Destabilization of a creeping flow by interfacial surfactant: linear theory extended to all wavenumbers

Abstract

Creeping flow of a two-layer system with a monolayer of an insoluble surfactant on the interface is considered. The linear-stability theory of plane Couette–Poiseuille flow is developed in the Stokes approximation. To isolate the Marangoni effect, gravity is excluded. The shear-flow instability due to the interfacial surfactant, uncovered earlier for long waves only (Frenkel & Halpern 2002), is studied with inclusion of all wavelengths, and over the entire parameter space of the Marangoni number $M$, the viscosity ratio $m$, the interfacial velocity shear $s$, and the thickness ratio $n$ (${\ge}\,1$). The complex wave speed of normal modes solves a quadratic equation, and the growth rate function is continuous at all wavenumbers and all parameter values. If $M\,{>}\,0$, $s\,{\ne}\,0$, $m\,{<}\,n^2$, and $n\,{>}\,1$, the small disturbances grow provided they are sufficiently long wave. However, the instability is not long wave in the following sense: the unstable waves are not necessarily much longer than the smaller of the two layer thicknesses. On the other hand, there are parametric regimes for which the instability has a mid-wave character, the flow being stable at both sufficiently large and small wavelengths and unstable in between. The critical (instability-onset) manifold in the parameter space is investigated. Also, it is shown that for certain parametric limits the convergence of the dispersion function is non-uniform with respect to the wavenumber. This is used to explain the parametric discontinuities of the long-wave growth-rate exponents found earlier.