Surfactant- and gravity-dependent instability of two-layer channel flows: linear theory covering all wavelengths. Part 2. Mid-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 ↓ 0. Due to the availability of explicit forms for the growth rates, in many instances the numerical results are corroborated with analytical asymptotics. For the less unstable branch, a mid-wave interval of unstable wavenumbers (Halpern and Frenkel [2003]) sometimes co-exists with a long-wave one. We study the instability landscape, determined by the threshold curve of the long-wave instability and the critical curve of the mid-wave instability in the (Ma, Bo)-plane. The changes of the extremal points of the critical curves with the variation of the other parameters, such as the viscosity ratio, and the extrema bifurcation points are investigated.

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

“…However it has been found by Halpern & Frenkel (2003) that when the surfactant is insoluble, it is possible for instability to exist in a finite interval of wavenumbers bounded below away from the origin (the so-called mid-wave instability), while long and short waves are stable. We find a similar result here for soluble surfactant (note that the mid-wave instability has also been reported for related systems in Picardo et al (2016) and Frenkel et al (2019b)). In figure 10 we take m = 17, n = 4 (i.e.…”

Instabilities at a sheared interface over a liquid laden with soluble surfactant

“…The dominant mode for the insoluble case (B = 0) is unstable over the range 0 ≤ k ≤ k c for k c ≈ 1.4, but is stabilised for any B > 0 at sufficiently small wavenumbers. For weakly soluble surfactant, that is sufficiently small B, mid-wave instability occurs over a window of wavenumbers away from zero (this type of instability has also been found in similar systems, see for example [9,20,22]). For larger values of the Biot number such as B = 0.1, the system is stable over the entire wavenumber range.…”

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

“…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.…”