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.
Highlights
Surfactants are surface-active compounds that play an important role as cleaning, wetting or foaming agents in a range of practical applications and everyday products
When the bulk concentration C is below the critical micelle concentration, CCMC, addition of surfactant in the bulk increases the interface concentration Γ, which in turn reduces the interfacial surface tension γ according to the Gibbs law
The degree of surfactant solubility is represented in the mathematical model by parameter Rb, with Rb 1 being highly soluble in the bulk and Rb 1 nearly insoluble
Summary
Surfactants are surface-active compounds that play an important role as cleaning, wetting or foaming agents in a range of practical applications and everyday products. There have been a number of studies considering the effect of soluble surfactants on the linear stability of falling liquid films (Ji & Setterwall 1994; Karapetsas & Bontozoglou 2013, 2014) or two-layer channel flows (Sun & Fahmy 2006; Zaisha et al 2008; You et al 2014; Picardo et al 2016) In the latter case, the papers were either based on the simplifying assumption of a non-deformable interface or considered surfactant soluble in both phases, with the surface tension linearly dependent on either bulk concentration. 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.
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