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Stability of the flow of a Bingham fluid in a channel: eigenvalue sensitivity, minimal defects and scaling laws of transition

Published online by Cambridge University Press:  08 December 2009

CHERIF NOUAR  and
ALESSANDRO BOTTARO
CHERIF NOUAR
Affiliation:
LEMTA, UMR 7563 (CNRS-INPL-UHP), 2 Avenue de la Forêt de Haye, BP 160 54504 Vandoeuvre Lès Nancy, France
ALESSANDRO BOTTARO*
Affiliation:
DICAT, Facoltà di Ingegneria, Università di Genova, Via Montallegro 1, 16145 Genova, Italy
*
†Email address for correspondence: [email protected], [email protected]
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Abstract

It has been recently shown that the flow of a Bingham fluid in a channel is always linearly stable (Nouar et al., J. Fluid Mech., vol. 577, 2007, p. 211). To identify possible paths of transition we revisit the problem for the case in which the idealized base flow is slightly perturbed. No attempt is made to reproduce or model the perturbations arising in experimental environments – which may be due to the improper alignment of the channel walls or to imperfect inflow conditions – rather a general formulation is given which yields the transfer function (the sensitivity) for each eigenmode of the spectrum to arbitrary defects in the base flow. It is first established that such a function, for the case of the most sensitive eigenmode, displays a very weak selectivity to variations in the spanwise wavenumber of the disturbance mode. This justifies a further look into the class of spanwise homogeneous modes. A variational procedure is set up to identify the base flow defect of minimal norm capable of optimally destabilizing an otherwise stable flow; it is found that very weak defects are indeed capable to excite exponentially amplified streamwise travelling waves. The associated variations in viscosity are situated mostly near the critical layer of the inviscid problem. Neutrally stable conditions are found as function of the Reynolds number and the Bingham number, providing scalings of critical values with the amplitude of the defect consistent with previous experimental and numerical studies. Finally, a structured pseudospectrum analysis is performed; it is argued that such a class of pseudospectra provides information well suited to hydrodynamic stability purposes.

JFM classification

Instability: Instability Instability: Transition to turbulence Non-Newtonian Flows: Viscoelasticity
Type
Papers
Information
Journal of Fluid Mechanics , Volume 642 , 10 January 2010 , pp. 349 - 372
Copyright
Copyright © Cambridge University Press 2009

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References

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