On computing the instability index of a non-selfadjoint differential operator associated with coating and rimming flows

TL;DR

This paper develops a method to compute the instability index of certain non-selfadjoint differential operators related to coating and rimming flows, reducing the problem to a finite-dimensional analysis using Lyapunov's method.

Contribution

It introduces a novel approach that reduces the instability index calculation to a finite-dimensional space via Lyapunov's method and elliptic PDE estimates.

Findings

Reduction of instability index computation to finite-dimensional space

Use of Lyapunov's method with quadratic forms

Numerical examples demonstrating the method's effectiveness

Abstract

We study the problem of finding the instability index of certain non-selfadjoint fourth order differential operators that appear as linearizations of coating and rimming flows, where a thin layer of fluid coats a horizontal rotating cylinder. The main result reduces the computation of the instability index to a finite-dimensional space of trigonometric polynomials. The proof uses Lyapunov's method to associate the differential operator with a quadratic form, whose maximal positive subspace has dimension equal to the instability index. The quadratic form is given by a solution of Lyapunov's equation, which here takes the form of a fourth order linear PDE in two variables. Elliptic estimates for the solution of this PDE play a key role. We include some numerical examples.