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# Effects of variable viscosity and variable permeability on fluid flow through porous media

## Abstract

In this work, we study the effects of variable viscosity and variable permeability on single-phase fluid flow through porous structures. This is accomplished by first deriving the equations governing fluid flow through porous structures in which porosity (hence permeability) is a function of position and viscosity of the fluid is pressure-dependent. The governing equations are derived using intrinsic volume averaging, and viscous effects are accounted for through Brinkman’s viscous shear term. When the Darcy resistance, Brinkman’s viscous shear effects and Lapwood’s macroscopic inertial terms are accounted for, the governing equation is known as the Darcy-Lapwood-Brinkman equation, and it governs the flow through a mushy zone undergoing rapid freezing, and is important in slurry transport. Three exact solutions to the Darcy-Lapwood-Brinkman equation with variable permeability are obtained in this work. Solutions are obtained for a given vorticity distribution, taken as a function of the streamfunction. Classification of the flow field is provided and comparison is made with the solutions obtained when permeability is constant. Interdependence of Reynolds number and the variable permeability is emphasized. Exact solutions are also obtained for this equation when the vorticity is proportional to the streamfunction, and a derivation of the permeability function that satisfies the governing equations is provided. The problem of laminar flow through a porous medium of variable permeability, behind a two-dimensional grid is considered in this work to further shed some light of the effects of permeability variations. Expressions for the permeability profiles are derived when the model equations are linearized and permeability is calculated at the stagnation points of the flow. Conditions on the parameters involved in the exact solution are analyzed and stated and the flow is classified and compared with the case of flow through constant permeability media. This work might be of interest in the stability analysis of flow through variable permeability media. In studying the effects of pressure-dependent viscosity on fluid flow, this work provided analysis involving viscosity stratification. Coupled parallel flow of fluids with viscosity stratification through two porous layers is initiated in this work. Conditions at the interface are discussed and appropriate viscosity stratification functions are selected in such a way that viscosity is highest at the bounding walls and decreases to reach its minimum at the interface. Velocity and shear stress at the interface are computed for different permeability and driving pressure gradient. Consideration is given to two-dimensional flow of a fluid with pressure-dependent viscosity through a variable permeability porous structure. Exact solutions are obtained for a Riabouchinsky type flow using a procedure that is based on an existing methodology that is implemented in the study of Navier-Stokes flow with pressure-dependent viscosity. Viscosity is considered proportional to fluid pressure due to the importance and uniqueness of validity of this type of relation in the study of Poiseuille flow. The effects of changing the proportionality constant on the pressure distribution are discussed. Since a variable permeability introduces an additional variable in the flow equations and renders the governing equations under-determined, the current work devises a methodology to determine the permeability function through satisfaction of a condition derived from the specified streamfunction. Illustrative examples are used to demonstrate how the variable permeability is determined, and how the arising parameters are determined. Although the current work considers flow in an infinite domain and does not handle a particular engineering problem, it nevertheless initiates the study of flow of fluids with pressure-dependent viscosity through variable-permeability media and sets the stage for future work in stability analysis of this type of flow. It is expected that the current work will be of value in transition layer analysis and the determination of variable permeability functions suitable for such analysis.