Applied Scientific Research

, Volume 56, Issue 1, pp 53–67 | Cite as

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

  • A. V. Kuznetsov
Article

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

DaH

Darcy number for a parallel-plate channel,K/H2

Dai

value of the Darcy number at the interface for the non-uniform porosity model

DaR

Darcy number for a cylindrical channel,K/R2

H

width of the fluid layer in a parallel-plate channel, m

Iv,Kv

modified Bessel functions of the orderv

K

permeability of the porous medium, m2

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

ũf

superficial average velocity, i.e. volumetric discharge of fluid per unit area, m s−1

u

dimensionless velocity, 〈ũ f 〉/U

ũi

superficial average velocity at the interface, m s−1

ui

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, (μeff f )1/2

γf

fluid viscosity, kg m−1 s{−1}

μeff

effective viscosity in the Brinkman term, kg m−1 s−1

ρ

density, kg m−3

σ

constant,\(L/\gamma H\sqrt {Da_H } \)

ΞR

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

ΞR

dimensionless pressure gradient inx-direction for a cylindrical channel,\( - (R^2 /\mu _f U)(d\left\langle {p_f } \right\rangle ^f /d\tilde x)\)

ψ1

constant,\(1/\gamma \sqrt {Da_R } \)

ψ2

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.

Unable to display preview. Download preview PDF.

References

  1. 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. 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. 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. 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. 5.
    Nield, D.A. and Bejan, A.,Convection in Porous Media. New York: Springer-Verlag (1992).Google Scholar
  6. 6.
    Kaviany, M.,Principles of Heat Transfer in Porous Media. New York: Springer-Verlag (1991).Google Scholar
  7. 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. 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. 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. 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. 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. 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. 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

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • A. V. Kuznetsov
    • 1
  1. 1.Mechanical Engineering Research Institute of Russian Academy of SciencesMoscowRussia

Personalised recommendations