Finite Element Approximation of Soluble Surfactant Spreading on a Thin Film

We consider a fully practical finite element approximation of the following system of nonlinear degenerate parabolic equations:{align*} ∂ u/∂ t + 12\,∇\,{·}\,(u^2 \,∇ [σ(v)]) - 13\, ∇ \,{·}\,(u^3 \,∇ w) &= 0, \\ w = - c \, Δ u - δ \, u-ν+\,a \, u⁻³\, ,{0.1cm}& \\ ∂ v/∂ t + ∇ \,{·}\, (u \,v \,∇ [σ(v)]) - 12\, ∇ \,{·}\,(u^2 \,v \,∇ w)- ρ_s \,Δ v - K (ψ - v) &= 0, \\ ∂ψ/∂ t + 12\,u\,∇ [σ(v)] \,{·}\, ∇ψ - 13\, u^2\,∇ w \,{·}\, ∇ψ - ρ_b\, u⁻¹ ∇ \,{·}\, (u ∇ψ) + β\, K\, u⁻¹(ψ - v) &= 0. {2cm} {align*} The above equations model a Marangoni driven thin film laden with a soluble surfactant, in which the bulk surfactant concentration has been vertically averaged. The model accounts for the presence of both attractive, a ≥ 0, and repulsive, delta > 0 with nu > 3, van der Waals forces. Here u is the height of the film, v is the concentration of the interfacial surfactant species, psi is the concentration of the surfactant species within the bulk phase, and sigma(v) := 1 - v is the typical surface tension. Moreover, rho s ≥ 0, rho b > 0, and c > 0 are the inverses of the surface Peclet number, the bulk Peclet number, and the modified capillary number, respectively; finally, beta > 0 and K > 0 are parameters that characterize the solubility and the rate of interfacial adsorption. In addition to showing stability bounds for our approximation, we prove convergence and hence existence of a solution to this nonlinear degenerate parabolic system (i) in one space dimension when rho s > 0 and, moreover, (ii) in two space dimensions if, in addition, nu ≥ 7. Furthermore, iterative schemes for solving the resulting nonlinear discrete system are discussed. Finally, some numerical experiments are presented.