Time integration and steady-state continuation method for lubrication equations

Beltrame P, Thiele U

Abstract

Lubrication equations describe many structuring processes of thin liquid films. We develop and apply a numerical framework suitable for their analysis employing a dynamical systems approach. In particular, we present a time integration algorithm based on exponential propagation and an algorithm for steady-state continuation. Both algorithms employ a Cayley transform to overcome numerical problems resulting from scale separation in space and time. An adaptive time-step allows one to study the dynamics close to hetero- or homoclinic connections. The developed framework is employed, on the one hand, to analyze different phases of the dewetting of a liquid film on a horizontal homogeneous substrate. On the other hand, we consider the depinning of drops pinned by a wettability defect. Time-stepping and path-following are used in both cases to analyze steady-state solutions and their bifurcations as well as dynamic processes on short and long time-scales. Both examples are treated for two- and three-dimensional (2d and 3d) physical settings and prove that the developed algorithms are reliable and efficient for 1d and 2d lubrication equations.