Asymptotic behavior of regularized shock solutions in coating flows

TL;DR

This paper analyzes the asymptotic behavior of regularized shock solutions in thin liquid film flows within rotating cylinders, revealing persistence, termination, and complex looping phenomena as surface tension approaches zero.

Contribution

It provides a detailed dynamical systems analysis and numerical investigation of shock solutions, uncovering their persistence, termination, and stability properties in the small surface tension limit.

Findings

Shock solutions persist in the small surface tension limit.

Existence curves form loops that grow as surface tension decreases.

Multiple solution branches are stable under small perturbations.

Abstract

We consider a model for thin liquid films in a rotating cylinder in the small surface tension limit. Using dynamical system methods, we show that the continuum of increasing shock solutions persists in the small surface tension limit, whereas the continuum of decreasing shock solutions terminates at the limit. Using delicate numerical computations, we show that the existence curves of regularized shock solutions on the mass-flux diagram exhibit loops. The number of loops increases and their locations move to infinity as the surface tension parameter decreases to zero. If $n$ is the number of loops in the mass-flux diagram with $2n+1$ solution branches, we show that $n+1$ solution branches are stable with respect to small perturbations.