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Convection–dissipation instability in the horizontal plane Couette flow of a highly viscous fluid

Published online by Cambridge University Press:  28 September 2010

A. BARLETTA  and
D. A. NIELD
A. BARLETTA*
Affiliation:
Dipartimento di Ingegneria Energetica, Nucleare e del Controllo Ambientale (DIENCA), Università di Bologna, Via dei Colli 16, I–40136 Bologna, Italy
D. A. NIELD
Affiliation:
Department of Engineering Science, University of Auckland, Private Bag 92019, Auckland 1142, New Zealand
*
†Email address for correspondence: antonio.barletta@unibo.it
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Abstract

The linear stability of the plane Couette flow against thermoconvective rolls is studied. The case of a flow without a boundary-imposed temperature gradient is investigated. The non-uniform, possibly unstable, basic temperature distribution is caused by the effect of the internal viscous heating. Asymmetric thermal boundary conditions are considered: the bottom boundary is adiabatic, while the top boundary is isothermal. The focus is on a fluid with a large, mathematically infinite, Prandtl number, although the two-dimensional transverse roll instability is discussed also for a finite Prandtl number. The transition to the instability is described through the governing parameter GePe2, where Ge is the Gebhart number and Pe is the Péclet number. The response of the basic Couette flow to arbitrarily oriented oblique rolls is tested, so that a complete set of disturbance modes is taken into account. It is shown that the Couette flow is more unstable to longitudinal rolls than to any other oblique roll mode.

JFM classification

Instability: Absolute/convective instability Convection: Buoyancy-driven instability Mathematical Foundations: General fluid mechanics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 662 , 10 November 2010 , pp. 475 - 492
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Barletta, A., Celli, M. & Rees, D. A. S. 2009 The onset of convection in a porous layer induced by viscous dissipation: a linear stability analysis. Intl J. Heat Mass Transfer 52, 337–344.CrossRefGoogle Scholar
Drazin, P. G. & Reid, W. H. 2004 Hydrodynamic Stability, 2nd edn. Cambridge University Press.Google Scholar
Finlayson, B. A. & Scriven, L. E. 1966 The method of weighted residuals – a review. Appl. Mech. Rev. 19, 735–748.Google Scholar
Fujimura, K. & Kelly, R. E. 1988 Stability of unstably stratified shear flow between parallel plates. Fluid Dyn. Res. 2, 281.Google Scholar
Hamabata, H. & Takashima, M. 1983 The effect of rotation on convective instability in a horizontal fluid layer with internal heat generation. J. Phys. Soc. Japan 52, 4145–4151.Google 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
Ingersoll, A. P. 1966 Convective instabilities in plane Couette flow. Phys. Fluids 9, 682–689.CrossRefGoogle Scholar
Johns, L. E. & Narayanan, R. 1997 Frictional heating in plane Couette flow. Proc. R. Soc. Lond. A 453, 1653–1670.Google Scholar
Joseph, D. D. 1965 Stability of frictionally-heated flow. Phys. Fluids 8, 2195–2200.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
Nouar, C. & Frigaard, I. 2009 Stability of plane Couette–Poiseuille flow of shear-thinning fluid. Phys. Fluids 21, 064104.Google Scholar
Roberts, P. H. 1967 Convection in horizontal layers with internal heat generation. Theory. J. Fluid Mech. 30, 33–49.Google Scholar
Romanov, V. A. 1973 Stability of plane-parallel Couette flow. Funct. Anal. Appl. 7, 137–146.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
Takashima, M. 1991 The stability of natural convection in an inclined fluid layer with internal heat generation. Part II. J. Phys. Soc. Japan 60, 455–465.CrossRefGoogle Scholar
Turcotte, D. L., Hsui, A. T., Torrance, K. E. & Schubert, G. 1974 Influence of viscous dissipation on Bénard convection. J. Fluid Mech. 64, 369–374.CrossRefGoogle Scholar
Tveitereid, M. & Palm, E. 1976 Convection due to internal heat sources. J. Fluid Mech. 76, 481–499.Google Scholar
Yueh, C. S. & Weng, C. I. 1996 Linear stability analysis of plane Couette flow with viscous heating. Phys. Fluids 8, 1802–1813.CrossRefGoogle Scholar