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Abstract
We consider patterns formed in a two-dimensional thin film on a planar substrate with a Derjaguin disjoining pressure and periodic wettability stripes. We rigorously clarify some of the results obtained numerically by Honischet al.[Langmuir31: 10618–10631, 2015] and embed them in the general theory of thin-film equations. For the case of constant wettability, we elucidate the change in the global structure of branches of steady-state solutions as the average film thickness and the surface tension are varied. Specifically we find, by using methods of local bifurcation theory and the continuation software package AUTO, both nucleation and metastable regimes. We discuss admissible forms of spatially non-homogeneous disjoining pressure, arguing for a form that differs from the one used by Honischet al., and study the dependence of the steady-state solutions on the wettability contrast in that case.
Highlights
Thin liquid films on solid substrates occur in many natural situations
Different forms of disjoining pressure are appropriate; these may incorporate long-range van der Waals forces and/or various types of short-range interaction terms such as Born repulsion; inclusion of a particular type of interaction can have significant effects on the wettability of the surface and the evolution of the film film, sometimes leading to dewetting phenomena, i.e. the rupture of the film and the appearance of dry spots. (Here and subsequently by “wettability” of the surface we mean the ability of a solid surface to reduce the surface tension of a liquid on contact with it such that it spreads over the surface and wets it.)
In the sinusoidally striped non-homogeneous substrate case, we offer a justification for using a form of the disjoining pressure that differs from the one used in these two papers
Summary
Thin liquid films on solid substrates occur in many natural situations. For example, they appear in tear films in the eye which protect the cornea [6] or in streams of lava from a volcanic eruption [19]. In a related more recent paper, Ji and Witelski [20] considered a different choice of disjoining pressure and investigated the finite-time rupture solutions in a model of thin film of liquid with evaporative effects They observed different types of finitetime singularities due to the non-conservative terms in the model. Different types of behaviour of the film are observed depending on the form of the disjoining pressure: finite-time singularities, self-similar solutions and coarsening He divides the evolution of dewetting processes into three phases: an initial linear instability that leads to finitetime rupture (short time dynamics), which is followed by the propagation of dewetting rims and instabilities of liquid ridges (intermediate time dynamics), and the eventual formation of quasi-steady droplets (long time dynamics). A detailed plan of the paper is given in the last paragraph of Section 2
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