Convergence of a finite-element approximation of surfactant spreading on a thin film in the presence of van der Waals forces

We consider a fully practical finite-element approximation of the following system of nonlinear degenerate parabolic equations: ∂u/∂t+1/2 ∇·(u2 ∇[σ(v)]) - 1/3 ∇·(u 3 ∇w) = 0, w = -c Δu - δu-ν + a u-3, ∂v/∂t + ∇·(u ∇v σ[(v)]) - ρ Δv - 1/2 ∇·(u2 v ∇w) = 0. The above models a surfactant-driven thin-film flow in the presence of both attractive, a > 0, and repulsive, δ > 0 with ν > 3, van der Waals forces; where u is the height of the film, v is the concentration of the insoluble surfactant monolayer and σ(v) := 1 - v is the typical surface tension. Here ρ ≥ 0 and c > 0 are the inverses of the surface Peclet number and the modified capillary number. 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 ρ > 0; and, moreover, (ii) in two space dimensions if in addition ν ≥ 7. Furthermore, iterative schemes for solving the resulting nonlinear discrete system are discussed. Finally, some numerical experiments are presented.