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Subcritical bifurcation of shear-thinning plane Poiseuille flows

Published online by Cambridge University Press:  29 September 2011

A. Chekila ,
C. Nouar ,
E. Plaut  and
A. Nemdili
A. Chekila
Affiliation:
Laboratoire de Rhéologie, Faculté de Génie Mécanique, Université des Sciences et de la Technologie d’Oran, B.P. 1505 El M’Naouer Oran 31000, Algeria LEMTA, UMR 7563 (CNRS – Nancy Université), 2 Avenue de la Forêt de Haye, BP 160, 54504 Vandoeuvre CEDEX, France
C. Nouar*
Affiliation:
LEMTA, UMR 7563 (CNRS – Nancy Université), 2 Avenue de la Forêt de Haye, BP 160, 54504 Vandoeuvre CEDEX, France
E. Plaut
Affiliation:
LEMTA, UMR 7563 (CNRS – Nancy Université), 2 Avenue de la Forêt de Haye, BP 160, 54504 Vandoeuvre CEDEX, France
A. Nemdili
Affiliation:
Laboratoire de Rhéologie, Faculté de Génie Mécanique, Université des Sciences et de la Technologie d’Oran, B.P. 1505 El M’Naouer Oran 31000, Algeria
*
†Email address for correspondence: [email protected]
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Abstract

In a recent article (Nouar, Bottaro & Brancher, J. Fluid. Mech., vol. 592, 2007, pp. 177–194), a linear stability analysis of plane Poiseuille flow of shear-thinning fluids has been performed. The authors concluded that the viscosity stratification delays the transition and that is important to account for the viscosity perturbation. The current paper focuses on the first-principles understanding of the influence of the viscosity stratification and the nonlinear variation of the effective viscosity with the shear rate on the flow stability with respect to a finite-amplitude perturbation. A weakly nonlinear analysis, using the amplitude expansion method, is adopted as a first approach to study nonlinear effects. The bifurcation to two-dimensional travelling waves is studied. For the numerical computations, the shear-thinning behaviour is described by the Carreau model. The rheological parameters are varied in a wide range. The results indicate that (i) the nonlinearity of the viscous terms tends to reduce the viscous dissipation and to accelerate the flow, (ii) the harmonic generated by the nonlinearity is smaller and in opposite phase to that generated by the quadratic nonlinear inertial terms and (iii) with increasing shear-thinning effects, the bifurcation becomes highly subcritical. Consequently, the magnitude of the threshold amplitude of the perturbation, beyond which the flow is nonlinearly unstable, decreases. This result is confirmed by computing higher order-Landau constants.

JFM classification

Waves/Free-surface Flows: Channel flow Instability: Nonlinear instability Non-Newtonian Flows: Non-Newtonian Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 686 , 10 November 2011 , pp. 272 - 298
Copyright
Copyright © Cambridge University Press 2011

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