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Surfactant- and gravity-dependent instability of two-layer channel flows: linear theory covering all wavelengths. Part 2. Mid-wave regimes

Published online by Cambridge University Press:  23 January 2019

Alexander L. Frenkel ,
David Halpern Open the ORCID record for David Halpern [Opens in a new window]  and
Adam J. Schweiger
Alexander L. Frenkel
Affiliation:
Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487, USA
David Halpern*
Affiliation:
Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487, USA
Adam J. Schweiger
Affiliation:
Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487, USA
*
†Email address for correspondence: dhalpern@ua.edu
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Abstract

The joint effects of an insoluble surfactant and gravity on the linear stability of a two-layer Couette flow in a horizontal channel are investigated. The inertialess instability regimes are studied for arbitrary wavelengths and with no simplifying requirements on the system parameters: the ratio of thicknesses of the two fluid layers; the viscosity ratio; the base shear rate; the Marangoni number $Ma$; and the Bond number $Bo$. As was established in the first part of this investigation (Frenkel, Halpern & Schweiger, J. Fluid Mech., vol. 863, 2019, pp. 150–184), a quadratic dispersion equation for the complex growth rate yields two, largely continuous, branches of the normal modes, which are responsible for the flow stability properties. This is consistent with the surfactant instability case of zero gravity studied in Halpern & Frenkel (J. Fluid Mech., vol. 485, 2003, pp. 191–220). The present paper focuses on the mid-wave regimes of instability, defined as those having a finite interval of unstable wavenumbers bounded away from zero. In particular, the location of the mid-wave instability regions in the ($Ma$, $Bo$)-plane, bounded by their critical curves, depending on the other system parameters, is considered. The changes of the extremal points of these critical curves with the variation of external parameters are investigated, including the bifurcation points at which new extrema emerge. Also, it is found that for the less unstable branch of normal modes, a mid-wave interval of unstable wavenumbers may sometimes coexist with a long-wave one, defined as an interval having a zero-wavenumber endpoint.

JFM classification

Instability: Instability Low-Reynolds-number flows: Low-Reynolds-number flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 863 , 25 March 2019 , pp. 185 - 214
Copyright
© 2019 Cambridge University Press 

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References

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