Finite element approximation of surfactant spreading on a thin film

We consider a fully practical finite element approximation of the following system of nonlinear degenerate parabolic equations: partial derivativeu/partial derivativet + 1/2 del.(u(2)del[sigma(v)]) - 1/3del.(u(3)delw) = 0, w = -cDeltau + au(-3) - delta u(-v), partial derivativev/partial derivativet + del.(uvdel[sigma(v)]) - rhoDeltav - 1/2 del.(u(2) vdelw) = 0. The above models a surfactant-driven thin film flow in the presence of both attractive, a > 0, and repulsive, delta > 0 with. > 3, van der Waals forces, where u is the height of the film, v is the concentration of the insoluble surfactant monolayer, and sigma(v) := 1 - v is the typical surface tension. Here rho greater than or equal to 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 in one space dimension when rho > 0 and either a = delta = 0 or delta > 0. Furthermore, iterative schemes for solving the resulting nonlinear discrete system are discussed. Finally, some numerical experiments are presented.