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Worst-case amplification of disturbances in inertialess Couette flow of viscoelastic fluids

Published online by Cambridge University Press:  16 April 2013

Binh K. Lieu ,
Mihailo R. Jovanović  and
Satish Kumar
Binh K. Lieu
Affiliation:
Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455, USA
Mihailo R. Jovanović*
Affiliation:
Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455, USA
Satish Kumar
Affiliation:
Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455, USA
*
†Email address for correspondence: [email protected]
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Abstract

Amplification of deterministic disturbances in inertialess shear-driven channel flows of viscoelastic fluids is examined by analysing the frequency responses from spatio-temporal body forces to the velocity and polymer stress fluctuations. In strongly elastic flows, we show that disturbances with large streamwise length scales may be significantly amplified even in the absence of inertia. For fluctuations without streamwise variations, we derive explicit analytical expressions for the dependence of the worst-case amplification (from different forcing to different velocity and polymer stress components) on the Weissenberg number ($\mathit{We}$), the maximum extensibility of the polymer chains ($L$), the viscosity ratio and the spanwise wavenumber. For the Oldroyd-B model, the amplification of the most energetic components of velocity and polymer stress fields scales as ${\mathit{We}}^{2} $ and ${\mathit{We}}^{4} $. On the other hand, the finite extensibility of polymer molecules limits the largest achievable amplification even in flows with infinitely large Weissenberg numbers: in the presence of wall-normal and spanwise forces, the amplification of the streamwise velocity and polymer stress fluctuations is bounded by quadratic and quartic functions of $L$. This high amplification signals low robustness to modelling imperfections of inertialess channel flows of viscoelastic fluids. The underlying physical mechanism involves interactions of polymer stress fluctuations with a base shear, and it represents a close analogue of the lift-up mechanism that initiates a bypass transition in inertial flows of Newtonian fluids.

JFM classification

Non-Newtonian Flows: Non-Newtonian Flows Instability: Transition to turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 723 , 25 May 2013 , pp. 232 - 263
Copyright
©2013 Cambridge University Press 

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