Numerical Solution of the Free Boundary Foam Drainage Equation

The numerical solution of the free boundary evolution equation for foams is investigated. The domain with free boundary is mapped to a fixed domain via a change of independent variable; the resulting singular nonlinear partial differential equation is transformed into a nonlinear partial differential equation by a change of dependent variable. After using domain truncation, convergence of the variable step size backward Euler discretization is discussed. Numerical algorithms based on the backward difference and implicit Crank-Nicolson are proposed for solving the field equations and comparative results are presented. Furthermore, it is shown that the reaction-diffusion problem with free drainage represented by a degenerate parabolic partial differential equation is covered by analytic semi-groups set in an abstract Banach space.