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The effect of viscous dissipation on the onset of convection in an inclined porous layer

Published online by Cambridge University Press:  18 May 2011

D. A. NIELD ,
A. BARLETTA  and
M. CELLI
D. A. NIELD
Affiliation:
Department of Engineering Science, University of Auckland, Private Bag 92019, Auckland 1142, New Zealand
A. BARLETTA*
Affiliation:
DIENCA, Alma Mater Studiorum – Università di Bologna, Viale Risorgimento 2, Bologna 40136, Italy
M. CELLI
Affiliation:
DIENCA, Alma Mater Studiorum – Università di Bologna, Viale Risorgimento 2, Bologna 40136, Italy
*
†Email address for correspondence: antonio.barletta@unibo.it
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Abstract

The linear stability of a basic forced and free convection flow in an inclined porous channel is analysed by using the Darcy law and the Oberbeck–Boussinesq approximation. The basic velocity and temperature distributions are influenced by the effect of viscous dissipation, as well as by the boundary conditions. The boundary planes are assumed to be impermeable and isothermal, with a temperature of the lower boundary higher than that of the upper boundary. The instability against longitudinal rolls is studied by employing a second-order weighted residual solution and an accurate sixth-order Runge–Kutta solution of the disturbance equations. The instability against transverse rolls is also investigated. It is shown that these disturbances are in every case less unstable than the longitudinal rolls.

JFM classification

Convection: Buoyancy-driven instability Convection: Convection in porous media Low-Reynolds-number flows: Porous media
Type
Papers
Information
Journal of Fluid Mechanics , Volume 679 , 25 July 2011 , pp. 544 - 558
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions. Dover.Google Scholar
Barletta, A., Celli, M. & Nield, D. A. 2010 Unstably stratified Darcy flow with impressed horizontal temperature gradient, viscous dissipation and asymmetric thermal boundary conditions. Intl J. Heat Mass Transfer 53, 1621–1627.CrossRefGoogle Scholar
Barletta, A., Celli, M. & Rees, D. A. S. 2009 a Darcy–Forchheimer flow with viscous dissipation in a horizontal porous layer: onset of convective instabilities. ASME J. Heat Transfer 131, 072602.CrossRefGoogle Scholar
Barletta, A., Celli, M. & Rees, D. A. S. 2009 b The onset of convection in a porous layer induced by viscous dissipation: a linear stability analysis. Intl J. Heat Mass Transfer 52, 337–344.Google Scholar
Barletta, A. & Nield, D. A. 2010 Instabilities of Hadley–Prats flow with viscous heating in a horizontal porous layer. Trans. Porous Med. 84, 241–256.Google Scholar
Caltagirone, J. P. & Bories, S. 1986 Solutions and stability criteria of natural convective flow in an inclined porous layer. J. Fluid Mech. 155, 267–287.Google Scholar
Cheng, K. C. & Wu, R. S. 1976 Viscous dissipation effects on convective instability and heat transfer in plane Poiseuille flow heated from below. Appl. Sci. Res. 32, 327–346.Google Scholar
Eldabe, N. T. M., El-Sabbagh, M. F. & El-Sayed (Hajjaj), M. A. S. 2007 The stability of plane Couette flow of a power-law fluid with viscous heating. Phys. Fluids 19, 094107.CrossRefGoogle Scholar
Finlayson, B. A. 1972 The Method of Weighted Residuals and Variational Principles. Academic Press.Google Scholar
Golubitsky, M. & Stewart, I. 1985 Hopf bifurcation in the presence of symmetry. Arch. Rat. Mech. Anal. 87, 107–165.CrossRefGoogle Scholar
Ho, T. C., Denn, M. M. & Anshus, B. E. 1977 Stability of low Reynolds number flow with viscous heating. Rheol. Acta 16, 61–68.Google Scholar
Johns, L. E. & Narayanan, R. 1997 Frictional heating in plane Couette flow. Proc. R. Soc. Lond. A 453, 1653–1670.CrossRefGoogle Scholar
Joseph, D. D. 1965 Stability of frictionally-heated flow. Phys. Fluids 8, 2195–2200.Google Scholar
Nield, D. A. 2011 A note on convection patterns in an inclined porous layer. Transp. Porous Med. 86, 23–25.Google Scholar
Nield, D. A. & Barletta, A. 2009 The Horton–Rogers–Lapwood problem revisited: the effect of pressure work. Transp. Porous Med. 77, 143–158.Google Scholar
Nield, D. A. & Barletta, A. 2010 Extended Oberbeck–Boussinesq approximation study of convective instabilities in a porous layer with horizontal flow and bottom heating. Intl J. Heat Mass Transfer 53, 577–585.Google Scholar
Nield, D. A. & Bejan, A. 2006 Convection in Porous Media, 3rd edn. Springer.Google Scholar
Nouar, C. & Frigaard, I. 2009 Stability of plane Couette–Poiseuille flow of shear-thinning fluid. Phys. Fluids 21, 064104.Google Scholar
Rees, D. A. S. & Bassom, A. P. 2000 The onset of Darcy–Bénard convection in an inclined layer heated from below. Acta Mechanica 144, 103–118.CrossRefGoogle Scholar
Storesletten, L. & Barletta, A. 2009 Linear instability of mixed convection of cold water in a porous layer induced by viscous dissipation. Intl J. Therm. Sci. 48, 655–664.Google Scholar
Subrahmaniam, N., Johns, L. E. & Narayanan, R. 2002 Stability of frictional heating in plane Couette flow at fixed power input. Proc. R. Soc. Lond. A 458, 2561–2569.CrossRefGoogle Scholar
Sukanek, P. C., Goldstein, C. A. & Laurence, R. L. 1973 The stability of plane Couette flow with viscous heating. J. Fluid Mech. 57, 651–670.CrossRefGoogle Scholar
Weber, J. E. 1975 Thermal convection in a tilted porous layer. Intl J. Heat Mass Transfer 18, 474–475.Google Scholar
White, J. M. & Muller, S. J. 2002 Experimental studies on the stability of Newtonian Taylor–Couette flow in the presence of viscous heating. J. Fluid Mech. 462, 133–159.Google Scholar
Wolfram, S. 2003 The Mathematica Book, 5th edn. Wolfram Media.Google Scholar
Yueh, C. S. & Weng, C. I. 1996 Linear stability analysis of plane Couette flow with viscous heating. Phys. Fluids 8, 1802–1813.Google Scholar