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On the onset of dissipation thermal instability for the Poiseuille flow of a highly viscous fluid in a horizontal channel

Published online by Cambridge University Press:  20 June 2011

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
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

Abstract

The thermal instability of the plane Poiseuille flow as a consequence of the effect of viscous dissipation is investigated. No external temperature difference is assumed in the environment; the lower boundary is considered adiabatic, while the upper boundary is isothermal. Thus, the sole cause of the unstable thermal stratification is the flow rate, through the volumetric heating induced by the viscous dissipation. A linear stability analysis is carried out numerically by the analysis of normal modes perturbing the basic flow with different inclinations. The study of cases with different Prandtl numbers and Gebhart numbers suggests that the most unstable perturbations are the longitudinal rolls, namely the normal modes with a wave vector perpendicular to the basic flow direction. A possible comparison with the hydrodynamic instability of the plane Poiseuille flow, described by the Orr–Sommerfeld eigenvalue problem is proposed. This comparison, when referred to high values of the Prandtl number, reveals that the dissipation instability may be effective at a Reynolds number smaller than that needed for the onset of the hydrodynamic instability.

Type
Papers
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Barletta, A. 2008 Comments on a paradox of viscous dissipation and its relation to the Oberbeck–Boussinesq approach. Intl J. Heat Mass Transfer 51, 63126316.CrossRefGoogle Scholar
Barletta, A. 2009 Local energy balance, specific heats and the Oberbeck–Boussinesq approximation. Intl J. Heat Mass Transfer 52, 52665270.CrossRefGoogle Scholar
Barletta, A. & Nield, D. A. 2010 Convection–dissipation instability in the horizontal plane Couette flow of a highly viscous fluid. J. Fluid Mech. 662, 475492.CrossRefGoogle Scholar
Bejan, A. 2004 Convection Heat Transfer, 3rd edn. John Wiley & Sons.Google Scholar
Carrière, P., Monkewitz, P. A. & Martinand, D. 2004 Envelope equations for the Rayleigh–Bénard–Poiseuille system. Part 1. Spatially homogeneous case. J. Fluid Mech. 502, 153174.CrossRefGoogle Scholar
Drazin, P. G. & Reid, W. H. 2004 Hydrodynamic Stability, 2nd edn. Cambridge University Press.CrossRefGoogle Scholar
Epperson, J. F. 2007 An Introduction to Numerical Methods and Analysis. Wiley.Google Scholar
Finlayson, B. A. & Scriven, L. E. 1966 The method of weighted residuals – A review. Appl. Mech. Rev. 19, 735748.Google Scholar
Grandjean, E. & Monkewitz, P. A. 2009 Experimental investigation into localized instabilities of mixed Rayleigh–Poiseuille convection. J. Fluid Mech. 640, 401419.CrossRefGoogle Scholar
Hu, J., Ben Hadid, H. & Henry, D. 2007 Linear stability analysis of Poiseuille–Rayleigh–Bénard flow in binary fluids with Soret effect. Phys. Fluids 19, 034101.CrossRefGoogle Scholar
Landau, L. D. & Lifschitz, E. M. 1987 Fluid Mechanics, 2nd edn., section 50, pp. 196 and 197, Pergamon.Google Scholar
Martinand, D., Carrière, P. & Monkewitz, P. A. 2004 Envelope equations for the Rayleigh–Bénard–Poiseuille system. Part 2. Linear global modes in the case of two-dimensional non-uniform heating. J. Fluid Mech. 502, 175197.CrossRefGoogle Scholar
Martinand, D., Carrière, P. & Monkewitz, P. A. 2006 Three-dimensional global instability modes associated with a localized hot spot in Rayleigh–Bénard–Poiseuille convection. J. Fluid Mech. 551, 275301.CrossRefGoogle Scholar
Nicolas, X. 2002 Bibliographic review on the Poiseuille–Rayleigh–Bénard flows; the mixed convection flows in horizontal rectangular ducts heated from below. Intl J. Therm. Sci. 41, 9611016.CrossRefGoogle Scholar
Nourollahi, M., Farhadi, M. & Sedighi, K. 2010 Numerical study of mixed convection and entropy generation in the Poiseuille–Bénard channel in different angles. Therm. Sci. 14, 329340.CrossRefGoogle Scholar
Orszag, S. A. 1971 Accurate solution of the Orr–Sommerfeld stability equation. J. Fluid Mech. 50, 689703.CrossRefGoogle Scholar
Pabiou, H., Mergui, S. & Bénard, C. 2005 Wavy secondary instability of longitudinal rolls in Rayleigh–Bénard–Poiseuille flows. J. Fluid Mech. 542, 175194.CrossRefGoogle Scholar
Squire, H. B. 1933 On the stability for three–dimensional disturbances of viscous fluid flow between parallel walls. Proc. R. Soc. Lond. A 142, 621628.Google Scholar
Turcotte, D. L., Hsui, A. T., Torrance, K. E. & Schubert, G. 1974 Influence of viscous dissipation on Bénard convection. J. Fluid Mech. 64, 369374.CrossRefGoogle Scholar
Xin, S. H., Nicolas, X. & Le Quéré, P. 2006 Stability analyses of longitudinal rolls of Poiseuille–Rayleigh–Bénard flows in air-filled channels of finite transversal extension. Numer. Heat Transfer A 50, 467490.CrossRefGoogle Scholar