On the dynamics of thin layers of viscous flows inside another viscous fluid

TL;DR

This paper models the dynamics of a thin viscous fluid layer within another viscous fluid using a free boundary problem derived from asymptotic analysis, ensuring well-posedness and physical consistency.

Contribution

It introduces the Geometric Free Boundary Problem for viscous layers and proves its well-posedness and the validity of the thickness equation.

Findings

The model accurately describes the fluid layer dynamics.

Solutions remain well-defined without layer breaking.

The approach is validated through mathematical proofs.

Abstract

In this work we will study the dynamics of a thin layer of a viscous fluid which is embedded in the interior of another viscous fluid. The resulting flow can be approximated by means of the solutions of a free boundary problem for the Stokes equation in which one of the unknowns is the shape of a curve which approximates the geometry of the thin layer of fluid. We also derive the equation yielding the thickness of this fluid. This model, that will be termed as the Geometric Free Boundary Problem, will be derived using matched asymptotic expansions. We will prove that the Geometric Free Boundary Problem is well posed and the solutions of the thickness equation are well defined (in particular they do not yield breaking of fluid layers) as long as the solutions of the Geometric Free Boundary Problem exist.