Abstract
We study gradient flow formulations of thin-film bilayer flows with triple-junctions between liquid/liquid/air phase. First we highlight the gradient structure in the Stokes free-boundary flow and identify its solutions with the well-known PDE with boundary conditions. Next we propose a similar gradient formulation for the corresponding reduced thin-film model and formally identify solutions with those of a PDE problem. A robust numerical algorithm for the thin-film gradient flow structure is then provided. Using this algorithm we compare the sharp triple-junction model with precursor models. For their stationary solutions a rigorous connection is established using \(\Gamma \)-convergence. For time-dependent solutions the comparison of numerical solutions shows a good agreement for small and moderate times. Finally we study spreading in the zero-contact angle case, where we compare numerical solutions with asymptotically exact source-type solutions.
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Acknowledgments
SJ and DP thank the German Research Foundation DFG for financial support through the project Structure formation in thin liquid–liquid films in the SPP 1506 and DFG Research Center MATHEON through the project C10. The work of GK was supported by the postdoctoral scholarship at the Max-Planck-Institute for Mathematics in the Sciences in Leipzig. We also thank Maciek Korzec (Technical University Berlin) for useful discussions.
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Huth, R., Jachalski, S., Kitavtsev, G. et al. Gradient flow perspective on thin-film bilayer flows. J Eng Math 94, 43–61 (2015). https://doi.org/10.1007/s10665-014-9698-1
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DOI: https://doi.org/10.1007/s10665-014-9698-1