Non-Newtonian two-phase thin-film problem: Local existence, uniqueness, and stability

TL;DR

This paper analyzes the mathematical behavior of a two-phase thin film involving a Newtonian and a non-Newtonian fluid, establishing local existence, uniqueness, and stability of solutions in a simplified asymptotic regime.

Contribution

It introduces a novel analysis of a coupled two-parabolic system modeling non-Newtonian two-phase thin films, proving key properties like existence, uniqueness, and stability.

Findings

Proved local existence of strong solutions for positive initial film heights.

Established uniqueness of solutions using energy methods.

Analyzed asymptotic stability of equilibrium states under additional regularity conditions.

Abstract

We study the flow of two immiscible fluids located on a solid bottom, where the lower fluid is Newtonian and the upper fluid is a non-Newtonian Ellis fluid. Neglecting gravitational effects, we consider the formal asymptotic limit of small film heights in the two-phase Navier--Stokes system. This leads to a strongly coupled system of two parabolic equations of fourth order with merely H\"older-continuous dependence on the coefficients. For the case of strictly positive initial film heights we prove local existence of strong solutions by abstract semigroup theory. Uniqueness is proved by energy methods. Under additional regularity assumptions, we investigate asymptotic stability of the unique equilibrium solution, which is given by constant film heights.