Hi, can anyone help me to program a code for the equation with initial and boundary conditions using finite difference method

Hi, I really need help from anyone help me with how to do a program code for governing equation with initial and boundary condition. Also, I want to plot graph velocity(u) against radius (r). The radius will be the x-axis and the range should be from 0 to 1 only. I will be really appreciated if anyone can help me. Thank you so much. Inside the coding there must be include discrerization of the governing equation. If u need , I wiil send it here right away.

The governing equation is:

Answer 1

[William Rose]

@Nur Nadhirah Syed Malik, See attached.

I have left out some parts, because this sounds like a homework assignment, and you will learn by filling in the unknown parts.

[Nur Nadhirah Syed Malik]

Actually this is not homework. This is for my final year project. And I need to send it by tomorrow. I have tried so many ways to program the code but none of them I can run. That is why I am asking for your help. Please, I hope you can help me

[William Rose]

@Nur Nadhirah Syed Malik,

Se attached file.

I also notice that this is the simplified version of the Navier Stokes equation for fluid flow in a circular pipe, subjected to a sinusoidal predssure gradient.

This was solved analytically by Womersley.

What values can you provide for A0, A1, b1, and w? What time range? And what are the units?

The reaosn I ask about the constants is that tried solving your PDE and I found the solution to be rather unstable. By unstable, I mean that velocity has a tendency to diverge to +/- infinity, if one is not careful.

[William Rose]

@Nur Nadhirah Syed Malik,

A script is attached that implements the equaitons in the document I posted earlier.

The script generates two figures, shown below:

[William Rose]

@Nur Nadhirah Syed Malik,

I believe that the constant in your equation corresponds to kinematic viscosity: =viscosity/density. I also believe that the units for A0 and A1 are pressure/length, and that the terms A0+A1*cos wt correspond to the pressure difference divided by the length of the tube. Are those beliefs correct? The Womersley number for pulsatile flow is . If I am correct about , then the Womersley number for this problem (in which R=1) is .