Hostname: page-component-848d4c4894-mwx4w Total loading time: 0 Render date: 2024-06-20T04:32:04.542Z Has data issue: false hasContentIssue false
  • English
  • Français

On the stability of shallow rivulets

Published online by Cambridge University Press:  25 September 2009

E. S. BENILOV
E. S. BENILOV*
Affiliation:
Department of Mathematics, University of Limerick, Limerick, Ireland
*
†Email address for correspondence: eugene.benilov@ul.ie
Get access Rights & Permissions [Opens in a new window]

Abstract

We examine the linear stability of a capillary rivulet under the assumption that it is shallow enough to be described by the lubrication approximation. It is shown that rivulets on a sloping plate are stable regardless of their parameters, whereas rivulets on the underside of a plate can be either stable or unstable, depending on their widths and the plate's slope. For the case of a horizontal plate, sufficiently narrow rivulets are shown to be stable and sufficiently wide ones unstable, with the threshold width being π/2(σ/gρ)1/2(ρ and σ are the liquid's density and surface tension, g is the acceleration due to gravity).

It is also shown that, even though the plate's slope induces in a rivulet a sheared flow (which would normally be viewed as a source of instability) – in the present problem, it is a stabilizing factor. The corresponding stability criterion involving the rivulet's width and the plate's slope is computed, and it is demonstrated that, if the latter is sufficiently strong, all rivulets are stable regardless of their widths.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 636 , 10 October 2009 , pp. 455 - 474
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

Benilov, E. S. 2003 Instability of quasigeostrophic vortices in a two-layer ocean with thin upper layer. J. Fluid Mech. 475, 303–331.CrossRefGoogle Scholar
Benilov, E. S. 2004 Stability of vortices in a two-layer ocean with uniform potential vorticity in the lower layer. J. Fluid Mech. 502, 207–232.CrossRefGoogle Scholar
Benilov, E. S., Naulin, V. & Rasmussen, J. J. 2002 Does a sheared flow stabilize inversely stratified fluid? Phys. Fluids 14, 1674–1680.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, 052106.CrossRefGoogle Scholar
Bertozzi, A. L. & Brenner, M. P. 1997 Linear stability and transient growth in driven contact lines. Phys. Fluids 9, 530–539.CrossRefGoogle Scholar
Davis, S. H. 1980 Moving contact lines and rivulet instabilities. Part I. The static rivulet. J. Fluid. Mech. 98, 225–242.CrossRefGoogle Scholar
Extrand, C. V. 2006 Hysteresis in contact angle measurements. In Encyclopedia of surface and colloid science, 2nd edn. (ed. Somasundaran, P. & Hubbard, A. T.), pp. 2414–2429. Taylor & Francis.Google Scholar
Hocking, L. M. 1981 Sliding and spreading of thin two-dimensional drops. Quart. J. Mech. Appl. Math. 34, 37–55.CrossRefGoogle Scholar
Killworth, P. D. 1980 Barotropic and baroclinic instability in rotating fluids. Dyn. Atmos. Oceans 4, 143–184.CrossRefGoogle Scholar
Langbein, D. 1990 The shape and stability of liquid menisci at solid edges. J. Fluid. Mech. 213, 251–265.CrossRefGoogle Scholar
Myers, T. G., Liang, H. X. & Wetton, B. 2004 The stability and flow of a rivulet driven by interfacial shear and gravity. Intl J. Nonlinear Mech. 39, 1239–1249.CrossRefGoogle Scholar
Roy, R. V. & Schwartz, L. W. 1999 On the stability of liquid ridges. J. Fluid Mech. 391, 293–318.CrossRefGoogle Scholar
Schmuki, P. & Laso, M. 1990 On the stability of rivulet flow. J. Fluid Mech. 215, 125–143.CrossRefGoogle Scholar
Simmons, A. J. 1974 The meridional scale of baroclinic waves. J. Atmos. Sci. 31, 1515–1525.2.0.CO;2>CrossRefGoogle Scholar
Sullivan, J. S., Wilson, S. K. & Duffy, B. R. 2008 A thin rivulet of perfectly wetting fluid subject to a longitudinal surface shear stress. Quart. J. Mech. Appl. Math. 61, 25–61.CrossRefGoogle Scholar
Weiland, R. H. & Davis, S. H. 1981 Moving contact lines and rivulet instabilities. Part II. Long waves on flat rivulets. J. Fluid Mech. 107, 261–280.CrossRefGoogle Scholar
Wilson, S. K. & Duffy, B. R. 1998 On the gravity-driven draining of a rivulet of viscous fluid down a slowly varying substrate with variation transverse to the direction of flow. Phys. Fluids 10, 13–22.CrossRefGoogle Scholar
Wilson, S. K. & Duffy, B. R. 2005 When is it energetically favourable for a rivulet of perfectly wetting fluid to split? Phys. Fluids 17, 078104-1–078104-3.CrossRefGoogle Scholar