Journal of Fluid Mechanics

Papers

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

CHERIF NOUARa1 and ALESSANDRO BOTTAROa2 c1

a1 LEMTA, UMR 7563 (CNRS-INPL-UHP), 2 Avenue de la Forêt de Haye, BP 160 54504 Vandoeuvre Lès Nancy, France

a2 DICAT, Facoltà di Ingegneria, Università di Genova, Via Montallegro 1, 16145 Genova, Italy

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.

(Received April 04 2009)

(Revised August 26 2009)

(Accepted August 27 2009)

(Online publication December 08 2009)

Key words:

  • instability;
  • transition to turbulence;
  • viscoelasticity

Correspondence:

c1 Email address for correspondence: cherif.nouar@ensem.inpl-nancy.fr, alessandro.bottaro@unige.it

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