Volume-averaged macroscopic equation for fluid flow in moving porous media

TL;DR

This paper derives a new macroscopic fluid flow equation for moving porous media that includes time derivative and nonlinear convective terms, extending traditional models like Darcy's law and Brinkman equation.

Contribution

A rigorously derived volume-averaged momentum equation incorporating additional terms, validated through lattice Boltzmann simulations for moving porous particles.

Findings

Intrinsic phase averaged velocity is suitable as superficial velocity.

The new equation reduces to Darcy's law and Brinkman equation under specific conditions.

Galilean invariance depends on the choice of velocity averaging method.

Abstract

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.