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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 ,
M. CELLI  and
D. A. NIELD
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: [email protected]
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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.

JFM classification

Instability: Absolute/convective instability Convection: Buoyancy-driven instability Low-Reynolds-number flows: Low-Reynolds-number flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 681 , 25 August 2011 , pp. 499 - 514
Copyright
Copyright © Cambridge University Press 2011

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References

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