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Dripping instability of a two-dimensional liquid film under an inclined plate

Published online by Cambridge University Press:  15 December 2021

Guangzhao Zhou Open the ORCID record for Guangzhao Zhou [Opens in a new window]  and
Andrea Prosperetti Open the ORCID record for Andrea Prosperetti [Opens in a new window]
Guangzhao Zhou
Affiliation:
Department of Mechanical Engineering, University of Houston, Houston, TX77204, USA
Andrea Prosperetti*
Affiliation:
Department of Mechanical Engineering, University of Houston, Houston, TX77204, USA Faculty of Science and Technology, and J. M. Burgers Center for Fluid Dynamics, University of Twente, 7500 AEEnschede, The Netherlands Department of Mechanical Engineering, Johns Hopkins University, Baltimore, MD21218, USA
*
†Email address for correspondence: [email protected]
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Abstract

It is known that the dripping of a liquid film on the underside of a plate can be suppressed by tilting the plate so as to cause a sufficiently strong flow. This paper uses two-dimensional numerical simulations in a closed-flow framework to study several aspects of this phenomenon. It is shown that, in quasi-equilibrium conditions, the onset of dripping is closely associated with the curvature of the wave crests approaching a well-defined maximum value. When dynamic effects become significant, this connection between curvature and dripping weakens, although the critical curvature remains a useful reference point as it is intimately related to the short length scales promoted by the Rayleigh–Taylor instability. In the absence of flow, when the film is on the underside of a horizontal plate, the concept of a limit curvature is relevant only for small liquid volumes close to a critical value. Otherwise, the drops that form have a smaller curvature and a large volume. The paper also illustrates the peculiarly strong dependence of the dripping transition on the initial conditions of the simulations. This feature prevents the development of phase maps dependent only on the governing parameters (Reynolds number, Bond number, etc.) similar to those available for film flow on the upper side of an inclined plate.

JFM classification

Interfacial Flows (free surface): Thin films Instability: Absolute/convective instability Materials Processing Flows: Coating
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 932 , 10 February 2022 , A49
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Abdelall, F.F., Abdel-Khalik, S.I., Sadowski, D.L., Shin, A. & Yoda, M. 2006 On the Rayleigh–Taylor instability for confined liquid films with injection through the bounding surfaces. Intl J. Heat Mass Transfer 49, 1529–1546.CrossRefGoogle Scholar
Alekseenko, S.V., Nakoryakov, V.E. & Pokusaev, B.G. 1994 Wave Flow of Liquid Films. Begell House.Google Scholar
Babchin, A.J., Frenkel, A.L., Levich, B.G. & Sivashinsky, G.I. 1983 Nonlinear saturation of Rayleigh–Taylor instability in thin films. Phys. Fluids 26, 3159–3161.CrossRefGoogle Scholar
Balestra, G., Nguyen, D.M.-P. & Gallaire, F. 2018 Rayleigh–Taylor instability under a spherical substrate. Phys. Rev. Fluids 3, 084005.CrossRefGoogle Scholar
Bertagni, M.B. & Camporeale, C. 2017 Nonlinear and subharmonic stability analysis in film-driven morphological patterns. Phys. Rev. E 96, 053115.CrossRefGoogle ScholarPubMed
Brackbill, J.U., Kothe, D.B. & Zemach, C. 1992 A continuum method for modeling surface tension. J. Comput. Phys. 100, 335–354.CrossRefGoogle Scholar
Brun, P.-T., Damiano, A., Pierre, R., Balestra, G. & Gallaire, F. 2015 Rayleigh–Taylor instability under an inclined plane. Phys. Fluids 27, 084107.CrossRefGoogle Scholar
Chang, H.-C. 1994 Wave evolution on a falling film. Annu. Rev. Fluid Mech. 26, 103–136.CrossRefGoogle Scholar
Chang, H.-C. & Demekhin, E.A. 2002 Complex Wave Dynamics on Thin Films. Elsevier.Google Scholar
Charogiannis, A., Denner, F., van Wachem, B.G.M., Kalliadasis, S., Scheid, B. & Markides, C.N. 2018 Experimental investigations of liquid falling films flowing under an inclined planar substrate. Phys. Rev. Fluids 3, 114002.CrossRefGoogle Scholar
Craster, R.V. & Matar, O.K. 2009 Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81, 1131–1198.CrossRefGoogle Scholar
Deissler, R.J. & Oron, A. 1992 Stable localized patterns in thin liquid films. Phys. Rev. Lett. 68, 2948–2951.CrossRefGoogle ScholarPubMed
Denner, F., Charogiannis, A., Pradas, M., Markides, C.N., van Wachem, B.G.M. & Kalliadasis, S. 2018 Solitary waves on falling liquid films in the inertia-dominated regime. J. Fluid Mech. 837, 491–519.CrossRefGoogle Scholar
Dietze, G.F. 2019 Effect of wall corrugations on scalar transfer to a wavy falling liquid film. J. Fluid Mech. 859, 1098–1128.CrossRefGoogle Scholar
Duprat, C., Ruyer-Quil, C., Kalliadasis, S. & Giorgiutti-Dauphiné, F. 2007 Absolute and convective instabilities of a film flowing down a vertical fiber. Phys. Rev. Lett. 98, 244502.CrossRefGoogle Scholar
Elgowainy, A. & Ashgriz, N. 1997 The Rayleigh–Taylor instability of viscous fluid layers. Phys. Fluids 9, 1635–1649.CrossRefGoogle Scholar
Fermigier, M., Limat, L., Wesfreid, J.E., Boudinet, P. & Quilliet, C. 1992 Two-dimensional patterns in Rayleigh–Taylor instability of a thin layer. J. Fluid Mech. 236, 349–383.CrossRefGoogle Scholar
François, M.M., Cummins, S.J., Dendy, E.D., Kothe, D.B., Sicilian, J.M. & Williams, M.W. 2006 A balanced-force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework. J. Comput. Phys. 213, 141–173.CrossRefGoogle Scholar
Gallino, G., Schneider, T.M. & Gallaire, F. 2018 Edge states control droplet breakup in subcritical extensional flows. Phys. Rev. Fluids 3, 073603.CrossRefGoogle Scholar
Indeikina, A., Veretennikov, I. & Chang, H.-C. 1997 Drop fall-off from pendent rivulets. J. Fluid Mech. 338, 173–201.CrossRefGoogle Scholar
Kalliadasis, S. & Chang, H.-C. 1994 Drop formation during coating of vertical fibres. J. Fluid Mech. 261, 135–168.CrossRefGoogle Scholar
Kalliadasis, S., Ruyer-Quil, C., Scheid, B. & Velarde, M.G. 2011 Falling Liquid Films. Springer.Google Scholar
Kapitza, P.L. 1948 Wave flow of thin layers of a viscous fluid. I. The free flow. Zh. Eksp. Teor. Fiz. 18, 3, English translation available in Collected Papers of P. L. Kapitza (ed. D. ter Haar), vol. 2, pp. 663–679, Pergamon, 1965.Google Scholar
Kapitza, P.L. & Kapitza, S.P. 1949 Wave flow of thin layers of viscous liquids. III. Experimental study of undulatory flow conditions. Zh. Eksp. Teor. Fiz. 19, 105, English translation available in Collected Papers of P. L. Kapitza (ed. D. ter Haar), vol. 2, pp. 690–709, Pergamon, 1965.Google Scholar
Kerswell, R.R. 2018 Nonlinear nonmodal stability theory. Annu. Rev. Fluid Mech. 50, 319–345.CrossRefGoogle Scholar
Kofman, N., Rohlfs, W., Gallaire, F., Scheid, B. & Ruyer-Quil, C. 2018 Prediction of two-dimensional dripping onset of a liquid film under an inclined plane. Intl J. Multiphase Flow 104, 286–293.CrossRefGoogle Scholar
Kondic, L. 2003 Instabilities in gravity driven flow of thin fluid films. SIAM Rev. 95, 45–115.Google Scholar
Ledda, P.G., Balestra, G., Lerisson, G., Scheid, B., Wyart, M. & Gallaire, F. 2021 Hydrodynamic-driven morphogenesis of karst draperies: spatio-temporal analysis of the two-dimensional impulse response. J. Fluid Mech. 910, A53.CrossRefGoogle Scholar
Ledda, P.G., Lerisson, G., Balestra, G. & Gallaire, F. 2020 Instability of a thin viscous film flowing under an inclined substrate: the emergence and stability of rivulets. J. Fluid Mech. 904, A23.CrossRefGoogle Scholar
Lerisson, G., Ledda, P.G., Balestra, G. & Gallaire, F. 2020 Instability of a thin viscous film flowing under an inclined substrate: steady patterns. J. Fluid Mech. 898, A6.CrossRefGoogle Scholar
Lichtenberg, A.J. & Lieberman, M.A. 1983 Regular and Stochastic Motion. Springer.CrossRefGoogle Scholar
Limat, L., Jenffer, P., Dagens, B., Touron, E., Fermigier, M. & Wesfreid, J.E. 1992 Gravitational instabilities of thin liquid layers: dynamics of pattern selection. Physica D 61, 166–182.CrossRefGoogle Scholar
Lin, T.-S. & Kondic, L. 2010 Thin films flowing down inverted substrates: two dimensional flow. Phys. Fluids 22, 052105.CrossRefGoogle Scholar
Lin, T.-S., Kondic, L. & Filippov, A. 2012 Thin films flowing down inverted substrates: three-dimensional flow. Phys. Fluids 24, 022105.CrossRefGoogle Scholar
Lister, J.R. & Kerr, R.C. 1989 The effect of geometry on the gravitational instability of a buoyant region of viscous fluid. J. Fluid Mech. 202, 577–594.CrossRefGoogle Scholar
Lister, J.R., Rallison, J.M., King, A.A., Cummings, L.J. & Jensen, O.E. 2006 Capillary drainage of an annular film: the dynamics of collars and lobes. J. Fluid Mech. 552, 311–343.CrossRefGoogle Scholar
Lister, J.R., Rallison, J.M. & Rees, S.J. 2010 The nonlinear dynamics of pendent drops on a thin film coating the underside of a ceiling. J. Fluid Mech. 647, 239–264.CrossRefGoogle Scholar
McCaughan, G. 2017 Equation of a ‘tilted’ sine, https://math.stackexchange.com/questions/2430564/equation-of-a-tilted-sine/2430837#2430837, accessed May 31, 2021.Google Scholar
Oron, A., Davis, S.H. & Bankoff, S.G. 1997 Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69, 931–980.CrossRefGoogle Scholar
Orsini, G. & Tricoli, V. 2017 A scaling theory of the free-coating flow on a plate withdrawn from a pool. Phys. Fluids 29, 052106.CrossRefGoogle Scholar
Ott, E. 2003 Chaos in Dynamical Systems, 2nd edn. Cambridge University Press.Google Scholar
Pitts, E. 1973 The stability of pendent liquid drops. Part 1. Drops formed in a narrow gap. J. Fluid Mech. 59, 753–767.CrossRefGoogle Scholar
Popinet, S. 2009 An accurate adaptive solver for surface-tension-driven interfacial flows. J. Comput. Phys. 228, 5838–5866.CrossRefGoogle Scholar
Pradas, M., Kalliadasis, S. & Tseluiko, D. 2012 Binary interactions of solitary pulses in falling liquid films. IMA J. Appl. Maths 77, 408–419.CrossRefGoogle Scholar
Pradas, M., Tseluiko, D. & Kalliadasis, S. 2011 Rigorous coherent-structure theory for falling liquid films: viscous dispersion effects on bound-state formation and self-organization. Phys. Fluids 23, 044104.CrossRefGoogle Scholar
Pumir, A., Manneville, T. & Pomeau, Y. 1983 On solitary waves running down an inclined plane. J. Fluid Mech. 135, 27–50.CrossRefGoogle Scholar
Reck, D. & Aksel, N. 2015 Recirculation areas underneath solitary waves on gravity-driven film flows. Phys. Fluids 27, 112107.CrossRefGoogle Scholar
Rietz, M., Scheid, B., Gallaire, F., Kofman, N., Kneer, R. & Rohlfs, W. 2017 Dynamics of falling films on the outside of a vertical rotating cylinder: waves, rivulets and dripping transitions. J. Fluid Mech. 832, 189–211.CrossRefGoogle Scholar
Rohlfs, W., Pischke, P. & Scheid, B. 2017 Hydrodynamic waves in films flowing under an inclined plane. Phys. Rev. Fluids 2, 044003.CrossRefGoogle Scholar
Rohlfs, W. & Scheid, B. 2015 Phase diagram for the onset of circulating waves and flow reversal in inclined falling films. J. Fluid Mech. 763, 322–351.CrossRefGoogle Scholar
Ruyer-Quil, C. & Manneville, P. 2000 Improved modeling of flows down inclined planes. Eur. Phys. J. B15, 357–369.CrossRefGoogle Scholar
Scardovelli, R. & Zaleski, S. 1999 Direct numerical simulation of free-surface and interfacial flow. Annu. Rev. Fluid Mech. 31, 567–603.CrossRefGoogle Scholar
Scheid, B., Kofman, N. & Rohlfs, W. 2016 Critical inclination for absolute/convective instability transition in inverted falling films. Phys. Fluids 28, 044107.CrossRefGoogle Scholar
Sterman-Cohen, E., Bestehorn, M. & Oron, A. 2017 Rayleigh–Taylor instability in thin liquid films subjected to harmonic vibration. Phys. Fluids 29, 052105, erratum: Phys. Fluids, vol. 29, 109901.CrossRefGoogle Scholar
Sterman-Cohen, E. & Oron, A. 2020 Dynamics of nonisothermal two-thin-fluid-layer systems subjected to harmonic tangential forcing under Rayleigh–Taylor instability conditions. Phys. Fluids 32, 082113.CrossRefGoogle Scholar
Takagi, D. & Huppert, H.E. 2010 Flow and instability of thin films on a cylinder and sphere. J. Fluid Mech. 647, 221–238.CrossRefGoogle Scholar
Talib, E. & Juel, A. 2007 Instability of a viscous interface under horizontal oscillation. Phys. Fluids 19, 092102.CrossRefGoogle Scholar
Tanasijczuk, A.J., Perazzo, C.A. & Gratton, J. 2010 Navier–Stokes solutions for steady parallel-sided pendent rivulets. Eur. J. Mech. B/Fluids 29, 465–471.CrossRefGoogle Scholar
Tomlin, R.J., Cimpeanu, R. & Papageorgiou, D.T. 2020 Instability and dripping of electrified liquid films flowing down inverted substrates. Phys. Rev. Fluids 5, 013703.CrossRefGoogle Scholar
Trinh, P., Kim, H., Hammoud, N., Howell, P., Chapman, S. & Stone, H. 2014 Curvature suppresses the Rayleigh–Taylor instability. Phys. Fluids 26, 051704.CrossRefGoogle Scholar
Weinstein, S.J. & Ruschak, K.J. 2004 Coating flows. Annu. Rev. Fluid Mech. 36, 29–53.CrossRefGoogle Scholar
Whitehead, J.A. 1988 Fluid models of geological hot spots. Annu. Rev. Fluid Mech. 20, 61–87.CrossRefGoogle Scholar
Wiggins, S. 1988 Global Bifurcations and Chaos. Springer.CrossRefGoogle Scholar
Wolf, G.H. 1969 The dynamic stabilization of the Rayleigh–Taylor instability and the corresponding dynamic equilibrium. Z. Phys. 227, 291–300.CrossRefGoogle Scholar
Yiantsios, S.G. & Higgins, B.G. 1989 Rayleigh–Taylor instability in thin viscous films. Phys. Fluids A1, 1484–1501.CrossRefGoogle Scholar
Zhou, G. & Prosperetti, A. 2020 Capillary waves on a falling film. Phys. Rev. Fluids 5, 114005.CrossRefGoogle Scholar

Zhou and Prosperetti supplementary movie 1

This movie shows the evolution of the wave shape (two periods) for the $135^\circ$ case of figure 8 in the paper. In the top panel, the scale in the normal direction is enlarged 6 times with respect to the horizontal scale, which permits a clearer view of the generation and absorption of the small disturbance in front of the main hump. The bottom panel is plotted using the same length unit for both directions. Here $Re_\perp=50$, $Bo_\perp=0.6$, $V/h_0^2=40$.

Download Zhou and Prosperetti supplementary movie 1(Video)
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Zhou and Prosperetti supplementary movie 2

This movie shows animations of two cases illustrated in figure 17(b), in which the gravity component normal to the plate is restored at $ \sqrt{g/h_0}\,t=10$ (red) and $ \sqrt{g/h_0}\,t=15$ (blue). In the former case, the dripping instability has sufficient time to develop and ultimately (two-dimensional) drops form. For the latter case, the Kapitza instability develops before the dripping instability and the film remains stable. Two wave periods are shown. Note the arrows in the lower right corner of the image, which indicate the gravity components acting on the film at the time of each frame. Initially, for both simulations, only the component parallel to the plate acts. The normal component is then restored, first for the red wave and later for the blue wave. Here $Re_\perp=50$, $Bo_\perp=0.6$, $V/h_0^2=40$.

Download Zhou and Prosperetti supplementary movie 2(Video)
Video 659.4 KB
Supplementary material: PDF

Zhou and Prosperetti supplementary material

Captions for movies 1-2

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