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Natural and Mixed Convection

# Onset of Convection in a Horizontal Porous Layer Saturated by a Power-Law Fluid

[+] Author and Article Information
Z. Alloui1

Ecole Polytechnique,  Université de Montréal, C.P. 6079, Succ. «Centre Ville», Montréal, PQ, H3C 3A7, Canadazineddine.alloui@polymtl.ca

N. Ben Khelifa

Laboratoire des Technologies Innovantes, Université de Picardie Jules Vernes d’Amiens, rue des Facultés le Bailly, 80025 Amiens Cedex, France; Laboratoire des procédés thermiques,  Centre des Recherches et des Technologies de l’Energie, Technopole Borj Cedria B.P N°95, Hammam Lif 2050, Tunisia

H. Beji

Laboratoire des Technologies Innovantes,  Université de Picardie Jules Vernes d’Amiens, rue des Facultés le Bailly, 80025 Amiens Cedex, France

P. Vasseur

Ecole Polytechnique, Université de Montréal, C.P. 6079, Succ. «Centre Ville», Montréal, PQ, H3C 3A7, Canada; Laboratoire des Technologies Innovantes,  Université de Picardie Jules Vernes d’Amiens, rue des Facultés le Bailly, 80025 Amiens Cedex, France

1

Corresponding author.

J. Heat Transfer 134(9), 092502 (Jul 09, 2012) (8 pages) doi:10.1115/1.4006244 History: Received November 02, 2011; Revised February 15, 2012; Published July 09, 2012; Online July 09, 2012

## Abstract

This paper investigates the onset of motion, and the subsequent finite-amplitude convection, in a shallow porous cavity filled with a non-Newtonian fluid. A power-law model is used to characterize the non-Newtonian fluid behavior of the saturating fluid. Constant fluxes of heat are imposed on the horizontal walls of the layer. The governing parameters of the problem under study are the Rayleigh number $R$, the power-law index n, and the aspect ratio of the cavity A. An analytical solution, valid for shallow enclosures (A ≫ 1), is derived on the basis of the parallel flow approximation. In the range of the governing parameters considered in this study, a good agreement is found between the analytical predictions and the numerical results obtained by solving the full governing equations. For dilatant fluids (n > 1), it is found that the onset of motion is linearly unstable, i.e., always occurs provided that the supercritical Rayleigh number $RCsup≥0$. For pseudoplastic fluids (n < 1), the supercritical Rayleigh number for the onset of motion is $RCsup=∞$. However, it is demonstrated, on the basis of the nonlinear parallel flow theory, that the onset of motion occurs above a subcritical Rayleigh number $RCsub$ which depends upon the power-law index n. For finite-amplitude convection, the heat and flow characteristics predicted by the analytical model are found to agree well with a numerical study of the full governing equations.

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## Figures

Figure 8

Effect of n and R on (a) flow intensity |Ψmax| and (b) Nusselt number Nu

Figure 7

Effect of the power-law index n, for dilatant fluids (n > 1), on the velocity profiles u versus y for (a) R = 15 and (b) R = 10

Figure 6

Effect of R and n on (a) flow intensity |Ψmax| and (b) Nusselt number Nu for dilatant fluids (n > 1)

Figure 5

Bifurcation diagram as a function of R for various values of n; (a) flow intensity |Ψmax| and (b) Nusselt number Nu for pseudoplasic fluids (n < 1)

Figure 4

Contour lines of stream function (top) and temperature (bottom) predicted by the numerical solution of the full governing equations for R=50 and (a) n = 0.6, Ψmax=4.29, Nu=4.78; (b) n = 1, Ψmax=2.44, Nu=2.73; (c) n=1.6, Ψmax=1.41, Nu=1.64

Figure 3

Values of the Nusselt number Nusub, maximum stream function |Ψmaxsub|, and constant temperature gradient Csub, at the point of subcritical bifurcation R=RCsub, as a function of the power-law index n

Figure 2

Critical Rayleigh numbers RCsup and RCsub for the onset of motion as a function of the power-law index n

Figure 1

Bifurcation diagram in term of Ψmax as a function of R for (a) a Newtonian fluid (n=1); (b) a dilatant fluid (n=3), and (c) a pseudoplastic fluid (n=1/3)

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