Nonlinear Sciences – Pattern Formation and Solitons
Scientific paper
2009-03-02
SIAM Journal on Applied Dynamical Systems, Vol.9, No.2, pp. 484-518 (2010)
Nonlinear Sciences
Pattern Formation and Solitons
33 pages, 16 figures
Scientific paper
10.1137/080718619
Lubrication equations allow to describe many structurin processes of thin liquid films. We develop and apply numerical tools 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. In both algorithms a Cayley transform is employed to overcome numerical problems resulting from scale separation in space and time. An adaptive time-step allows to study the dynamics close to hetero- or homoclinic connections. The developed framework is employed on the one hand to analyse 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 analyse 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 physical settings and prove that the developed algorithms are reliable and efficient for 1d and 2d lubrication equations, respectively.
Beltrame Ph.
Thiele Uwe
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