Heuristic derivation of Brinkman's seepage equation

Abstract

Brinkman’s law is describing the seepage of viscous fluid through a porous medium and is more acurate than the classical Darcy’s law. Namely, Brinkman’s law permits to conform the flow through a porous medium to the free Stokes’ flow. However, Brinkman’s law, similarly as Schro¨dinger’s equation was only devined. Fluid in its motion through a porous solid is interacting at every point with the walls of pores, but the interactions of the fluid particles inside pores are different than the interactions at the walls, and are described by Stokes’ equation. Here, we arrive at Brinkman’s law from Stokes’ flow equation making use of successive iterations, in type of Born’s approximation method, and using Darcy’s law as a zero-th approximation.