Abstract
Despite the widespread use of the Darcy equation to model porous flow, it is well known that this equation is inconsistent with commonly prescribed no slip conditions at flow domain walls or interfaces between different sections. Therefore, in cases where the wall effects on the flow regime are expected to be significant, the Darcy equation which is only consistent with perfect slip at solid boundaries, cannot predict velocity and pressure profiles properly and alternative models such as the Brinkman equation need to be considered. This paper is devoted to the study of the flow of a Newtonian fluid in a porous medium between two impermeable parallel walls at different Darcy parameters (Da). The flow regime is considered to be isothermal and steady. Three different flow regimes can be considered using the Brinkman equation: free flow (Da > 1), porous flow (high permeability, 1 > Da > 10−6) and porous flow (low permeability Da < 10−6). In the present work the described bench mark problem is used to study the effects of solid walls for a range of low to high Darcy parameters. Both no-slip and slip conditions are considered and the results of these two cases are compared. The range of the applicability of the Brinkman equation and simulated results for different cases are shown.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
F.M. Allan M.H. Hamdan (2002) ArticleTitleFluid mechanics of the interface region between two porous layers Appl. Math. Comp. 128 37–43 Occurrence Handle10.1016/S0096-3003(01)00016-9
S. Chikh A. Boumedien K. Bouhadef G. Lauriat (1995) ArticleTitleAnalytical solution of non-Darcian forced convection in an annular duct partially filled with a porous medium Int. J. Heat Mass Transfer 38 1543–1551 Occurrence Handle10.1016/0017-9310(94)00295-7
C.F. Gerald P.O. Wheatley (1984) Applied Numerical Analysis Addison-Wesley Reading, MA 258–261
M.H.R. Ghoreishy V. Nassehi (1997) ArticleTitleModelling the transient flow of rubber compounds in the dispersive section of an internal mixer with slip-stick boundary conditions Adv. Polymer Technol. 16 45–68
J.T. Hong C.L. Tien M. Kaviany (1985) ArticleTitleNon-Darcian effects on vertical -plate natural convection in porous media with high porosities Int. J. Heat Mass Transfer 28 2149–2157 Occurrence Handle10.1016/0017-9310(85)90109-7
C.T. Hsu P. Cheng (1985) ArticleTitleThe Brinkman model for natural convection about a semi-infinite vertical flat plate in a porous medium Int. J. Heat Mass Transfer 28 683–697 Occurrence Handle10.1016/0017-9310(85)90190-5
B.M. Irons (1970) ArticleTitleA frontal solution for finite element analysis Int. J. Numer. Methods Engng 2 5–32 Occurrence Handle10.1002/nme.1620020104
M. Kaviany (1985) ArticleTitleLaminar flow through a porous channel bounded by isothermal parallel plates Int. J. Heat Mass Transfer 28 851–858
M. Kaviany (1986) ArticleTitleNon-Darcian effects on natural convection in porous media confined between horizontal cylinders Int. J. Heat Mass Transfer 29 1513–1519
D.V. Krishna D.R.V. Prasada V. Sugunamma (1999) ArticleTitleFinite element analysis of convection flow through a porous medium in a horizontal channel Transport Porous Media 36 69–83 Occurrence Handle10.1023/A:1006513000488
A.A. Merrikh A.A. Mohmmad (2002) ArticleTitleNon-Darcy effects in boundary driven flows in an enclosure filled with vertically layered porous media Int. J. Heat Mass Transfer 45 4305–4313 Occurrence Handle10.1016/S0017-9310(02)00135-7
V. Nassehi (2002) Practical Aspects of Finite Element Modelling of Polymer Processing Wiley Chichester
D.A. Nield A. Bejan (1992) Convection in Porous Media Springer-Verlag New York
J.F.T. Pittman (1989) Finite elements for field problems C.L. Tucker SuffixIII (Eds) Computer Modelling for Polymer Processing Hanser Publishers Munich 269–273
J.N. Reddy (1986) Applied Functional Analysis and Variational Methods in Engineering McGraw-Hill New York
B.V.K. Satya Sai K.N. Sitaramu P.A. Aswathanarayana (1997) ArticleTitleFinite element analysis of heat transfer by natural convection in porous media in vertical enclosures Int. J. Numerical Methods Heat Fluid Flow 7 367–400
K. Vafai C.L. Tien (1981) ArticleTitleBoundary and inertia effects on flow and heat transfer in porous media Int. J. Heat Mass Transfer 24 195–203 Occurrence Handle10.1016/0017-9310(81)90027-2
K. Vafai R. Thiagaraja (1987) ArticleTitleAnalysis of flow and heat transfer at the interface region of a porous medium Int. J. Heat Mass Transfer 30 1391–1405 Occurrence Handle10.1016/0017-9310(87)90171-2
O.C. Zienkiewicz R.L. Taylor (1994) The Finite Element Method McGraw-Hill London
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Parvazinia, M., Nassehi, V., Wakeman, R.J. et al. Finite Element Modelling of Flow Through a Porous Medium Between Two Parallel Plates Using The Brinkman Equation. Transp Porous Med 63, 71–90 (2006). https://doi.org/10.1007/s11242-005-2721-2
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s11242-005-2721-2