Surfactant- and gravity-dependent instability of two-layer channel flows: linear theory covering all wavelengths. Part 1. ‘Long-wave’ regimes

Abstract

A linear stability analysis of a two-layer plane Couette flow of two immiscible fluid layers with different densities, viscosities and thicknesses, bounded by two infinite parallel plates moving at a constant relative velocity to each other, with an insoluble surfactant monolayer along the interface and in the presence of gravity is carried out. The normal modes approach is applied to the equations governing flow disturbances in the two layers. These equations, together with boundary conditions at the plates and the interface, yield a linear eigenvalue problem. When inertia is neglected the velocity amplitudes are the linear combinations of certain hyperbolic functions, and a quadratic dispersion equation for the increment, that is the complex growth rate, is obtained, where coefficients depend on the aspect ratio, the viscosity ratio, the basic velocity shear, the Marangoni number $Ma$ that measures the effects of surfactant and the Bond number $Bo$ that measures the influence of gravity. An extensive investigation is carried out that examines the stabilizing or destabilizing influences of these parameters. Since the dispersion equation is quadratic in the growth rate, there are two continuous branches of the normal modes: a robust branch that exists even with no surfactant, and a surfactant branch that, to the contrary, vanishes when $Ma\downarrow 0$. Regimes have been uncovered with crossings of the two dispersion curves, their reconnections at the point of crossing and separations as $Bo$ changes. Due to the availability of the explicit forms for the growth rates, in many instances the numerical results are corroborated with analytical asymptotics.

Linear stability and nonlinear dynamics in a long-wave model of film flows inside a tube in the presence of surfactant

“…Other terminology used in the literature includes the ‘robust mode’ and ‘surfactant mode’ as in, e.g., Frenkel & Halpern (2017), who note that the robust mode, like the interface mode, does not vanish as ; see, e.g., Frenkel et al. (2019 a ) for use of this terminology and discussion of the mode branches in a planar geometry case with base flow.…”

“…The distinction between the two modes can become blurred as parameters vary (particularly in the case of a base flow); nevertheless this terminology will be adopted for the rest of the paper and in most cases reported here is fairly unambiguous. Other terminology used in the literature includes the 'robust mode' and 'surfactant mode' as in, e.g., Frenkel & Halpern (2017), who note that the robust mode, like the interface mode, does not vanish as M → 0; see, e.g., Frenkel et al (2019a) for use of this terminology and discussion of the mode branches in a planar geometry case with base flow. It is important to note that the long-wave model was derived under the assumption of aspect ratio =R 0 /λ 1, and so this linear stability analysis conducted on the model should be a faithful representation of the full system's linear dynamics in the limit k → 0 (in the parameter regimes considered here).…”

“…These phase differences are modified by changes to Bo, a and M, although in the current study, for many of the parameter regimes studied and reported on the modes explored have phase differences somewhat similar to those in figure 4(b,c) for the interfacial and surfactant modes, respectively. There are, however, also interesting cases where the phase differences can be quite different, and the distinction between mode types is somewhat blurred; see Frenkel et al (2019a) for an exploration of these eigenmodes in two-layer Couette flow. In order to compare how surfactant affects these growth rates relative to the clean-interface case, the ratio of maximum growth rate at M = 0 to maximum growth rate for various M is plotted in figure 4(d); the maximum growth rate is taken to be the largest growth rate in either of the two modes.…”

“…Frenkel & Halpern (2017) incorporated the effects of gravity in horizontal Couette flow with surfactants and negligible inertia and, using lubrication theory, showed that in some parametric regimes, arbitrarily strong gravity cannot completely stabilize the flow. Frenkel, Halpern & Schweiger (2019 a , b ) extended the work of Frenkel & Halpern (2017) and Halpern & Frenkel (2003) by both incorporating gravity and considering arbitrary wavenumbers. The two eigenmodes that solve the linear stability problem were studied extensively, and parameter regimes were found where (i) both modes are stable, (ii) exactly one mode is unstable and (iii) for some values of Bond number, both modes are unstable.…”

Nonlinear dynamics of two-layer channel flow with soluble surfactant below or above the critical micelle concentration

“…It was found that the presence of surfactants gives rise to destabilising Marangoni stresses and induces interfacial instability even under conditions supporting a stable clean interface, namely when the fluids’ viscosities are equal or when inertial effects are negligible. The combined effects of gravity and Marangoni forces on flow stability were examined in Frenkel, Halpern & Schweiger (2019 a , b ). More recently, a number of studies considered the surfactant to be soluble in one or both fluids and analysed the impact of surfactant solubility on the linear stability of two-layer channel flows.…”