# Analytical investigation of the fluid flow in the interface region between a porous medium and a clear fluid in channels partially filled with a porous medium

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## Abstract

In this paper analytical solutions for the steady fully developed laminar fluid flow in the parallel-plate and cylindrical channels partially filled with a porous medium and partially with a clear fluid are presented. The Brinkman-extended Darcy equation is utilized to model the flow in a porous region. The solutions account for the boundary effects and for the stress jump boundary condition at the interface recently suggested by Ochoa-Tapia and Whitaker. The dependence of the velocity on the Darcy number and on the adjustable coefficient in the stress jump boundary condition is investigated. It is shown that accounting for a jump in the shear stress at the interface essentially influences velocity profiles.

### Key words

porous media fluid flow interface region### Nomenclature

- Da
_{H} Darcy number for a parallel-plate channel,

*K*/*H*^{2}- Da
_{i} value of the Darcy number at the interface for the non-uniform porosity model

- Da
_{R} Darcy number for a cylindrical channel,

*K*/*R*^{2}*H*width of the fluid layer in a parallel-plate channel, m

*I*_{v},*K*_{v}modified Bessel functions of the order

*v**K*permeability of the porous medium, m

^{2}*L*width of the porous layer in a parallel-plate channel, m

- \(\left\langle {\tilde p_f } \right\rangle ^f \)
intrinsic average pressure, Pa

- \(\tilde r\)
radial coordinate, m

*r*dimensionless radial coordinate,\(\tilde r/R\)

*R*radius of the interface in a cylindrical channel, m

*R**the outer radius of a cylindrical channel, m

- 〈
*ũ*_{f}〉 superficial average velocity, i.e. volumetric discharge of fluid per unit area, m s

^{−1}*u*dimensionless velocity, 〈

*ũ*_{ f }〉/*U**ũ*_{i}superficial average velocity at the interface, m s

^{−1}*u*_{i}dimensionless velocity at the interface,

*ũ*_{ i }/*U**U*reference velocity, m s

^{−1}- \(\dot x\)
streamwise coordinate, m

- \(\dot y\)
transverse coordinate, m

*y*dimensionless transverse coordinate,\(\dot y/H\)

- \(\dot z\)
transverse coordinate, m

*z*dimensionless transverse coordinate,\(\dot z/H\)

### Greek letters

*β*the adjustable coefficient in the stress boundary condition

*ɛ*porosity

*γ*constant, (μ

_{eff}/μ_{ f })^{1/2}*γ*_{f}fluid viscosity, kg m

^{−1}s{−1}*μ*_{eff}effective viscosity in the Brinkman term, kg m

^{−1}s^{−1}*ρ*density, kg m

^{−3}*σ*constant,\(L/\gamma H\sqrt {Da_H } \)

- Ξ
_{R} dimensionless pressure gradient in

*x*-direction for a parallel-plate channel,\( - (H^2 /\mu _f U)(d\left\langle {p_f } \right\rangle ^f /d\tilde x)\)- Ξ
_{R} dimensionless pressure gradient in

*x*-direction for a cylindrical channel,\( - (R^2 /\mu _f U)(d\left\langle {p_f } \right\rangle ^f /d\tilde x)\)*ψ*_{1}constant,\(1/\gamma \sqrt {Da_R } \)

*ψ*_{2}constant,\(\left( {R^* /R} \right)\left( {1/\gamma \sqrt {Da_R } } \right)\)

### Subscripts

*f*fluid

*η*identifies a quantity associated with the clear fluid region

- ω
identifies a quantity associated with the porous region

### Other symbols

- 〈...〉
volume average

- 〈...〉
^{f} intrinsic volume average

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