91667px; transform-origin: 47.4583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e define the line \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y = 0\" style=\"width: 37px; height: 18px;\" width=\"37\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 143.508px 7.91667px; transform-origin: 143.508px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e as a line of symmetry; the transverse velocity \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ev\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 128.733px 7.91667px; transform-origin: 128.733px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and the shear stress are zero there. The condition \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f'(oo) = 0\" style=\"width: 69px; height: 19px;\" width=\"69\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 110.45px 7.91667px; transform-origin: 110.45px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e states that the streamwise velocity \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eu\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 203.767px 7.91667px; transform-origin: 203.767px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is zero far from the source. The integral constraint states that the momentum flux of the jet is constant. The value \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"a = 4/3\" style=\"width: 51.5px; height: 19px;\" width=\"51.5\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 200.7px 7.91667px; transform-origin: 200.7px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e corresponds to the physical problem of the jet. Other values are included for variety.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function f = lamjetODE(eta,a)\r\n  f = g(eta,a);\r\nend","test_suite":"%%\r\na = 4/3; \r\neta = 0:0.2:1;\r\nf_correct = [0 0.197375320224904 0.379948962255225 0.537049566998035 0.664036770267849 0.761594155955765];\r\nassert(all(abs(lamjetODE(eta,a)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 4/3; \r\neta = log(2);\r\nf_correct = 0.6;\r\nassert(abs(lamjetODE(eta,a)-f_correct)\u003c1e-10)\r\n\r\n%%\r\na = 1;\r\neta = pi;\r\nf_correct = 0.902552583791843;\r\nassert(abs(lamjetODE(eta,a)-f_correct)\u003c1e-10)\r\n\r\n%%\r\na = 0.5;\r\neta = 1.5;\r\nf_correct = 0.572446107496431;\r\nassert(abs(lamjetODE(eta,a)-f_correct)\u003c1e-10)\r\n\r\n%%\r\na = 0.75;\r\neta = -4:0;\r\nf_correct = [-0.823247562011528 -0.813902901498664 -0.766864130068021 -0.559711742572462 0];\r\nassert(all(abs(lamjetODE(eta,a)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = rand;\r\neta = 4*rand;\r\nassert(abs(lamjetODE(eta,a)+lamjetODE(-eta,a))\u003c1e-10)","published":true,"deleted":false,"likes_count":2,"comments_count":2,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":"2021-03-21T03:15:00.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2021-03-21T02:07:31.000Z","updated_at":"2021-03-22T00:42:37.000Z","published_at":"2021-03-21T02:24:49.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn the solution for a two-dimensional laminar jet, the following nonlinear ordinary differential equation arises\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f’’’ + 2(f’2+ff”) = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\\\\prime\\\\prime\\\\prime + 2(f\\\\prime^2+ff\\\\prime\\\\prime) = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere primes denote differentiation with respect to the independent variable \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The problem has the conditions \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f(0) = f''(0) = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef(0) = f\\\\prime\\\\prime(0) = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f'(oo) = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\\\\prime(\\\\infty ) = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and the constraint         \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"integral of f'^2 from -infinity to infinity = a\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\int_{-\\\\infty}^\\\\infty\\\\left[f\\\\prime(\\\\eta)\\\\right]^2d\\\\eta=a\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e                                     \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is a constant.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to solve this problem—that is, return values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at specified values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The test suite allows MATLAB’s functions for numerical solution of ODEs, but the equation has a relatively simple analytical solution. If you would like a hint for the analytical solution, execute this command:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003echar('Zxj%ymj%uwtizhy%wzqj%tk%inkkjwjsynfynts%y|t%ynrjx%fsi%nsyjlwfyj%ymwjj%ynrjx3'-5)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eBackground\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe physical problem involves a jet in the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x, y\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex, y\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e plane emanating from a source at \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"(x,y) = (0,0)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e(x,y) = (0,0)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. It can be solved by expressing the velocities \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"u\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eu\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"v\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ev\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e in terms of a streamfunction and employing a similarity solution that combines the spatial coordinates into a single variable \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. In the problem above, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f(eta)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef(\\\\eta)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is proportional to the streamfunction. The conditions at \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e define the line \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e as a line of symmetry; the transverse velocity \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"v\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ev\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and the shear stress are zero there. The condition \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f'(oo) = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\\\\prime(\\\\infty ) = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e states that the streamwise velocity \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"u\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eu\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is zero far from the source. The integral constraint states that the momentum flux of the jet is constant. The value \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a = 4/3\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea = 4/3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e corresponds to the physical problem of the jet. Other values are included for variety.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"}],"term":"tag:\"fluid 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