Finite Element Approximation of Surfactant Spreading on a Thin Film

Abstract

We consider a fully practical finite element approximation of the following system of nonlinear degenerate parabolic equations: \begin{alignat}{2} \textstyle{\frac{\partial u}{\partial t}} + \textstyle \frac{1}{2} \,\nabla . (u^2 \,\nabla [\sigma(v)]) - \textstyle \frac{1}{3}\, \nabla .(u^3 \,\nabla w) &= 0, &&%\quad\mbox{in} \;\;\Omega_T, \qquad %\hspace{2cm} \nonumber \\ w = - c \, \Delta u + a \, u^{-3} - \delta \, u^{-\nu}, \nonumber \\ \textstyle{\frac{\partial v}{\partial t}} + \nabla . (u\,v\,\nabla [\sigma(v)]) - \rho \,\Delta v - \textstyle \frac{1}{2}\, \nabla .(u^2\,v \,\nabla w) &= 0. &&%\quad\mbox{in} \;\;\Omega_T. \nonumber %\\ \end{alignat} The above models a surfactant-driven thin film flow in the presence of both attractive, a >0, and repulsive, $\delta >0$ with $\nu >3$, 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 $\rho \geq 0$ 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.