Spreading equilibria under mildly singular potentials: pancakes versus droplets. (English) Zbl 1503.35158
Summary: We study global minimizers of a functional modeling the free energy of thin liquid layers over a solid substrate under the combined effect of surface, gravitational, and intermolecular potentials. When the latter ones have a mild repulsive singularity at short ranges, global minimizers are compactly supported and display a microscopic contact angle of \(\pi /2\). Depending on the form of the potential, the macroscopic shape can either be droplet-like or pancake-like, with a transition profile between the two at zero spreading coefficient for purely repulsive potentials. These results generalize, complete, and give mathematical rigor to de Gennes’ formal discussion of spreading equilibria. Uniqueness and non-uniqueness phenomena are also discussed.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 34C60 | Qualitative investigation and simulation of ordinary differential equation models |
| 34E10 | Perturbations, asymptotics of solutions to ordinary differential equations |
| 35J91 | Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian |
| 49J05 | Existence theories for free problems in one independent variable |
| 49J10 | Existence theories for free problems in two or more independent variables |
| 76A20 | Thin fluid films |
| 76D03 | Existence, uniqueness, and regularity theory for incompressible viscous fluids |
| 76D08 | Lubrication theory |
| 35R35 | Free boundary problems for PDEs |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
Keywords:
singular minimization problem; mass constraint; singular potential; attractive-repulsive potential; inter-molecular potential; partial wetting; complete wetting; spreading coefficient; precursor; asymptotic behavior; scaling law; droplet; pancake; macroscopic contact angle; effective contact angle; uniqueness; lubrication theory; thin-film equation; free boundary problems; Alt-Phillips functional; Alt-Caffarelli functionalReferences:
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