Patterns formed in a thin film with spatially homogeneous and non-homogeneous Derjaguin disjoining pressure

Alshaikhi, Abdulwahed S. and Grinfeld, Michael and Wilson, Stephen K. (2022) Patterns formed in a thin film with spatially homogeneous and non-homogeneous Derjaguin disjoining pressure. European Journal of Applied Mathematics, 33 (5). pp. 894-918. ISSN 0956-7925 (https://doi.org/10.1017/S0956792521000267)

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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 Honisch et al. [Langmuir 31: 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 Honisch et al.and study the dependence of the steady-state solutions on the wettability contrast in that case.

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