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The role of soluble surfactants in the linear stability of two-layer flow in a channel

Published online by Cambridge University Press:  18 June 2019

A. Kalogirou Open the ORCID record for A. Kalogirou [Opens in a new window]  and
M. G. Blyth Open the ORCID record for M. G. Blyth [Opens in a new window]
A. Kalogirou*
Affiliation:
School of Mathematics, University of East Anglia, Norwich Research Park, Norwich NR4 7TJ, UK School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK
M. G. Blyth
Affiliation:
School of Mathematics, University of East Anglia, Norwich Research Park, Norwich NR4 7TJ, UK
*
†Email address for correspondence: Anna.Kalogirou@nottingham.ac.uk
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Abstract

The linear stability of Couette–Poiseuille flow of two superposed fluid layers in a horizontal channel is considered. The lower fluid layer is populated with surfactants that appear either in the form of monomers or micelles and can also get adsorbed at the interface between the fluids. A mathematical model is formulated which combines the Navier–Stokes equations in each fluid layer, convection–diffusion equations for the concentration of monomers (at the interface and in the bulk fluid) and micelles (in the bulk), together with appropriate coupling conditions at the interface. The primary aim of this study is to investigate when the system is unstable to arbitrary wavelength perturbations, and in particular, to determine the influence of surfactant solubility and/or sorption kinetics on the instability. A linear stability analysis is performed and the growth rates are obtained by solving an eigenvalue problem for Stokes flow, both numerically for disturbances of arbitrary wavelength and analytically using long-wave approximations. It is found that the system is stable when the surfactant is sufficiently soluble in the bulk and if the fluid viscosity ratio $m$ and thickness ratio $n$ satisfy the condition $m<n^{2}$. On the other hand, the effect of surfactant solubility is found to be destabilising if $m\geqslant n^{2}$. Both of the aforementioned results are manifested for low bulk concentrations below the critical micelle concentration; however, when the equilibrium bulk concentration is sufficiently high (and above the critical micelle concentration) so that micelles are formed in the bulk fluid, the system is stable if $m<n^{2}$ in all cases examined.

JFM classification

Waves/Free-surface Flows: Channel flow Multiphase and Particle-laden Flows: Particle/fluid flow

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JFM Papers
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Journal of Fluid Mechanics , Volume 873 , 25 August 2019 , pp. 18 - 48
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© 2019 Cambridge University Press 

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