Capillarity driven Stokes flow: the one-phase problem as small viscosity limit

Georg Prokert; Bogdan-Vasile Matioc

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arXiv · 2209.13376

Capillarity driven Stokes flow: the one-phase problem as small viscosity limit

Georg Prokert·Bogdan-Vasile Matioc

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Abstract

We consider the quasistationary Stokes flow that describes the motion of a two-dimensional fluid body under the influence of surface tension effects in an unbounded, infinite-bottom geometry. We reformulate the problem as a fully nonlinear parabolic evolution problem for the function that parameterizes the boundary of the fluid with the nonlinearities expressed in terms of singular integrals. We prove well-posedness of the problem in the subcritical Sobolev spaces $H^s(\mathbb{R})$ up to critical regularity, and establish parabolic smoothing properties for the solutions. Moreover, we identify the problem as the singular limit of the two-phase quasistationary Stokes flow when the viscosity of one of the fluids vanishes.

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Georg Prokert, Bogdan-Vasile Matioc. 2022-09-27. Capillarity driven Stokes flow: the one-phase problem as small viscosity limit. https://arxiv.org/abs/2209.13376

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