Momentum Conservation
There are numerous ways for deriving the momentum equation in porous media. Here, start from the assumption that the pressure drop in a fluid results from viscous losses only and transient and inertial effects are neglected:
The pressure p is the fluid pore pressure, and for low velocities the viscous force depends linearly on the velocity (Equation 2-11) according to ∇p ∼ u. This relation is well known as Darcy’s law (Equation 2-1). The situation becomes more complicated when transient and inertial effects must be considered. For a pure fluid, the governing equations are the Navier–Stokes equations (see the section General Single-Phase Flow Theory in the COMSOL Multiphysics Reference Manual). The momentum conservation (Equation 13-2) rewritten for the fluid in a porous medium reads
(2-13)
Remember that v denotes the interstitial velocity and K is the stress tensor. Equation 2-13 describes the momentum conservation per unit volume of the porous medium. It is more common to formulate the equation in terms of Darcy velocity. Dividing by εp and using Equation 2-12 yields
The pressure drop due to viscous losses is described in Equation 2-2 and one obtains
(2-14)
A mass source or sink term adds a momentum source/sink as well and adds the term -εpQv to the right hand side of Equation 2-13. In terms of Darcy velocity Equation 2-14 can be rewritten as
with
Thus it results in the Brinkman equations (Equation 4-35) used by The Brinkman Equations Interface.