Porous media fluid motion equations

The governing flow equation describing flow through as porous medium is known as Darcy s law, which is a relationship between the volumetric flow rate of a fluid flowing linearly through a porous medium and the energy loss of the fluid in motion. 

All these different mechanisms of mass transport through a porous medium can be studied experimentally and theoretically through classical models (Darcy s law, Knudsen diffusion, molecular dynamics, Stefan-Maxwell equations, dusty-gas model etc.) which can be coupled or not with the interactions or even reactions between the solid structure and the fluid elements. Another method for the analysis of the species motion inside a porous structure can be based on the observation that the motion occurs as a result of two or more elementary evolutions that are randomly connected. This is the stochastic way for the analysis of species motion inside a porous body. Some examples that will be analysed here by the stochastic method are the result of the particularisations of the cases presented with the development of stochastic models in Sections 4.4 and 4.5. 

Problem 9-22. Flow in a Brinkman Medium. Fluid flow in a packed bed or porous medium can be modeled as flow in a Brinkman medium, which we may envision as a bed of spherical particles. Each particle in the bed (there are n particles per unit volume) exerts a drag force on the fluid proportional to fluid velocity relative to the particle given by Stokes law, i.e., ( —Gtt/hiu, where a is the characteristic size of a bed particle). Thus the equations describing the fluid motion on an averaged scale (averaged over many bed particles, for example) are... 

Problem 12-17. Buoyancy-Driven Instability of a Fluid Layer in a Porous Medium Based on Darcy s Law. We consider the classical Rayleigh-Benard problem of a fluid layer that is heated from below, except in this case, the fluid is within a porous medium so that the equations of motion are replaced with the Darcy equations, which were discussed in Subsection Cl of this chapter. Hence the averaged velocity within the porous medium is given by Darcy s law... 

Problem 12-18. Buoyancy-Driven Instability of a Fluid Layer in a Porous Medium Based on the Darcy-Brinkman Equations. A more complete model for the motion of a fluid in a porous medium is provided by the so-called Darcy Brinkman equations. In the following, we reexamine the conditions for buoyancy-driven instability when the fluid layer is heated from below. We assume that inertia effects can be neglected (this has no effect on the stability analysis as one can see by reexamining the analysis in Section H) and that the Boussinesq approximation is valid so that fluid and solid properties are assumed to be constant except for the density of the fluid. The Darcy Brinkman equations can be written in the form... 

A porous medium consists of a packed bed of solid particles in which the fluid in the pores between particles is free to move. The superficial fluid velocity V is defined as the volumetric flow rate of the fluid per unit of cross-sectional area normal to the motion. It is the imbalance between the pressure gradient (VP) and the hydrostatic pressure gradient (pg) that drives the fluid motion. The relation that includes both viscous and inertial effects is the Forscheimer equation [47]... 

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