Hostname: page-component-7bb8b95d7b-lvwk9 Total loading time: 0 Render date: 2024-09-17T10:27:18.606Z Has data issue: false hasContentIssue false
  • English
  • Français

On inertial effects in the Moffatt–Pukhnachov coating-flow problem

Published online by Cambridge University Press:  25 August 2009

MARK A. KELMANSON
MARK A. KELMANSON*
Affiliation:
Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK
*
†Email address for correspondence: mark@maths.leeds.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

The effects are investigated of including inertial terms, in both small- and large-surface-tension limits, in a remodelling of the influential and fundamental problem first formulated by Moffatt and Pukhnachov in 1977: that of viscous thin-film free-surface Stokes flow exterior to a circular cylinder rotating about its horizontal axis in a vertical gravitational field.

An analysis of the non-dimensionalizations of previous related literature is made and the precise manner in which different rescalings lead to the asymptotic promotion or demotion of pure-inertial flux terms over gravitational-inertial terms is highlighted. An asymptotic mass-conserving evolution equation for a perturbed-film thickness is derived and solved using two-timescale asymptotics with a strained fast timescale. By using an algebraic manipulator to automate the asymptotics to high orders in the small expansion parameter of the ratio of the film thickness to the cylinder radius, consistent a posteriori truncations are obtained.

Via two-timescale and numerical solutions of the evolution equation, new light is shed on diverse effects of inertia in both small- and large-surface-tension limits, in each of which a critical Reynolds number is discovered above which the thin-film evolution equation has no steady-state solution due to the strength of the destabilizing inertial centrifugal force. Extensions of the theory to the treatment of thicker films are discussed.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 633 , 25 August 2009 , pp. 327 - 353
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Acrivos, A. & Jin, B. 2004 Rimming flows within a rotating horizontal cylinder: asymptotic analysis of the thin-film lubrication equations and stability of their solutions. J. Engng Math. 50, 99–120.CrossRefGoogle Scholar
Ashmore, J., Hosoi, A. E. & Stone, H. A. 2003 The effect of surface tension on rimming flows in a partially filled rotating cylinder. J. Fluid Mech. 479, 65–98.CrossRefGoogle Scholar
Benilov, E. S. & O'Brien, S. B. G. 2005 Inertial instability of a liquid film inside a rotating horizontal cylinder. Phys. Fluids 17, 1–16.CrossRefGoogle Scholar
Benjamin, T. B., Pritchard, W. G. & Tavener, S. J. 1993 Steady and unsteady flows of a highly viscous liquid inside a rotating horizontal cylinder. Report No. AM 122. Department of Mathematics, Penn State University.Google Scholar
Brandrup, J., Immergut, E. H., Bloch, D. R. & Grulke, E. A. (Eds.) 2003 The Polymer Handbook, 4th ed., Wiley.Google Scholar
Evans, P. L., Schwarz, L. W. & Roy, R. V. 2004 Steady and unsteady solutions for coating flow on a rotating horizontal cylinder: two-dimensional theoretical and numerical modelling. Phys. Fluids 16, 2742–2756.CrossRefGoogle Scholar
Hansen, E. B. & Kelmanson, M. A. 1994 Steady, viscous, free-surface flow on a rotating cylinder. J. Fluid Mech. 272, 91–107.CrossRefGoogle Scholar
Hinch, E. J. & Kelmanson, M. A. 2003 On the decay and drift of free-surface perturbations in viscous, thin-film flow exterior to a rotating cylinder. Proc. Roy. Soc. Lond. A 459, 1193–1213.CrossRefGoogle Scholar
Hinch, E. J., Kelmanson, M. A. & Metcalfe, P. D. 2004 Shock-like free-surface perturbations in low-surface-tension, viscous, thin-film flow exterior to a rotating cylinder. Proc. Roy. Soc. Lond. A 460, 2975–2991.CrossRefGoogle Scholar
Hocking, L. M. & Davis, S. H. 2002 Inertial effects in time-dependent motion of thin films and drops. J. Fluid Mech. 467, 1–17.CrossRefGoogle Scholar
Hosoi, A. E. & Mahadevan, L. 1999 Axial instability of a free-surface front in a partially filled horizontal rotating cylinder. Phys. Fluids 11, 97–106.CrossRefGoogle Scholar
Karabut, E. A. 2007 Two régimes of liquid film flow on a rotating cylinder. J. Appl. Mech. Tech. Phys. 48 (1), 55–64.CrossRefGoogle Scholar
Kelmanson, M. A. 1995 Theoretical and experimental analyses of the maximum-supportable fluid load on a rotating cylinder. J. Engng Math., 29 (3), 271–285.CrossRefGoogle Scholar
Kelmanson, M. A. 2009 Pseudo-three-timescale approximation of exponentially modulated free-surface waves. J. Fluid Mech. 625, 435–443.CrossRefGoogle Scholar
Kevorkian, J. & Cole, J. D. 1996 Multiple Scale and Singular Perturbation Methods. Springer-Verlag.CrossRefGoogle Scholar
Moffatt, H. K. 1977 Behaviour of a viscous film on the outer surface of a rotating cylinder. J. Méc. 187, 651–673.Google Scholar
Murdock, J. A. 1991 Perturbations: Theory and Methods. SIAM, Wiley.Google Scholar
Noakes, C. J., King, J. R. & Riley, D. S. 2006 On the development of rational approximations incorporating inertial effects in coating and rimming flows: a multiple-scales approach. Q. J. Mech. Appl. Math. 59, 163–190.CrossRefGoogle Scholar
Peterson, R. C., Jimack, P. K. & Kelmanson, M. A. 2001 On the stability of viscous free-surface flow supported by a rotating cylinder. Proc. Roy. Soc. Lond. A 457, 1427–1445.CrossRefGoogle Scholar
Pukhnachev, V. V. 1977 Motion of a liquid film on the surface of a rotating cylinder in a gravitational field. J. Appl. Mech. Tech. Phys. 18, 344–351.CrossRefGoogle Scholar
Pukhnachov, V. V. 2005 a On the equation of a rotating film. Siber. Math. J. 46 (5), 913–924.CrossRefGoogle Scholar
Pukhnachov, V. V. 2005 b Capillary/gravity film flows on the surface of a rotating cylinder. J. Math. Sci. 130 (4), 4871–4883.CrossRefGoogle Scholar
Wilson, S. K., Hunt, R. & Duffy, B. R. 2002 On the critical solutions in coating and rimming flow in a uniformly rotating horizontal cylinder Q. J. Mech. Appl. Math. 55, 357–383.CrossRefGoogle Scholar