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Oscillatory Kelvin–Helmholtz instability. Part 1. A viscous theory

Published online by Cambridge University Press:  23 March 2011

HARUNORI N. YOSHIKAWA  and
JOSÉ EDUARDO WESFREID
HARUNORI N. YOSHIKAWA*
Affiliation:
Physique et Mécanique des Milieux Hétérogènes (PMMH) UMR 7636 CNRS – ESPCI – UPMC Université de Paris 06 – UPD Université de Paris 07 10, rue Vauquelin, 75231 Paris CEDEX 5, France
JOSÉ EDUARDO WESFREID
Affiliation:
Physique et Mécanique des Milieux Hétérogènes (PMMH) UMR 7636 CNRS – ESPCI – UPMC Université de Paris 06 – UPD Université de Paris 07 10, rue Vauquelin, 75231 Paris CEDEX 5, France
*
†Present address: Laboratoire J.-A. Dieudonné, Université de Nice Sophia-Antipolis, Parc Valrose - 06108 Nice CEDEX 2, France. Email address for correspondence: [email protected]
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Abstract

The stability of oscillatory two-layer flows is investigated with a linear perturbation analysis. An asymptotic case is considered where the oscillation amplitude is small when compared to the perturbation wavelength. The focus of the analysis is on the influence of viscosity and its contrast at the interface. The flows are unstable when the relative velocity of the layers is larger than a critical value. Depending on the oscillation frequency, the flows are in different dynamical regimes, which are characterized by the relative importance of the capillary wavelength and the thicknesses of the Stokes boundary layers developed on the interface. A particular regime is found in which instability occurs at a substantially lower critical velocity. The mechanism behind the instability is studied by identifying the velocity- and shear-induced components in the disturbance growth rate. They interchange dominance depending on the frequency and the viscosity contrast. Results of the analysis are compared with the experiments in the literature. Good agreement is found with the experiments that have a small oscillation amplitude. The validity condition of the asymptotic theory is estimated.

JFM classification

Boundary Layers: Boundary layer stability Multiphase and Particle-laden Flows: Multiphase flow Nonlinear Dynamical Systems: Pattern formation
Type
Papers
Information
Journal of Fluid Mechanics , Volume 675 , 25 May 2011 , pp. 223 - 248
Copyright
Copyright © Cambridge University Press 2011

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