Theoretical model of the boundary condition at a fluid‐porous interface

An equation for the volume average fluid velocity in a boundary layer region between a fluid and a porous medium is derived. The result is valid for a general anisotropic porous medium subject to the restrictions imposed by the assumptions made in the analysis. The basic equations of motion for an incompressible Newtonian fluid with negligible inertia are used and a linear map M ij from the volume average velocity 〈 v 〉 into the point velocity v is assumed to exist. Using scaling arguments, an equation for 〈 v 〉 is found which reduces to the Darcy equation for a point entirely within the porous medium below the boundary region and to the usual Navier‐Stokes equation for a point entirely within the fluid above the boundary region. The form of the equation contains two of the terms that were originally hypothesized and tested experimentally by Beavers and Joseph (1967). In particular, the following equation is obtained:\documentclass{article}\pagestyle{empty}\begin{document}$$ \left\langle {v_i } \right\rangle = - \frac{{K_{ij} }}{\mu }\frac{\partial }{{\partial x_j }}\left\langle P \right\rangle + K_{ij} L_{jmk} \frac{\partial }{{\partial x_k }}\left\langle {v_m } \right\rangle + K_{ij} N_{jm} \nabla ^2 \left\langle {v_m } \right\rangle $$\end{document} where the tensors K ij , L jmk and N jm are mathematically defined in terms of the map M ij .