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I'm trying to solve the flow profile inside an inhomogeneous porous material between two parallel moving plates (essentially Couette flow with a deviation), and I model my system by the following equations:

[itex]

\nabla^2 \mathbf{u} = p(x,y,z) \mathbf{u}\\

\nabla \cdot \mathbf{u} = 0

[/itex]

where p(x,y,z) is a given function describing the resistance of the porous material. The boundary conditions on u are of Dirichlet type. I am looking for a numerical solution to this problem. I have tried to solve the viscous equations by a relaxation technique (Jacobi method, etc.) and then impose the incompressibility by some other means such as:

1) Pressure projection method. The problem I encounter is that I have a hard time solving the pressure Poisson equation with Neumann boundary conditions (grad of pressure is the velocity at the boundary).

2) Penalty function. To my first equation I add a penalty term proportional to the local divergence. This effectivelly couples the flow in various directions, but the total of the divergence keeps increasing and my solution never settles to some stationary state.

My main problem is to enforce the incompressibility (zero divergence). Any ideas? I think this is not a difficult problem, I just can't seem to find any examples in the literature...

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# Numerical solution to stationary viscous flow with a Darcy term

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