Viscous stress-driven vortex instability in a surfactant layer

Abstract

The presence of a surfactant layer at the interphase boundary significantly complicates the hydrodynamics of both the phase boundary itself and the main bulk phase (subphase). Interest in such systems is dictated by their widespread occurrence in nature and numerous technological applications. Due to the complexity and diversity of the mechanisms governing surface dynamics, their theoretical studies are often limited to a two-dimensional description. However, the presence of an additional degree of freedom in the form of another dimension can qualitatively alter the structure of the surface flow. This paper studies the stability of a two-dimensional insoluble surfactant film at the liquid-gas interface under conditions of volumetric steady-state flow. The film motion develops in two dimensions and can involve surface circulation and the formation of a periodic vortex structure. Experimental data have revealed a threshold nature of vortex formation: vortices appear when the shear stresses generated at the subphase reach a critical value determined by the flow velocity and the rheological properties of the interface and the lower liquid phase. There is a formal analogy between this instability and the Rayleigh–Benard convection: the momentum flow across the interface, like the energy flow through a fluid layer heated from below, experiences a crisis, resulting in the development of vortex motion. Therefore, the surfactant layer is treated as an independent hydrodynamic object, rather than being reduced to boundary conditions for the volumetric equations of motion. A theoretical model of the motion of a layer located above a steady-state volumetric flow is constructed based on the Boussinesq–Scriven equations. The interaction with the liquid phase is taken into account by correlating the equations to the effective mode localized near the surface, which allows us to reduce the three-dimensional problem to a closed system of two-dimensional equations. A linear analysis of equilibrium stability shows that a vortex structure with a given wave number is separated in a threshold manner, with the critical Reynolds number increasing with increase of the surface viscosity. The results of the linear theory are confirmed by a direct numerical solution of the complete nonlinear system of equations. Nonlinear analysis reveals complex dynamics of the flow including oscillations of the vortex structure, the creation and annihilation of vortices, as well as their advective transport by the main flow.