A Model for the Compressible Flow Through a Porous Medium

Abstract

A model for a continuum gas flowing through a porous matrix is proposed where the gas kinetics is governed by the Boltzmann equation and the solid phase by the energy equation. In the Boltzmann equation the integral relative to the gas-solid collisions is evaluated as for the collisions of hard spheres molecules against much heavier and longer straight particles (Lebowitz model of a sticks gas), randomly distributed in space according to a Maxwellian function with zero mean velocity. The mean flow is one-dimensional but the molecules are free to move in all three space dimensions. In the continuum limit, the moments of the Boltzmann equation provide the mass continuity, energy and momentum equations, the last one expressing the Darcy law for a compressible gas. The transport coefficients are analytically evaluated and a few examples are dealt with.

Rarefied gas flow over an in-line array of circular cylinders

“…[46][47][48][49] On the other hand, the macroscopic models for rarefied gas flows in porous media have been classically derived on the basis of the dusty gas model 50 or more recently on the basis of the modified Boltzmann equations with additional collision terms which take into account the effect of the collisions of the gas molecules with the solid matrix. [51][52][53] It should be stressed that, in the present approach, the effect of the solid boundary is included in the effective diffusion tensor entirely through the boundaryvalue problem ͑or the cell auxiliary problem͒ for ⌽. In this connection, it should be mentioned that, although the diffuse reflection condition is assumed in the present analysis, the extension to more general boundary conditions is straightforward ͑see, e.g., Ref.…”

Topology optimization of fluid domains: kinetic theory approach

“…Therefore, we try to employ similar ideas to control the gases in our situation. Over the years, researchers have proposed several models for computing flows of gases through porous media based on effective properties of the domain (as opposed to high-fidelity resolution of the complicated geometries natural for porous media): see for example works [18,19,41,44,57] among many others. It should also be possible to apply homogenization ideas for transport equations [20] to a linearized BGK equation (6) in the same manner as it was done for incompressible [1] or compressible [40] Navier-Stokes equations.…”

“…Consider the optimization problem (19). Assume that the set of design constraints Z is weakly * closed, and the objective functional F is lower semi-continuous w.r.t.…”

A Kinetic Approach to the Plane Poiseuille Flow Over a Porous Matrix

“…As we will see later in greater details, a particular model for the gas flow in a porous matrix was already proposed in a paper (de Socio et al, 2001) which is based on a description of the molecular collisions at different microscopic scales between light molecules and massive particles.…”

“…The kinetic model cited above (de Socio et al, 2001(de Socio et al, , 2003a assumes that the Boltzmann equation holds also for the fluid in this region; then a second collision integral appears on its right hand side in addition to the integral corresponding to the particle-particle collisions. This second integral takes care of the collisions between the particles and the solid matrix.…”