Time Integration and Steady-State Continuation for 2d Lubrication Equations

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 timestep 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 timescales. 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.

Self-organized dip-coating patterns of simple, partially wetting, nonvolatile liquids

“…It is known that on horizontal substrates in the absence of lateral driving terms, fronts of spinodal dewetting are often transversally unstable [71][72][73]. Our numerical time simulations suggest that this is not the case in the considered dip-coating geometry as the dragging of the plate stabilises the transversal instability modes.…”

Numerical Bifurcation Methods and their Application to Fluid Dynamics: Analysis beyond Simulation

“…Here, most publications focus on studies of thin film dynamics by time integration, e.g., for heated films [11,52,84], and the dewetting of partially wetting films [10,85,110]. To our knowledge only [6] have developed suitable continuation tools, presenting a common framework to perform time-stepping and continuation tasks for thin film equations containing a bi-Laplacian. The time stepping is based on an exponential propagation scheme and requires the computation of the exponential of the Jacobian matrix.…”

“…The package developed by [6] has up to now only been applied to three-dimensional liquid ridges and drops on an inclined heterogeneous substrate [4,5]. Typical examples of a bifurcation diagram and corresponding steady profiles of stable and unstable drops are given in Fig.…”

Kinetic roughening of a soft dewetting line under quenched disorder: A numerical study

“…The continuous emergence of newer and newer schemes in the topic suggests that the demand for a convenient approach for modeling thin liquid layers' dynamics is still unsatisfied [16][17][18][19][20][21]. Based on the revolutionary paper of J. F. Joanny and P. G. de Gennes on the perturbed contact line shape [32], a series of depinning type models were constructed that aimed to describe interface dynamics in presence of disorder [33][34][35].…”