# What are acceptable boundary conditions for porous media flow?

I am attempting to simulate fluid flow through a porous foam. I would like to have no-slip boundary conditions on part of the boundary and free flow conditions on the inlet and outlet. Right now I am using the Darcy-Brinkman equations:

$-\mu\Delta\mathbf{u} + \mathbf{K}\mathbf{u} + \nabla p = \mathbf{0}$

$\nabla\cdot\mathbf{u} = 0$

where $\mathbf{K}$ is a permeability tensor. Mathematically the boundary conditions I have here make sense, however Whitaker (http://link.springer.com/article/10.1007%2FBF01036523) uses a length scale argument to say that the Darcy-Brinkman equations cannot be used to impose no slip boundary conditions. In literature this is still a common technique however. Is there a justification for using the Darcy-Brinkman equations as I wish to do? Ideally I'd like a mathematical justification, but that would apparently directly contradict Whitaker, so an experimental justification is acceptable.

If there is no justification, can I reformulate my problem somehow using regular Darcy flow?

## 1 Answer

The velocity $\mathbf u$ is in $H^1$, so imposing all or some components of the velocity as boundary conditions is allowed. This includes no-slip or tangential flow boundary conditions.

However, from a physical perspective, this only makes sense if the ratio of the Stokes to the Brinkman terms is sufficient large, i.e., in particular if $\frac{\mu}{|\mathbf K| L^2} \gg 1$ where $L$ is a length scale that compensates that the Stokes term has two derivatives whereas the Brinkman term does not. On the other hand, if $\frac{\mu}{|\mathbf K| L^2} \ll 1$, then the porous media aspect dominates the Stokes flow aspect, and the solution degenerates to one that is only a slightly regularized function in $H_\text{div}$ where the solution of the mixed Laplace lives. In that case, you can no longer impose all components of the velocity at the boundary, but only the normal component.

• Thanks. What is the physical interpretation of $L$? – Lukas Bystricky Jun 27 '15 at 5:28
• Some length scale of the flow, say the size of an eddy if you were using the Navier-Stokes equations, or the length scale over which the boundary conditions for the velocity change significantly, or simply the diameter of the domain. – Wolfgang Bangerth Jun 27 '15 at 11:28
• OK, $L$ is the macro length scale. In the Whitaker paper he uses multiple length scales (pore, solid, averaging volume and domain). Do you have a reference I could look at regarding your original reply? – Lukas Bystricky Jun 27 '15 at 12:19
• No reference. I just made this up as an educated answer. – Wolfgang Bangerth Jun 28 '15 at 22:03