MATLAB PDE TOOL for Newtonian Fluid

For a Newtonian fluid (like water) in a duct with cross section, A and Length L = 2.0m. The governing equation is given by:

Mu*d2u/dx2 = Delta P/L

Mu = dynamic viscosity = 10*10E-04 Pa-s Delta P/L = 10 kPa/m is the pressure gradient that drives the flow.

What will be the coefficients c,a and f? What will be the Neumann and Dirichlet B.C?

Answer 1

[Mukul Rao]

Hello,

As far as I can see, your equation is a one dimensional description of Newtonian Fluid flow (unless the d stands for some other operator). For this simple scenario, especially when there is no time derivative, I would recommend using the MATLAB ODE suite to solve this problem. A solver like ode45 should suffice.

As far as the boundary conditions are concerned, it would depend on the problem at hand. Most pipe flows will have the inlet velocity specified. Essentially, u is known at the inlet and the mass flux d(rho*A*u)/dx will probably be specified.

If you would still like to use the PDE tool, you can refer the following page to learn how to specify coefficients :

https://www.mathworks.com/help/pde/ug/specifycoefficients.html

In your case, formatting the governing equation to the standard PDE toolbox equation form, leaves us with ( m = 0 , d = 0, c = -Mu, a = 0, f = DeltaP/L

[Rumit Doshi]

Hello Mukul,

Thanks for responding. All the data that is specified is being mentioned above. It does not mention mass flux or inlet/outlet velocity. Only dynamic viscosity and pressure gradient are given. So for generating velocity contour can you please tell me what should be the coefficients c,a and f. And what should be the Neumann and Dirichlet b.cs?

[Rumit Doshi]

I saw the updated comment. I need to use the elliptic system so coefficients c = Mu , a=0 and f= pressure gradient seems fine. Then I suppose the top face and bottom face should have u =0 as the Dirichlet boundary condition. I am unsure for the inlet and outlet Neumann b.c. please correct me if I am wrong with any of the above stated things and provide an answer to the other question.

Answer 2

[Rumit Doshi]

I saw the updated comment. I need to use the elliptic system so coefficients c = Mu , a=0 and f= pressure gradient seems fine. Then I suppose the top face and bottom face should have u =0 as the Dirichlet boundary condition. I am unsure for the inlet and outlet Neumann b.c. please correct me if I am wrong with any of the above stated things and provide an answer to the other question.