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Stability of pressure-driven flow in a deformable neo-Hookean channel

Published online by Cambridge University Press:  09 July 2010

GAURAV  and
V. SHANKAR
GAURAV
Affiliation:
Department of Chemical Engineering, Indian Institute of Technology, Kanpur 208 016, India
V. SHANKAR*
Affiliation:
Department of Chemical Engineering, Indian Institute of Technology, Kanpur 208 016, India
*
†Email address for correspondence: vshankar@iitk.ac.in
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Abstract

The stability of pressure-driven flow in a rectangular channel with deformable neo-Hookean viscoelastic solid walls is analysed for a wide range of Reynolds numbers (from Re ≪ 1 to Re ≫ 1) by considering both sinuous and varicose modes for the perturbations. Pseudospectral numerical and asymptotic methods are employed to uncover the various unstable modes, and their stability boundaries are determined in terms of the solid elasticity parameter Γ = Vη/(ER) and the Reynolds number Re = RV ρ/η; here V is the maximum velocity of the laminar flow, R is the channel half-width, η and ρ are respectively the viscosity and density of the fluid and E is the shear modulus of the solid layer. We show that for small departures from a rigid solid, wall deformability could have a destabilizing or stabilizing effect on the Tollmien–Schlichting (TS) instability (a sinuous mode) depending on the solid-layer thickness. Upon further increase in solid deformability, the TS mode coalesces with another unstable mode (absent in rigid channels) giving rise to a single unstable mode which extends to very low Reynolds number (<1) for highly deformable walls. There are other types of instabilities that exist only due to wall deformability. In the absence of inertia (Re = 0), there is a short-wave instability of both sinuous and varicose modes arising due to the discontinuity of the first normal stress difference across the fluid–solid interface. For both sinuous and varicose modes, it is shown that inclusion of inertia is important even for Re ≪ 1, wherein a new class of long-wavelength unstable modes are predicted which are absent at Re = 0. These unstable modes are a type of shear waves in an elastic solid which are destabilized by the flow. These long- and short-wave instabilities are absent if a simple linear elastic model is used for the solid. At intermediate and high Re, upstream and downstream travelling waves of both sinuous and varicose modes become unstable. We show that sinuous and varicose modes become critical in different parameter regimes, thereby demonstrating the importance of capturing all the unstable modes. Inclusion of dissipative effects in the neo-Hookean model is generally shown to play a stabilizing role on the instabilities due to both sinuous and varicose modes. The predicted instabilities will be important for the flow of liquids (with viscosity ≥ 10−3 Pa s) in deformable channels of width ≤1 mm, and with shear modulus ≤ 105 Pa.

JFM classification

Waves/Free-surface Flows: Elastic waves Instability: Instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 659 , 25 September 2010 , pp. 318 - 350
Copyright
Copyright © Cambridge University Press 2010

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