Darcy’s law from lattice-gas hydrodynamics

Abstract

Within the hexagonal lattice-gas model, we obtain Darcy’s law for flow in the presence of scatterers. The associated momentum dissipation is described by an effective damping term in the Navier-Stokes equation, which we relate to the density of scatterers. The kinematic viscosity can be obtained from the Darcy velocity profile, once the permeability is determined. We also check that in the hexagonal lattice model, after coarse graining, velocity decay and plane-parallel Poiseuille flow occur as described by the macroscopic equations.