# 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

- 467 Downloads
- 62 Citations

## 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

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.Beavers, G.S. and Joseph, D.D., Boundary conditions at a naturally permeable wall.
*J. Fluid Mechanics*30 (1967) 197–207.Google Scholar - 2.Larson, R.E. and Higdon, J.J.L., Microscopic flow near the surface of two-dimensional porous media — I. Axial flow.
*J. Fluid Mechanics*166 (1987) 449–472.Google Scholar - 3.Larson, R.E. and Higdon, J.J.L., Microscopic flow near the surface of two-dimensional porous media — II. Transverse flow.
*J. Fluid Mechanics*178 (1987) 119–136.Google Scholar - 4.Sahraoui, M. and Kaviany, M., Slip and no-slip velocity boundary conditions at interface of porous, plain media.
*Int. J. Heat Mass Transfer*35 (1992) 927–943.Google Scholar - 5.Nield, D.A. and Bejan, A.,
*Convection in Porous Media*. New York: Springer-Verlag (1992).Google Scholar - 6.Kaviany, M.,
*Principles of Heat Transfer in Porous Media*. New York: Springer-Verlag (1991).Google Scholar - 7.Vafai, K. and Thiyagaraja, R., Analysis of flow and heat transfer at the interface region of a porous medium.
*Int. J. Heat Mass Transfer*30 (1987) 1391–1405.Google Scholar - 8.Vafai, K. and Kim, S.J., Fluid mechanics of the interface region between a porous medium and a fluid layer — An exact solution.
*Int. J. Heat and Fluid Flow*11 (1990) 254–256.Google Scholar - 9.Poulikakos, D. and Kazmierczak, M., Forced convection in a duct partially filled with a porous material.
*ASME Journal of Heat Transfer*109 (1987) 653–662.Google Scholar - 10.Nield, D.A., The limitations of the Brinkman-Forchheimer equation in modeling flow in a saturated porous medium and at an interface.
*Int. J. Heat and Fluid Flow*12 (1991) 269–272.Google Scholar - 11.Ochoa-Tapia, J.A. and Whitaker, S., Momentum transfer at the boundary between a porous medium and a homogeneous fluid — I. Theoretical development.
*Int. J. Heat Mass Transfer*38 (1995) 2635–2646.Google Scholar - 12.Ochoa-Tapia, J.A. and Whitaker, S., Momentum transfer at the boundary between a porous medium and a homogeneous fluid — II. Comparison with experiment.
*Int. J. Heat Mass Transfer*38 (1995) 2647–2655.Google Scholar - 13.Givler, R.C. and Altobelli, S.A., A determination of the effective viscosity for the Brinkman-Forchheimer flow model.
*J. Fluid Mechanics*258 (1994) 355–370.Google Scholar