title: Gravity-driven instability of a thin liquid film underneath a soft solid.
abstract: The gravity-driven instability of a thin liquid film located underneath a soft solid material is considered. The equations and boundary conditions governing the solid deformation are systematically converted from a Lagrangian representation to an Eulerian representation, which is the natural framework for describing the liquid motion. This systematic conversion reveals that the continuity-of-velocity boundary condition at the liquid-solid interface is more complicated than has previously been assumed, even in the small-strain limit. We then make clear the conditions under which the commonly used simplified version of this boundary condition is valid. The small-strain approximation, lubrication theory, and linear stability analysis are applied to derive an expression for the growth rate of small-amplitude perturbations. Asymptotic analysis reveals that the coupling between the liquid and solid manifests itself as a lower effective liquid-air interfacial tension that leads to larger instability growth rates. Although this suggests that it is more difficult to maintain a stable liquid coating underneath a soft solid, the effect is expected to be weak for cases of practical interest.
authors.affiliation: Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455, USA.
authors.forename: S H
authors.initials: SH
authors.lastname: Lee
authors.affiliation: School of Mathematical Sciences, Rochester Institute of Technology, Rochester, New York 14623, USA.
authors.forename: K L
authors.initials: KL
authors.lastname: Maki
authors.affiliation: Department of Mathematics, Statistics, and Computer Science, Macalester College, St. Paul, Minnesota 55105, USA.
authors.forename: D
authors.initials: D
authors.lastname: Flath
authors.affiliation: Department of Chemical Engineering, Rochester Institute of Technology, Rochester, New York 14623, USA.
authors.forename: S J
authors.initials: SJ
authors.lastname: Weinstein
authors.affiliation: Department of Mathematics, Beloit College, Beloit, Wisconsin 53511, USA.
authors.forename: C
authors.initials: C
authors.lastname: Kealey
authors.affiliation: Department of Mathematics, Statistics, and Computer Science, Macalester College, St. Paul, Minnesota 55105, USA.
authors.forename: W
authors.initials: W
authors.lastname: Li
authors.affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269, USA.
authors.forename: C
authors.initials: C
authors.lastname: Talbot
authors.affiliation: Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455, USA.
authors.forename: S
authors.initials: S
authors.lastname: Kumar