Authors: E S Benilov M S Benilov S B G O’Brien
Publish Date: 2008/04/05
Volume: 63, Issue: 2-4, Pages: 197-212
Abstract
This paper is concerned with regularization of shock solutions of nonlinear hyperbolic equations ie 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 nonregularized equations eg 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 ie 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
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