Numerical studies of the fingering phenomena for the thin film equation

We present a new interpretation of the fingering phenomena of the thin liquid film layer through numerical investigations. The governing partial differential equation is h_t + (h²-h³)_x = -∇·(h³∇Δh), which arises in the context of thin liquid films driven by a thermal gradient with a counteracting gravitational force, where h = h(x, y, t) is the liquid film height. A robust and accurate finite difference method is developed for the thin liquid film equation. For the advection part (h²-h³)_x, we use an implicit essentially non-oscillatory (ENO)-type scheme and get a good stability property. For the diffusion part -∇·(h³∇Δh), we use an implicit Euler's method. The resulting nonlinear discrete system is solved by an efficient nonlinear multigrid method. Numerical experiments indicate that higher the film thickness, the faster the film front evolves. The concave front has higher film thickness than the convex front. Therefore, the concave front has higher speed than the convex front and this leads to the fingering phenomena.