  Darcy's law and the Brinkman equation are two main models used for creeping
fluid flows inside moving permeable particles. For these two models, the time
derivative and the nonlinear convective terms of fluid velocity are neglected
in the momentum equation. In this paper, a new momentum equation including
these two terms are rigorously derived from the pore-scale microscopic
equations by the volume-averaging method, which can reduces to Darcy's law and
the Brinkman equation under creeping flow conditions. Using the lattice
Boltzmann equation method, the macroscopic equations are solved for the problem
of a porous circular cylinder moving along the centerline of a channel.
Galilean invariance of the equations are investigated both with the intrinsic
phase averaged velocity and the phase averaged velocity. The results
demonstrate that the commonly used phase averaged velocity cannot serve as the
superficial velocity, while the intrinsic phase averaged velocity should be
chosen for porous particulate systems.
