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: ∂u/∂t + 1/2 ∇ · (u2 ∇[σ(v)]) - 1/3 ∇ · (u3 ∇w) = 0, w = -c Δu - δ u-v + a u-3, ∂v/∂t + ∇ · (u v ∇[u v ∇[σ(v)]) - 1/2 ∇ · (u2 v ∇w) - ρs Δv - K (ψ - v) = 0, ∂ψ/∂t + 1/2 u ∇[σ(v)] · ∇ψ - 1/3 u2 ∇w · ∇ψ - ρb u-1∇ · (u∇ψ) + β K u-1 (ψ - v) = 0. 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, δ > 0 with v > 3, van der Waals forces. Here u is the height of the film, v is the concentration of the interfacial surfactant species, ψ is the concentration of the surfactant species within the bulk phase, and σ(v):= 1 - v is the typical surface tension. Moreover, ρs ≥ 0, ρb > 0, and c > 0 are the inverses of the surface Peclet number, the bulk Peclet number, and the modified capillary number, respectively; finally, β > 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 ρs > 0 and, moreover, (ii) in two space dimensions if, in addition, v ≥ 7. Furthermore, iterative schemes for solving the resulting nonlinear discrete system are discussed. Finally, some numerical experiments are presented. © 2006 Society for Industrial and Applied Mathematics.