Is this a correct application of a loop to measure a derivative?

Hi All,

I have a velocity profile (v_phi) for a radial distance (r). From this velocity profile I am trying calculate the shear rate (y) which can be calculated by ( Eq 2 in Image):

y = ( d(v_phi) / d(r) ) - ( (v_phi) / r ) = r * ( ( d ( v_phi / r) / d(r) )

Please find the image of the equation from the papers attached,

I have implemented the equation as such,

~~~~

for i = 1: size(v_phi,1) 

y (i)  = ( r(i) ) * (( ( v_phi (i) / r(i) ) - ...
                            ( v_phi (i+1) / r(i+1) ) ) / ( r(i+1) - r(i)) );

y1 (i)  = ( ( ( v_phi(i) / r(i) ) - ( v_phi(i+1) / r(i+1) ) ) * ...
                            ( v_phi (i) / r(i) ) );
end

~~~~

However it seems that y and y1 values are very different. I am just wondering, whether I am doing something wrong in the implementation and looking for help if possible.

Many thanks.

Data >>>>

v_phi = [36.99 37.00 35.78 31.43 26.91 23.32 19.46 16.97 14.49 12.12 10.42 8.52 7.25 6.06 5.11 3.92 3.30 2.54 2.11 1.42 1.46 1.46 0.97 0.63 0.63 0.51 0.40 0.91 0.24 0.36 0.63 0.71 0.45 0.26 0.74 ];

r = [3.59 3.76 3.94 4.12 4.29 4.47 4.65 4.82 5.00 5.18 5.35 5.53 5.71 5.88 6.06 6.24 6.41 6.59 6.76 6.94 7.12 7.29 7.47 7.65 7.82 8.00 8.18 8.35 8.53 8.71 8.88 9.06 9.24 9.41 9.50 ];

Answer 1

[Arvind Narayanan]

Hi Abhi,

I understand that you are trying to use the finite difference method to calculate the derivative of v_phi w.r.t r. A few salient points I noticed about your code:

1) Your representation of the partial derivative of v_phi w.r.t r is only valid if r is the only independent variable of which v_phi is defined. If there are other variables involved, then d(v_phi) is not the same as partial derivative of v_phi.

2) Assuming that your representation is valid, I understand that y is the RHS of the equation and y1 is the LHS of the equation. If so, the representation of y1 is incorrect. The correct formula should be:

y1(i)= ( ( v_phi(i+1)-v_phi(i) / r(i+1)-r(i) ) - ( v_phi(i) / r(i) ) );

3) Ensure that the limits of the for loop are defined correctly between 1 and n-1, where n is the size of the array v_phi. Something like this:

v_phi = [36.99 37.00 35.78 31.43 26.91 23.32 19.46 16.97 14.49 12.12 10.42 8.52 7.25 6.06 5.11 3.92 3.30 2.54 2.11 1.42 1.46 1.46 0.97 0.63 0.63 0.51 0.40 0.91 0.24 0.36 0.63 0.71 0.45 0.26 0.74 ];
r = [3.59 3.76 3.94 4.12 4.29 4.47 4.65 4.82 5.00 5.18 5.35 5.53 5.71 5.88 6.06 6.24 6.41 6.59 6.76 6.94 7.12 7.29 7.47 7.65 7.82 8.00 8.18 8.35 8.53 8.71 8.88 9.06 9.24 9.41 9.50 ];
y=[];
y1=[];
for i = 1:1:size(v_phi,2)-1  
  y(i) = (r(i))*((( v_phi(i)/ r(i)) -( v_phi(i+1)/r(i+1)) )/(r(i+1)-r(i)) );
  y1(i)= ( ( v_phi(i+1)-v_phi(i) / r(i+1)-r(i) ) - ( v_phi(i) / r(i) ) );
end

Ensure to define y and y1 as empty arrays before the for loop.

Regards,

Arvind