Steady rimming flows with surface tension

Benilov, E. S.; Benilov, M. S.; Kopteva, Natalia

doi:10.1017/s0022112007009585

Cited by 45 publications

(46 citation statements)

References 10 publications

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“…Finally, it can be shown that the net mass of shock solutions can never exceed a certain threshold value, M * * ≈ 6.93; thus, initial conditions with M > M * * appear to have no steady state to evolve to. We shall not discuss what happens in such cases in this paper, but mention only that they have been examined in [8,10].…”

Section: Flows Without Surface Tensionmentioning

confidence: 99%

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

Benilov

O’Brien

2008

J Eng Math

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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 ε, then taking the limit ε → 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.

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Section: Flows Without Surface Tensionmentioning

confidence: 99%

“…[3,4]) and, in particular, rimming flows [5][6][7][8][9][10]-i.e., flows of a thin film of viscous liquid on the inside of a horizontal rotating cylinder (see Fig. 1).…”

Section: Introductionmentioning

confidence: 99%

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

Benilov

O’Brien

2008

J Eng Math

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“…Some stability properties of its linearization were studied in [2,4,7]. The authors came to the conclusion that almost all modes, except some first ones, become stable even if the surface tension is relatively weak.…”

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confidence: 99%

On the Nature of Ill-Posedness of the Forward-Backward Heat Equation

Chugunova

Karabash

Pyatkov³

2009

Integr. equ. oper. theory

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We study the Cauchy problem with periodic initial data for the forward-backward heat equation defined by a J-self-adjoint linear operator L depending on a small parameter. The problem originates from the lubrication approximation of a viscous fluid film on the inner surface of a rotating cylinder. For a certain range of the parameter we rigorously prove the conjecture, based on numerical evidence, that the complete set of eigenvectors of the operator L does not form a Riesz basis in L 2 (−π, π). Our method can be applied to a wide range of evolution problems given by P T -symmetric operators. Mathematics Subject Classification (2000). Primary 35P10; Secondary 35Q35, 35K15, 76A20. Keywords. Viscous fluid film, forward-backward diffusion, parabolic equation of mixed type, highly non-self-adjoint differential operator, Riesz basis property, completeness.

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“…As a result there is a considerable and steadily growing literature on both coating and rimming flows. Following the work by Pukhnachev (1977) and Moffatt (1977), important contributions have been made by many authors, including Johnson (1988), Preziosi & Joseph (1988), Melo & Douady (1993), Thoroddsen & Mahadevan (1997), Hosoi & Mahadevan (1999), Duffy & Wilson (1999), Peterson et al (2001), Wilson et al (2002b), Ashmore et al (2003), Hinch & Kelmanson (2003), Acrivos & Jin (2004), Evans et al (2004), Benilov & O'Brien (2005), Evans et al (2005), Villegas-Díaz et al (2005), Benilov (2006), Noakes et al (2006), Chen et al (2007), Benilov et al (2008), Hunt (2008), Benilov et al (2009), Kelmanson (2009), Shrager et al (2009), Tougher et al (2009), Chicharro et al (2011), Pougatch & Frigaard (2011), Thiele (2011), Benilov et al (2012), Leslie et al (2012), and Williams et al (2012). The recent review article by Seiden & Thomas (2011) focuses on pattern formation and segregation in rimming flows of dilute suspensions and granular materials, but also provides probably the most complete overview of the literature on coating and rimming flows of pure fluids currently available.…”

Section: Introductionmentioning

confidence: 99%

Three-dimensional coating and rimming flow: a ring of fluid on a rotating horizontal cylinder

2013

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This version is available at https://strathprints.strath.ac.uk/42349/ Strathprints is designed to allow users to access the research output of the University of Strathclyde. Unless otherwise explicitly stated on the manuscript, Copyright © and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. Please check the manuscript for details of any other licences that may have been applied. You may not engage in further distribution of the material for any profitmaking activities or any commercial gain. You may freely distribute both the url (https://strathprints.strath.ac.uk/) and the content of this paper for research or private study, educational, or not-for-profit purposes without prior permission or charge.Any correspondence concerning this service should be sent to the Strathprints administrator: strathprints@strath.ac.ukThe Strathprints institutional repository (https://strathprints.strath.ac.uk) is a digital archive of University of Strathclyde research outputs. It has been developed to disseminate open access research outputs, expose data about those outputs, and enable the management and persistent access to Strathclyde's intellectual output.Under consideration for publication in J. Fluid Mech. The steady three-dimensional flow of a thin, slowly varying ring of Newtonian fluid on either the outside or the inside of a uniformly rotating large horizontal cylinder is investigated. Specifically, we study "full-ring" solutions, corresponding to a ring of continuous, finite and non-zero thickness that extends all the way around the cylinder. In particular, it is found that there is a critical solution corresponding to either a critical load above which no full-ring solution exists (if the rotation speed is prescribed) or a critical rotation speed below which no full-ring solution exists (if the load is prescribed). We describe the behaviour of the critical solution and, in particular, show that the critical flux, the critical load, the critical semi-width and the critical ring profile are all increasing functions of the rotation speed. In the limit of small rotation speed, the critical flux is small and the critical ring is narrow and thin, leading to a small critical load. In the limit of large rotation speed, the critical flux is large and the critical ring is wide on the upper half of the cylinder and thick on the lower half of the cylinder, leading to a large critical load. We also describe the behaviour of the non-critical full-ring solution, and, in particular, show that the semi-width and the ring profile are increasing functions of the load but, in general, non-monotonic functions of the rotation speed. In the limit of large rotation speed, the ring approaches a limiting non-uniform shape, whereas in the limit of small load, the ring is narrow and thin with a uniform parabolic profile. Finally, we show that, while for most values of the rotation speed and the load the azimuthal velocity is in the same direction as the rotation of the cylinder, there ...

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Steady rimming flows with surface tension

Cited by 45 publications

References 10 publications

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

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

On the Nature of Ill-Posedness of the Forward-Backward Heat Equation

Three-dimensional coating and rimming flow: a ring of fluid on a rotating horizontal cylinder

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