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

Abstract

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