Flow of surfactant-laden thin films down an inclined plane

Abstract

A theory is formulated to describe the dynamics of a thin film flowing down an inclined plane laden with insoluble surfactant, present in dilute concentrations. Use of lubrication theory yields a coupled pair of partial differential equations for the film height and surfactant monolayer concentration. The contact line singularity is relieved by assuming the presence of a thin precursor layer ahead of the advancing film. Base flow solutions for a flow of constant flux are examined over various inclination angle, precursor-layer thickness, Peclet number, and capillary parameter ranges. Application of a transient growth analysis highlights the presence of an instability and the vulnerability of the flow to transverse disturbances of intermediate wavenumber. Our results reveal that several key features of the much-studied uncontaminated film flow, including stability, are modified qualitatively by the inclusion of surfactant.

The Motion of a Thin Liquid Film Driven by Surfactant and Gravity

“…In the presence of surfactant, however, the Marangoni force is not constant; the density and motion of the surfactant molecules are modeled by an additional equation. The full model, derived using lubrication theory in [3], consists of a system of two nonlinear coupled PDEs for the film height and the surfactant concentration.The PDE system exhibits a complicated combination of wave-like structures in the solution of initial value and boundary value problems [6,7]. The complexity comes…”

Growing surfactant waves in thin liquid films driven by gravity

“…In a series of recent numerical studies, Edmonstone, Matar, and Craster [6,7] considered one-dimensional flow with surfactant down an inclined plane, and explored the stability to transverse perturbations. An interesting observation arising from their study is that when surfactant is supplied from an upstream reservoir, it accumulates at the leading edge of the flow, resulting in a steady increase of the maximum surfactant concentration (see Figure 2).…”

“…Let h = h(x, t) ≥ 0 be the nondimensional height of the free surface of the fluid film, and let Γ = Γ(x, t) ≥ 0 denote the scaled concentration of insoluble surfactant on the free surface, where x is the position measured down the inclined plane. The system of governing equations is [6,7,16,17,28] …”

“…dI/dt = 0; this is sometimes called the constant volume (of surfactant) case [6,7]. For constant Γ L > 0 the mass of surfactant grows linearly in time, i.e.,…”

Simple waves and shocks in a thin film of a perfectly soluble anti-surfactant solution

“…The resulting model can describe both surfactants, molecules of which accumulate preferentially at the free surface, so that Γ * > 0, and anti-surfactants, molecules of which preferentially accumulate in the bulk of the fluid, so that Γ * < 0. In particular, if the surface concentration of solute is much greater than the bulk concentration, then the classical models for surfactants [14][15][16][17][18][19][20][21][22] are recovered.…”