Existence and stability of regularized shock solutions, with applications to rimming flows
 E. S. Benilov,
 M. S. Benilov,
 S. B. G. O’Brien
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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 nonregularized 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.
 Ockendon JR, Howison SD, Lacey AA, Movchan AB (2003) Applied partial differential equations. Oxford University Press, Oxford
 Lax PD (1957) Hyperbolic conservation laws II. Comm Pure Appl Math 10: 537–566 CrossRef
 Bertozzi AL, Münch A, Fanton X, Cazabat AM (1998) Contact line stability and “undercompressive shocks” in driven thin film flow. Phys Rev Lett 81: 5169– CrossRef
 Bertozzi AL, Münch A, Shearer M (1999) Undercompressive shocks in thin film flows. Phys D 134: 431– CrossRef
 Benjamin TB, Pritchard WG, Tavener SJ (1993) Steady and unsteady flows of a highly viscous liquid inside a rotating horizontal cylinder (unpublished, can be obtained from the authors of the present paper on request)
 O’Brien SBG, Gath EG (1998) The location of a shock in rimming flow. Phys Fluids 10: 1040– CrossRef
 O’Brien SBG (2002) A mechanism for two dimensional instabilities in rimming flow. Q Appl Math 60: 283–
 Ashmore J, Hosoi AE, Stone HA (2003) The effect of surface tension on rimming flows in a partially filled rotating cylinder. J Fluid Mech 479: 65– CrossRef
 Acrivos A, Jin B (2004) Rimming flows within a rotating horizontal cylinder: asymptotic analysis of the thinfilm lubrication equations and stability of their solutions. J Eng Math 50: 99– CrossRef
 Benilov ES, Benilov MS, Kopteva N (2007) Steady rimming flows with surface tension. J Fluid Mech 597: 91–
 Moffatt HK (1977) Behaviour of a viscous film on the outer surface of a rotating cylinder. J Mec 16: 651–673
 Ladyzhenskaya OA (1957) On the construction of discontinuous solutions of quasilinear hyperbolic equations as limits of solutions of the corresponding parabolic equations when the ‘coefficient of viscosity’ tends towards zero. Proc Mosc Math Soc 6:465–480 (in Russian)
 Oleinik OA (1957) Discontinuous solutions of nonlinear differential equations. Usp Mat Nauk 12: 3–
 Hopf E (1950) The partial differential equation u _{ t } + uu _{ x } = μ u _{ xx }. Comm Pure Appl Math 3: 201– CrossRef
 Cole JD (1951) A quasilinear parabolic equation in aerodynamics. Quart Appl Math 9: 225–
 Wilson SK, Hunt R, Duffy BR (2002) On the critical solutions in coating and rimming flow on a uniformly rotating horizontal cylinder. Q J Mech Appl Math 55: 357– CrossRef
 Benilov ES, O’Brien SBG (2005) Inertial instability of a liquid film inside a rotating horizontal cylinder. Phys Fluids 17: 052106 CrossRef
 Duffy BR, Wilson SK (1999) Thinfilm and curtain flows on the outside of a rotating horizontal cylinder. J Fluid Mech 394: 29– CrossRef
 Press WH, Teulkolsky SA, Vetterling WT, Flannery BP (1992) Numerical recipes. Cambridge University Press, Cambridge
 LAPACK—Linear Algebra PACKage, http://www.netlib.org/lapack
 Title
 Existence and stability of regularized shock solutions, with applications to rimming flows
 Journal

Journal of Engineering Mathematics
Volume 63, Issue 24 , pp 197212
 Cover Date
 20090401
 DOI
 10.1007/s1066500892271
 Print ISSN
 00220833
 Online ISSN
 15732703
 Publisher
 Springer Netherlands
 Additional Links
 Topics
 Keywords

 Liquid films
 Rimming flows
 Shocks
 Stability
 Surface tension
 Industry Sectors
 Authors

 E. S. Benilov ^{(1)}
 M. S. Benilov ^{(2)}
 S. B. G. O’Brien ^{(1)}
 Author Affiliations

 1. Department of Mathematics, University of Limerick, Limerick, Ireland
 2. Department of Physics, University of Madeira, Funchal, Portugal