@janhardo sir thanks alot your support 

@janhardo sir, can you please help me to fix this issue? I think the problem is in this part. 

#-------------------------compute nusselt num ate y=1--------------------#######
                     

      Nu_results[i, j] := eval(-diff(theta(y), y), dsol(1));
end do;

This diff(theta(y), y), dsol(1))  is not working properly. Besides this, if put 0 at place of 1, the same file works good and give different values for Nuesselt numbers.

Same_values_why_help.mw

@janhardo I changed some values and functions in the previous code that you uploaded, but after this it did not provide any values. I followed all of the steps you wrote. Can you please help me correct the file?

@janhardo  Ok sir, thanks a lot for your help. My last query: I am calculating the Nusselt number, but it gives the same values for all parameter sets. Can you please help me why the code is giving the same results for the Nusselt number?

Same_values_why_help.mw

@janhardo Sir, does the file work on your computer? Because it gives error in my side

• @janhardo Sir, thanks for your help. Is it possible to generate single values of skin friction rather at any point of y=0.1 and k=0.1, other parameters remaining the same, then tables or an Excel sheet? 

@janhardo sir, attention is to view nuemric values in tabular form at one point of y such as 0.1 for each k=0.1,0.3,0.5,0.7,0.9 and also looking for to generate 1001 data points of skin friction in the range of y=-1...1 for each k values.

@janhardo Thanks, sir, for your help, but in my system, there is an error.

@janhardo, Thank you, sir; it's working perfectly. I had also submitted another question regarding how to export data points of a different quantity (from the same system) to an Excel sheet. However, it seems that my question was deleted by the system, possibly because it was marked as a duplicate.

I need further assistance on how to generate tables for the skin friction parameter and correctly save the data points of another quantity into Excel. Although the line graphs are generating correctly, there is an error when exporting the data to Excel. Also, in addition to line graphs, is it possible to create 3D graphs for this quantity? Please have a look at my Maple worksheet.

Data_points_help.mw

@janhardo Good day, sir, and thank you very much for your help. In the first part of the file, the tables and graphs of the Nusselt number might have been skipped by you, and they were also showing errors in my file, which I uploaded. If you don't mind, could you please help me with that as well?

@janhardo Dear sir, Thank you very much for your detailed responses and kind assistance. I sincerely appreciate the way you have explained everything thoroughly. Following your guidance, I attempted to run a different system; however, I am currently facing issues with the boundary conditions, and the results are not being generated properly. Could you please assist me with this?

Additionally, is it possible to export 1001 data points of the Nusselt number to an Excel sheet at a desired place in the computer? Specifically, I want the first column to represent the Brinkman number (Br) ranging from 0 to 1, and the second column to contain the corresponding Nusselt number (Nu) values for each fixed value of kkk, such as k=0.1, 0.3,…, 0.9.

Nusselt_Number_help.mw

@dharr, sir, The expected results for velocity look like this, means u=1 at x=0 in the range of y=0..1 The actual BCs for u(y) is u(1 + x^2/2) =1  and D(u)(0) = 0, but when we convert these BCs for Psi, it becomes D(psi)(1 + x^2/2) = 1 and D(D(psi))(0) = 0. After converting system in the form of psi by using stream function, the final equation becomes 4th order, and two more Bcs are needed, which are psi(1 + x^2/2) = H, psi(0) = 0.

@dharr sir, here is the final output of my code, and the curves for u(y) started from zero, not 1. 

@dharr thanks I got this point but could you ease resolve issue regarding u(y).

@acer I am solving a system of three coupled ordinary differential equations (ODEs), where the first equation for u(y) contains the unknown pressure gradient term dp/dx​. To eliminate this term and simplify the system, I introduced a stream function such that u(y)=dψ/dy​. This reformulation allows me to rewrite the entire system in terms of ψ(y) and solve for it numerically. Once ψ(y) is determined, compute u(y). However, I observed a problem: according to boundary conditions, the solutions for u(y), θ(y), and ϕ(y) should start from 1, but in my current code, the plots for these variables begin from zero.

The Bcs for u(y) are u(1 + x^2/2) = 1, D(u)(0) = 0, but in terms of psi(y),  is D(psi)(1 + x^2/2) = 1, D(D(psi))(0) = 0.