Instability of two-layer film flows due to the interacting effects of surfactants, inertia, and gravity

Abstract

We consider a two-fluid shear flow where the interface between the two fluids is coated with an insoluble surfactant. An asymptotic model is derived in the thin-layer approximation, consisting of a set of nonlinear PDEs describing the evolution of the film and surfactant disturbances at the interface. The model includes important physical effects such as Marangoni forces (caused by the presence of surfactant), inertial forces arising in the thick fluid layer, as well as gravitational forces. The aim of this study is to investigate the effect of density stratification or gravity -represented through the Bond number Bo -on the flow stability, and the interplay between the different (de)stabilisation mechanisms. It is found that gravity can either stabilise or destabilise the interface (depending on fluid properties), but not always as intuitively anticipated. Different travelling-wave branches are presented for varying Bo and the destabilising mechanism associated with the Marangoni forces is discussed.

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

“…It accounts for all the salient physical effects, including inertia, density stratification, viscosity contrast, Marangoni stresses, surface and bulk diffusion, adsorption and desorption kinetics, and micellar dis/assembly kinetics. In the limit of vanishing desorption or rapid adsorption rates, the model reduces to that of Kalogirou (2018) for a two-layer flow with an insoluble surfactant.…”

“…The impact of surfactants on two-layer flows has been analysed predominantly for the case of insoluble surfactants. Linear stability studies indicated that the interface can be destabilised by insoluble surfactants even in the Stokes flow limit (Frenkel & Halpern 2002;Halpern & Frenkel 2003;Blyth & Pozrikidis 2004b;Frenkel et al 2019a,b), and eventually becomes saturated to non-uniform states in the nonlinear regime (Pozrikidis 2004a;Blyth & Pozrikidis 2004b;Wei 2005;Frenkel & Halpern 2006;Bassom et al 2010;Samanta 2013;Kalogirou & Papageorgiou 2016;Frenkel & Halpern 2017;Kalogirou 2018).…”

“…The derived model integrates a number of salient physical properties that play an important role in the dynamics of surfactant-laden multilayer flows, such as inertia, gravity, surface tension, Marangoni stresses, diffusion, and mass transfer between the different surfactant forms. This model reduces to that of Kalogirou (2018) (which is the most general model presented so far and includes the effects of inertia and density stratification) for a two-layer flow with an insoluble surfactant in appropriate limits. We perform a linear stability analysis that yields an eigenvalue problem for the wave speed, and obtain analytical expressions for the two dominant growth rates valid in the long-wave approximation.…”

Asymptotic modelling and direct numerical simulations of multilayer pressure-driven flows

“…Here we are interested in the early stages of the instability when the disturbances are small and we will accordingly derive a weakly nonlinear equation for the evolution of the film thickness perturbation. Following the work of Bassom et al [30], Kalogirou et al [28] and Kalogirou [31], we proceed by introducing a perturbation to the interface defined by…”

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

“…x . The evolution of the interface and the interfacial surfactant concentration are described by the following equations (Bassom et al 2010;Kalogirou 2018)…”