Journal
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
Volume 218, Issue 2, Pages 589-646Publisher
SPRINGER
DOI: 10.1007/s00205-015-0868-8
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Funding
- NSF [DMS-1211806]
- [SFB 611]
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [1211806] Funding Source: National Science Foundation
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We consider the spreading of a thin two-dimensional droplet on a solid substrate. We use a model for viscous fluids where the evolution is governed by Darcy's law. At the contact point where air and liquid meet the solid substrate, a constant, non-zero contact angle (partial wetting) is assumed. We show local and global well-posedness of this free boundary problem in the presence of the moving contact point. Our estimates are uniform in the contact angle assumed by the liquid at the contact point. In the so-called lubrication approximation (long-wave limit) we show that the solutions converge to the solution of a one-dimensional degenerate parabolic fourth order equation which belongs to a family of thin-film equations. The main technical difficulty is to describe the evolution of the non-smooth domain and to identify suitable spaces that capture the transition to the asymptotic model uniformly in the small parameter .
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