Existence and stability of regularized shock solutions, with applications to rimming flows

Abstract

This paper is concerned with regularization of shock solutions of nonlinear hyperbolic equations, i.e., introduction of a smoothing term with a coefficient ɛ, then taking the limit ɛ → 0. In addition to the classical use of regularization for eliminating physically meaningless solutions which always occur in non-regularized equations (e.g. waves of depression in gas dynamics), we show that it is also helpful for stability analysis. The general approach is illustrated by applying it to rimming flows, i.e., flows of a thin film of viscous liquid on the inside of a horizontal rotating cylinder, with or without surface tension (which plays the role of the regularizing effect). In the latter case, the spectrum of available linear eigenmodes appears to be continuous, but in the former, it is discrete and, most importantly, remains discrete in the limit of infinitesimally weak surface tension. The regularized (discrete) spectrum is fully determined by the point where the velocity of small perturbations vanishes, with the rest of the domain, including the shock region, being unimportant.