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Gravity-driven coatings on curved substrates: a differential geometry approach

Published online by Cambridge University Press:  03 October 2022

Pier Giuseppe Ledda*
Laboratory of Fluid Mechanics and Instabilities, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland Dipartimento di Ingegneria Civile, Ambientale e Architettura, University of Cagliari, 09123 Cagliari, Italy
M. Pezzulla
Slender Structures Lab, Department of Mechanical and Production Engineering, Århus University, Inge Lehmanns Gade 10, 8000 Århus C, Denmark
E. Jambon-Puillet
Department of Chemical and Biological Engineering, Princeton University, Princeton, NJ 08540, USA
P.-T. Brun
Department of Chemical and Biological Engineering, Princeton University, Princeton, NJ 08540, USA
F. Gallaire
Laboratory of Fluid Mechanics and Instabilities, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland
Email address for correspondence:


Drainage and spreading processes in thin liquid films have received considerable attention in the past decades. Yet, our understanding of three-dimensional cases remains sparse, with only a few studies focusing on flat and axisymmetric substrates. Here, we exploit differential geometry to understand the drainage and spreading of thin films on curved substrates, under the assumption of negligible surface tension and hydrostatic gravity effects. We develop a solution for the drainage on a local maximum of a generic substrate. We then investigate the role of geometry in defining the spatial thickness distribution via an asymptotic expansion in the vicinity of the maximum. Spheroids with a much larger (respectively smaller) height than the equatorial radius are characterized by an increasing (respectively decreasing) coating thickness when moving away from the pole. These thickness variations result from a competition between the variations of the substrate's slope and mean curvature. The coating of a torus presents larger thicknesses and a faster spreading on the inner region than on the outer region, owing to the different curvatures in these two regions. In the case of an ellipsoid with three different axes, spatial modulations in the drainage solution are observed as a consequence of a faster drainage along the short principal axis, faithfully reproduced by a three-dimensional asymptotic solution. Leveraging the conservation of mass, an analytical solution for the average spreading front is obtained. The solutions are in agreement with numerical simulations and experimental measurements obtained from the coating of a curing polymer on diverse substrates.

JFM Papers
© The Author(s), 2022. Published by Cambridge University Press

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Acheson, D.J. 1990 Elementary Fluid Dynamics. Oxford University Press.Google Scholar
Ancey, C. 2007 Plasticity and geophysical flows: a review. J. Non-Newtonian Fluid Mech. 142 (1–3), 435.CrossRefGoogle Scholar
Balestra, G., Badaoui, M., Ducimetière, Y.-M. & Gallaire, F. 2019 Fingering instability on curved substrates: optimal initial film and substrate perturbations. J. Fluid Mech. 868, 726761.CrossRefGoogle Scholar
Balestra, G., Kofman, N., Brun, P.-T., Scheid, B. & Gallaire, F. 2018 a Three-dimensional Rayleigh–Taylor instability under a unidirectional curved substrate. J. Fluid Mech. 837, 1947.CrossRefGoogle Scholar
Balestra, G., Nguyen, D.M.-P. & Gallaire, F. 2018 b Rayleigh–Taylor instability under a spherical substrate. Phys. Rev. Fluids 3 (8), 084005.CrossRefGoogle Scholar
Balmforth, N.J., Burbidge, A.S., Craster, R.V., Salzig, J. & Shen, A. 2000 Visco-plastic models of isothermal lava domes. J. Fluid Mech. 403, 3765.CrossRefGoogle Scholar
Balmforth, N.J., Craster, R.V., Rust, A.C. & Sassi, R. 2006 Viscoplastic flow over an inclined surface. J. Non-Newtonian Fluid Mech. 139 (1–2), 103127.CrossRefGoogle Scholar
Balmforth, N.J., Craster, R.V. & Sassi, R. 2002 Shallow viscoplastic flow on an inclined plane. J. Fluid Mech. 470, 129.CrossRefGoogle Scholar
Balmforth, N.J. & Kerswell, R.R. 2005 Granular collapse in two dimensions. J. Fluid Mech. 538, 399428.CrossRefGoogle Scholar
Bertagni, M.B. & Camporeale, C. 2021 The hydrodynamic genesis of linear karren patterns. J. Fluid Mech. 913, A34.CrossRefGoogle Scholar
Bertagni, M.B. & Camporeale, C. 2017 Nonlinear and subharmonic stability analysis in film-driven morphological patterns. Phys. Rev. E 96 (5), 053115.CrossRefGoogle ScholarPubMed
Bertozzi, A.L. & Brenner, M.P. 1997 Linear stability and transient growth in driven contact lines. Phys. Fluids 9 (3), 530539.CrossRefGoogle Scholar
Camporeale, C. 2015 Hydrodynamically locked morphogenesis in karst and ice flutings. J. Fluid Mech. 778, 89119.CrossRefGoogle Scholar
Camporeale, C. & Ridolfi, L. 2012 Hydrodynamic-driven stability analysis of morphological patterns on stalactites and implications for cave paleoflow reconstructions. Phys. Rev. Lett. 108 (23), 238501.CrossRefGoogle ScholarPubMed
Chandrasekhar, S. 2013 Hydrodynamic and Hydromagnetic Stability. Courier Corporation.Google Scholar
Couder, Y., Fort, E., Gautier, C.-H. & Boudaoud, A. 2005 From bouncing to floating: noncoalescence of drops on a fluid bath. Phys. Rev. Lett. 94, 177801.CrossRefGoogle ScholarPubMed
Craster, R.V. & Matar, O.K. 2009 Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81 (3), 1131.CrossRefGoogle Scholar
Deserno, M. 2004 Notes on differential geometry. Available at: Scholar
Didden, N. & Maxworthy, T. 1982 The viscous spreading of plane and axisymmetric gravity currents. J. Fluid Mech. 121, 2742.CrossRefGoogle Scholar
Duruk, S., Boujo, E. & Sellier, M. 2021 Thin liquid film dynamics on a spinning spheroid. Fluids 6 (9), 318.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, 349383.CrossRefGoogle Scholar
Gratton, J. & Minotti, F. 1990 Self-similar viscous gravity currents: phase-plane formalism. J. Fluid Mech. 210, 155182.CrossRefGoogle Scholar
Hosoi, A.E. & Bush, J.W.M. 2001 Evaporative instabilities in climbing films. J. Fluid Mech. 442, 217.CrossRefGoogle Scholar
Hoult, D.P. 1972 Oil spreading on the sea. Annu. Rev. Fluid Mech. 4 (1), 341368.CrossRefGoogle Scholar
Howell, P.D. 2003 Surface-tension-driven flow on a moving curved surface. J. Engng Maths 45 (3), 283308.CrossRefGoogle Scholar
Huppert, H.E. 1982 a Flow and instability of a viscous current down a slope. Nature 300 (5891), 427429.CrossRefGoogle Scholar
Huppert, H.E. 1982 b The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface. J. Fluid Mech. 121, 4358.CrossRefGoogle Scholar
Huppert, H.E. 1986 The intrusion of fluid mechanics into geology. J. Fluid Mech. 173, 557594.CrossRefGoogle Scholar
Huppert, H.E. 2006 Gravity currents: a personal perspective. J. Fluid Mech. 554, 299322.CrossRefGoogle Scholar
Huppert, H.E. & Simpson, J.E. 1980 The slumping of gravity currents. J. Fluid Mech. 99 (4), 785799.CrossRefGoogle Scholar
Irgens, F. 2019 Tensor Analysis. Springer.CrossRefGoogle Scholar
Jambon-Puillet, E., Ledda, P.G., Gallaire, F. & Brun, P.-T. 2021 Drops on the underside of a slightly inclined wet substrate move too fast to grow. Phys. Rev. Lett. 127, 044503.CrossRefGoogle Scholar
Jeffreys, H. 1930 The draining of a vertical plate. Math. Proc. Camb. Phil. Soc. 26 (2), 204205.CrossRefGoogle Scholar
Jones, T.J., Jambon-Puillet, E., Marthelot, J. & Brun, P.-T. 2021 Bubble casting soft robotics. Nature 599 (7884), 229233.CrossRefGoogle ScholarPubMed
Kapitza, P.L. 1948 Wave flow of thin layers of viscous liquid. Zh. Eksp. Teor. Fiz. 18, 328.Google Scholar
Kapitza, P.L. & Kapitza, S.P. 1965 Wave flow of thin layers of a viscous fluid: experimental study of undulatory flow conditions. In Collected Papers of PL Kapitza (ed. D. Ter Haar), vol. 2. Pergamon.Google Scholar
Keulegan, G.H. 1957 An experimental study of the motion of saline water from locks into fresh water channels. Nat. Bur. Stand. Rept. Tech. Rep. 5168.Google Scholar
Kondic, L. 2003 Instabilities in gravity driven flow of thin fluid films. SIAM Rev. 45 (1), 95115.CrossRefGoogle Scholar
Kondic, L. & Diez, J. 2002 Flow of thin films on patterned surfaces: controlling the instability. Phys. Rev. E 65, 045301.CrossRefGoogle ScholarPubMed
Landau, L.D. & Levich, B. 1942 Dragging of a liquid by a moving plate. Acta Physicochim. USSR 17, 141153.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. & Gallaire, F. 2021 Secondary instability in thin film flows under an inclined plane: growth of lenses on spatially developing rivulets. Proc. R. Soc. Lond. A 477 (2251), 20210291.Google 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
Lee, A., Brun, P.-T., Marthelot, J., Balestra, G., Gallaire, F. & Reis, P.M. 2016 Fabrication of slender elastic shells by the coating of curved surfaces. Nat. Commun. 7 (1), 11155.CrossRefGoogle ScholarPubMed
Lerisson, G., Ledda, P.G., Balestra, G. & Gallaire, F. 2019 Dripping down the rivulet. Phys. Rev. Fluids 4, 100504.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
Lin, T.-S., Dijksman, J.A. & Kondic, L. 2021 Thin liquid films in a funnel. J. Fluid Mech. 924, A26.CrossRefGoogle Scholar
Lister, J.R. 1992 Viscous flows down an inclined plane from point and line sources. J. Fluid Mech. 242, 631653.CrossRefGoogle Scholar
Mayo, L.C., McCue, S.W., Moroney, T.J., Forster, W.A., Kempthorne, D.M., Belward, J.A. & Turner, I.W. 2015 Simulating droplet motion on virtual leaf surfaces. R. Soc. Open. Sci. 2 (5), 140528.CrossRefGoogle ScholarPubMed
Meakin, P. & Jamtveit, B. 2010 Geological pattern formation by growth and dissolution in aqueous systems. Proc. R. Soc. Lond. A 466 (2115), 659694.Google Scholar
Oron, A. 2000 Nonlinear dynamics of three-dimensional long-wave Marangoni instability in thin liquid films. Phys. Fluids 12 (7), 16331645.CrossRefGoogle Scholar
Oron, A., Davis, S.H. & Bankoff, S.G. 1997 Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69 (3), 931.CrossRefGoogle Scholar
Qin, J., Xia, Y.-T. & Gao, P. 2021 Axisymmetric evolution of gravity-driven thin films on a small sphere. J. Fluid Mech. 907, A4.CrossRefGoogle Scholar
Rayleigh, , 1882 Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. Proc. Lond. Math. Soc. s1-14 (1), 170177.CrossRefGoogle Scholar
Roberts, A.J. & Li, Z. 2006 An accurate and comprehensive model of thin fluid flows with inertia on curved substrates. J. Fluid Mech. 553, 3373.CrossRefGoogle Scholar
Roy, R.V., Roberts, A.J. & Simpson, M.E. 2002 A lubrication model of coating flows over a curved substrate in space. J. Fluid Mech. 454, 235261.CrossRefGoogle Scholar
Scheid, B. 2013 Rivulet Structures in Falling Liquid Films, pp. 435441. Springer.Google Scholar
Schwartz, L.W. & Roy, R.V. 2004 Theoretical and numerical results for spin coating of viscous liquids. Phys. Fluids 16 (3), 569584.CrossRefGoogle Scholar
Scriven, L.E. 1988 Physics and applications of dip coating and spin coating. MRS Online Proc. Library (OPL) 121, 717729.CrossRefGoogle Scholar
Short, M.B., Baygents, J.C., Beck, J.W., Stone, D.A., Toomey III, R.S. & Goldstein, R.E. 2005 Stalactite growth as a free-boundary problem: a geometric law and its platonic ideal. Phys. Rev. Lett. 94 (1), 018501.CrossRefGoogle ScholarPubMed
Simpson, J.E. 1982 Gravity currents in the laboratory, atmosphere, and ocean. Annu. Rev. Fluid Mech. 14 (1), 213234.CrossRefGoogle Scholar
Smith, S.H. 1969 On initial value problems for the flow in a thin sheet of viscous liquid. Z. Angew. Math. Phys. 20 (4), 556560.CrossRefGoogle Scholar
Takagi, D. & Huppert, H.E. 2010 Flow and instability of thin films on a cylinder and sphere. J. Fluid Mech. 647, 221238.CrossRefGoogle Scholar
Taylor, G.I. 1950 The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I. Proc. R. Soc. Lond. A 201 (1065), 192196.Google Scholar
Thiffeault, J.-L. & Kamhawi, K. 2006 Transport in thin gravity-driven flow over a curved substrate. arXiv:nlin/0607075.Google Scholar
Troian, S.M., Herbolzheimer, E., Safran, S.A. & Joanny, J.F. 1989 a Fingering instabilities of driven spreading films. Europhys. Lett. 10 (1), 25.CrossRefGoogle Scholar
Troian, S.M., Wu, X.L. & Safran, S.A. 1989 b Fingering instability in thin wetting films. Phys. Rev. Lett. 62 (13), 1496.CrossRefGoogle ScholarPubMed
Weinstein, S.J. & Ruschak, K.J. 2004 Coating flows. Annu. Rev. Fluid Mech. 36, 2953.CrossRefGoogle Scholar
Wray, A.W., Papageorgiou, D.T. & Matar, O.K. 2017 Reduced models for thick liquid layers with inertia on highly curved substrates. SIAM J. Appl. Maths 77 (3), 881904.CrossRefGoogle Scholar
Xue, N., Pack, M.Y. & Stone, H.A. 2020 Marangoni-driven film climbing on a draining pre-wetted film. J. Fluid Mech. 886, A24.CrossRefGoogle Scholar
Xue, N. & Stone, H.A. 2020 Self-similar draining near a vertical edge. Phys. Rev. Lett. 125 (6), 064502.CrossRefGoogle Scholar
Xue, N. & Stone, H.A. 2021 Draining and spreading along geometries that cause converging flows: viscous gravity currents on a downward-pointing cone and a bowl-shaped hemisphere. Phys. Rev. Fluids 6 (4), 043801.CrossRefGoogle Scholar

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