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Static rivulet instabilities: varicose and sinuous modes

Published online by Cambridge University Press:  05 January 2018

J. B. Bostwick Open the ORCID record for J. B. Bostwick [Opens in a new window]  and
P. H. Steen Open the ORCID record for P. H. Steen [Opens in a new window]
J. B. Bostwick*
Affiliation:
Department of Mechanical Engineering, Clemson University, Clemson, SC 29634, USA
P. H. Steen
Affiliation:
School of Chemical and Biomolecular Engineering and Center for Applied Mathematics, Cornell University, Ithaca, NY 14853, USA
*
†Email address for correspondence: jbostwi@clemson.edu
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Abstract

A static rivulet is subject to disturbances in shape, velocity and pressure fields. Disturbances to interfacial shape accommodate a contact line that is either (i) fixed (pinned) or (ii) fully mobile (free) and preserves the static contact angle. The governing hydrodynamic equations for this inviscid, incompressible fluid are derived and then reduced to a functional eigenvalue problem on linear operators, which are parametrized by axial wavenumber and base-state volume. Solutions are decomposed according to their symmetry (varicose) or anti-symmetry (sinuous) about the vertical mid-plane. Dispersion relations are then computed. Static stability is obtained by setting growth rate to zero and recovers existing literature results. Critical growth rates and wavenumbers for the varicose and sinuous modes are reported. For the varicose mode, typical capillary break-up persists and the role of the liquid/solid interaction on the critical disturbance is illustrated. There exists a range of parameters for which the sinuous mode is the dominant instability mode. The sinuous instability mechanism is shown to correlate with horizontal centre-of-mass motion and illustrated using a toy model.

JFM classification

Interfacial Flows (free surface): Capillary flows Interfacial Flows (free surface): Contact lines Interfacial Flows (free surface): Liquid bridges
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 837 , 25 February 2018 , pp. 819 - 838
Copyright
© 2018 Cambridge University Press 

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References

Aktershev, S. P. & Alekseenko, S. V. 2015 Wave flow of a vertical rivulet. Tech. Phys. Lett. 41 (1), 25–28.Google Scholar
Alekseenko, S. V., Aktershev, S. P., Bobylev, A. V., Kharlamov, S. M. & Markovich, D. M. 2015 Nonlinear forced waves in a vertical rivulet flow. J. Fluid Mech. 770, 350–373.Google Scholar
Alekseenko, S. V., Markovich, D. M. & Shtork, S. I. 1996 Wave flow of rivulets on the outer surface of an inclined cylinder. Phys. Fluids 8 (12), 3288–3299.Google Scholar
Arfken, G. B. & Weber, H. J. 2001 Mathematical Methods for Physicists. Harcourt Academic Press.Google Scholar
Benilov, E. S. 2009 On the stability of shallow rivulets. J. Fluid Mech. 636, 455–474.Google Scholar
Birnir, B., Mertens, K., Putkaradze, V. & Vorobieff, P. 2008a Meandering fluid streams in the presence of flow-rate fluctuations. Phys. Rev. Lett. 101 (11), 114501.Google Scholar
Birnir, B., Mertens, K., Putkaradze, V. & Vorobieff, P. 2008b Morphology of a stream flowing down an inclined plane. Part 2. Meandering. J. Fluid Mech. 607, 401–411.Google Scholar
Bonn, D., Eggers, J., Indekeu, J., Meunier, J. & Rolley, E. 2009 Wetting: statics and dynamics. Rev. Mod. Phys. 81, 739–805.Google Scholar
Bostwick, J. B. & Steen, P. H. 2009 Capillary oscillations of a constrained liquid drop. Phys. Fluids 21, 032108.Google Scholar
Bostwick, J. B. & Steen, P. H. 2010 Stability of constrained cylindrical interfaces and the torus lift of Plateau–Rayleigh. J. Fluid Mech. 21, 201–219.Google Scholar
Bostwick, J. B. & Steen, P. H. 2013 Coupled oscillations of deformable spherical-cap droplets. Part 1. Inviscid motions. J. Fluid Mech. 714, 312–335.Google Scholar
Bostwick, J. B. & Steen, P. H. 2014 Dynamics of sessile drops. Part 1. Inviscid theory. J. Fluid Mech. 760, 5–38.Google Scholar
Bostwick, J. B. & Steen, P. H. 2015a Liquid-bridge shape stability by energy bounding. IMA J. Appl. Maths 80 (6), 1759–1775.Google Scholar
Bostwick, J. B. & Steen, P. H. 2015b Stability of constrained capillary surfaces. Annu. Rev. Fluid Mech. 47, 539–568.Google Scholar
Brown, R. A. & Scriven, L. E. 1980 On the multiple equilibrium shapes and stability of an interface pinned on a slot. J. Colloid Interface Sci. 78, 528–542.Google Scholar
Couvreur, S. & Daerr, A. 2012 The role of wetting heterogeneities in the meandering instability of a partial wetting rivulet. Europhys. Lett. 99 (2), 24004.Google Scholar
Culkin, J. B. & Davis, S. H. 1983 Meandering of water rivulets. AIChE J. 30, 263–267.Google Scholar
Daerr, A., Eggers, J., Limat, L. & Valade, N. 2011 General mechanism for the meandering instability of rivulets of Newtonian fluids. Phys. Rev. Lett. 106 (18), 184501.Google Scholar
Davis, S. H. 1980 Moving contact lines and rivulet instabilities. Part 1. The static rivulet. J. Fluid Mech. 98, 225–242.Google Scholar
Diez, J. A., González, A. G. & Kondic, L. 2009 On the breakup of fluid rivulets. Phys. Fluids 21 (8), 082105.Google Scholar
Diez, J. A., González, A. G. & Kondic, L. 2012 Instability of a transverse liquid rivulet on an inclined plane. Phys. Fluids 24 (3), 032104.Google Scholar
Dupré, A. 1869 Théorie Méchanique de La Chaleur. Gauthier-Villars.Google Scholar
Dussan, V. E. B. 1979 On the spreading of liquid on solid surfaces: static and dynamic contact lines. Annu. Rev. Fluid Mech. 11, 371–400.Google Scholar
Grand-Piteira, N. L., Daerr, A. & Limat, L. 2006 Meandering rivulets on a plane: a simple balance between inertia and capillarity. Phys. Rev. Lett. 96, 254503.Google Scholar
Hocking, L. M. 1992 Rival contact-angle models and the spreading of drops. J. Fluid Mech. 239, 671–681.Google Scholar
Kim, H. Y., Kim, J. H. & Kang, B. H. 2004 Meandering instability of a rivulet. J. Fluid Mech. 498, 245–256.Google Scholar
Kreyszig, E. 1991 Differential Geometry. Dover.Google Scholar
Langbein, D. 1990 The shape and stability of liquid menisci at solid edges. J. Fluid Mech. 213, 251–265.Google Scholar
Mertens, K., Putkaradze, V. & Vorobieff, P. 2005 Morphology of a stream flowing down an inclined plane. Part 1. Braiding. J. Fluid Mech. 531, 49–58.Google 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. Non-Linear Mech. 39, 1239–1249.Google Scholar
Nakagawa, T. & Nakagawa, R. 1996 A novel oscillation phenomenon of the water rivulet on a smooth hydrophobic surface. Acta Mechanica 115, 27–37.Google Scholar
Nakagawa, T. & Scott, J. C. 1984 Stream meanders on a smooth hydrophobic surface. J. Fluid Mech. 149, 89–99.Google Scholar
Nakagawa, T. & Scott, J. C. 1992 Rivulet meanders on a smooth hydrophobic surface. Intl J. Multiphase Flow 18, 455–463.Google Scholar
Oron, A., Davis, S. H. & Bankoff, S. G. 1997 Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69, 931–980.Google Scholar
Paterson, C., Wilson, S. K. & Duffy, B. R. 2013 Pinning, de-pinning and re-pinning of a slowly varying rivulet. Eur. J. Mech. (B/Fluids) 41, 94–108.Google Scholar
Paterson, C., Wilson, S. K. & Duffy, B. R. 2015 Strongly coupled interaction between a ridge of fluid and an inviscid airflow. Phys. Fluids 27 (7), 072104.Google Scholar
Plateau, J. A. F. 1863 Experimental and theoretical researches on the figures on equilibrium of a liquid mass withdrawn from the action of gravity. In Annual Report of the Board of Regents of the Smithsonian Institution, pp. 207–285. Government Printing Office.Google Scholar
Rayleigh, Lord 1879 On the capillary phenomenon of jets. Proc. R. Soc. Lond. 29, 71–97.Google Scholar
Roy, R. V. & Schwartz, L. W. 1999 On the stability of liquid ridges. J. Fluid Mech. 391, 293–318.Google Scholar
Schmuki, P. & Laso, M. 1990 On the stability of rivulet flow. J. Fluid Mech. 215, 125–143.Google Scholar
Segel, L. A. 1987 Mathematics Applied to Continuum Mechanics. Dover.Google Scholar
Sekimoto, K., Oguma, R. & Kawasaki, K. 1987 Morphological stability analysis of partial wetting. Ann. Phys. 176 (2), 359–392.Google Scholar
Thiele, U. & Knobloch, E. 2003 Front and back instability of a liquid film on a slightly inclined plate. Phys. Fluids 15 (4), 892–907.Google Scholar
Towell, G. D. & Rothfeld, L. B. 1966 Hydrodynamics of rivulet flow. AIChE J. 12, 972–980.Google Scholar
Weiland, R. H. & Davis, S. H. 1981 Moving contact lines and rivulet instabilities. Part 2. Long waves on flat rivulets. J. Fluid Mech. 107, 261–280.Google Scholar
Wilson, S. K. & Duffy, B. R. 2005 When is it energetically favorable for a rivulet of perfectly wetting fluid to split? Phys. Fluids 17, 078104.Google Scholar
Wilson, S. K., Sullivan, J. M. & Duffy, B. R. 2011 The energetics of the breakup of a sheet and of a rivulet on a vertical substrate in the presence of a uniform surface shear stress. J. Fluid Mech. 674, 281–306.Google Scholar
Young, G. W. & Davis, S. H. 1987 Rivulet instabilities. J. Fluid Mech. 176, 1–31.Google Scholar
Young, T. 1805 An essay on the cohesion of fluids. Phil. Trans. R. Soc. Lond. 95, 65–87.Google Scholar