Matched asymptotic expansion calculation of the equilibrium shape of a hole in a thin liquid film

O’Brien, S. B. G.

doi:10.1090/qam/1704451

Author: S. B. G. O’Brien
Journal: Quart. Appl. Math. 57 (1999), 453-464
MSC: Primary 76A20; Secondary 34E05, 76B45, 76M45
DOI: https://doi.org/10.1090/qam/1704451
MathSciNet review: MR1704451
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Abstract: We consider the occurrence of small axisymmetric pinholes in an otherwise uniform infinite thin liquid film. Corresponding to any particular undisturbed film thickness there exists precisely one (unstable) equilibrium solution reflecting a balance between surface tension and gravity effects. If a pinhole is smaller than this critical size the pinhole tends to close over and “heal". If a pinhole is larger it tends to open out. So determination of this critical hole size is crucial. We examine this problem in the case of a “small” pinhole where the fundamental length-scale in the film is much smaller than the capillary length. Solutions are obtained using matched asymptotic expansions for which several different scalings are necessary.

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