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Linear stability of ultrathin spherical coatings

D. Moreno-Boza* and A. Sevilla†

  • Grupo de Mecánica de Fluidos, Departamento de Ingeniería Térmica y de Fluidos, Universidad Carlos III de Madrid, Avda. de la Universidad 30, 28911 Leganés, Madrid, Spain

  • *damoreno@ing.uc3m.es
  • †alejandro.sevilla@uc3m.es
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Phys. Rev. Fluids 9, 014004 – Published 24 January, 2024

DOI: https://doi.org/10.1103/PhysRevFluids.9.014004

Abstract

We unravel the linear stability properties of an otherwise stagnant ultrathin nonwetting liquid film of thickness ho coating a spherical substrate of radius R. The configuration is known to be unstable due to the competition of the destabilizing van der Waals (vdW) forces and the stabilizing surface tension force. The governing equations of motion written in the Stokes limit of negligible liquid inertia and an accompanying lubrication model are linearized about the zero-velocity base state and decomposed into normal modes to obtain the temporal dispersion relation. Discrete unstable modes are identified and tracked as a function of a capillary number Ca measuring the relative importance of surface tension to vdW forces and the film aspect ratio η=R/ho. For small enough values of the capillary number only the first polar mode is unstable, and the corresponding maximum growth rate is shown to be a universal function of η. Lubrication theory is seen to provide a good quantitative prediction of the film stability properties for η≫1.

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References (49)

  1. H. E. Huppert, Flow and instability of a viscous current down a slope, Nature (London) 300, 427 (1982).
  2. S. Kalliadasis, C. Ruyer-Quil, B. Scheid, and M. G. Velarde, Falling Liquid Films, Vol. 176 (Springer Science & Business Media, Berlin, 2011)
  3. R. Pattle, Diffusion from an instantaneous point source with a concentration-dependent coefficient, Q. J. Mech. Appl. Math. 12, 407 (1959).
  4. H. E. Huppert, The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface, J. Fluid Mech. 121, 43 (1982).
  5. H. E. Huppert, The intrusion of fluid mechanics into geology, J. Fluid Mech. 173, 557 (1986).
  6. C. Ancey, Plasticity and geophysical flows: A review, J. Non-Newtonian Fluid Mech. 142, 4 (2007).
  7. S. V. Makarov, V. A. Milichko, I. S. Mukhin, I. I. Shishkin, D. A. Zuev, A. M. Mozharov, A. E. Krasnok, and P. A. Belov, Controllable femtosecond laser-induced dewetting for plasmonic applications, Laser Photonics Rev. 10, 91 (2016).
  8. R. A. Hughes, E. Menumerov, and S. Neretina, When lithography meets self-assembly: A review of recent advances in the directed assembly of complex metal nanostructures on planar and textured surfaces, Nanotechnology 28, 282002 (2017).
  9. L. Kondic, A. G. González, J. A. Diez, J. D. Fowlkes, and P. R., Liquid-state dewetting of pulsed-laser-heated nanoscale metal films and other geometries, Annu. Rev. Fluid Mech. 52, 235 (2019).
  10. R. J. Braun, Dynamics of the tear film, Annu. Rev. Fluid Mech. 44, 267 (2012).
  11. G. G. Fuller and J. Vermant, Complex fluid-fluid interfaces: Rheology and structure, Annu. Rev. Chem. Biomol. Eng. 3, 519 (2012).
  12. E. Hermans, M. S. Bhamla, P. Kao, G. G. Fuller, and J. Vermant, Lung surfactants and different contributions to thin film stability, Soft Matter 11, 8048 (2015).
  13. M. Dey, A. S. V., H. N. Dixit, A. Richhariya, and J. J. Feng, A model of tear-film breakup with continuous mucin concentration and viscosity profiles, J. Fluid Mech. 858, 352 (2019).
  14. O. E. Jensen and J. B. Grotberg, Insoluble surfactant spreading on a thin viscous film: Shock evolution and film rupture, J. Fluid Mech. 240, 259 (1992).
  15. D. Halpern and J. B. Grotberg, Dynamics and transport of a localized soluble surfactant on a thin film, J. Fluid Mech. 237, 1 (1992).
  16. O. E. Jensen and J. B. Grotberg, The spreading of heat or soluble surfactant along a thin liquid film, Phys. Fluids A: Fluid Dyn. 5, 58 (1993).
  17. D. Halpern and J. B. Grotberg, Surfactant effects on fluid-elastic instabilities of liquid-lined flexible tubes: A model of airway closure, J. Bio. Eng. 115, 271 (1993).
  18. J. B. Grotberg, Pulmonary flow and transport phenomena, Annu. Rev. Fluid Mech. 26, 529 (1994).
  19. D. Halpern, O. E. Jensen, and J. B. Grotberg, A theoretical study of surfactant and liquid delivery into the lung, J. App. Phys. 85, 333 (1998).
  20. K. J. Cassidy, D. Halpern, B. G. Ressler, and J. B. Grotberg, Surfactant effects in model airway closure experiments, J. App. Phys. 87, 415 (1999).
  21. P.-G. De Gennes, Wetting: Statics and dynamics, Rev. Mod. Phys. 57, 827 (1985).
  22. A. Oron, S. H. Davis, and S. G. Bankoff, Long-scale evolution of thin liquid films, Rev. Mod. Phys. 69, 931 (1997).
  23. D. Bonn, J. Eggers, J. Indekeu, J. Meunier, and E. Rolley, Wetting and spreading, Rev. Mod. Phys. 81, 739 (2009).
  24. R. V. Craster and O. K. Matar, Dynamics and stability of thin liquid films, Rev. Mod. Phys. 81, 1131 (2009).
  25. R. Blossey, Thin Liquid Films: Dewetting and Polymer Flow (Springer Science & Business Media, 2012).
  26. L. Schwartz and D. Weidner, Modeling of coating flows on curved surfaces, J. Eng. Math. 29, 91 (1995).
  27. R. V. Roy, A. Roberts, and M. Simpson, A lubrication model of coating flows over a curved substrate in space, J. Fluid Mech. 454, 235 (2002).
  28. N. Xue and H. A. Stone, 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, 043801 (2021).
  29. P. G. Ledda, M. Pezzulla, E. Jambon-Puillet, P.-T. Brun, and F. Gallaire, Gravity-driven coatings on curved substrates: A differential geometry approach, J. Fluid Mech. 949, A38 (2022).
  30. S. Bakshi, I. V. Roisman, and C. Tropea, Investigations on the impact of a drop onto a small spherical target, Phys. Fluids 19, 032102 (2007).
  31. D. Takagi and H. E. Huppert, Flow and instability of thin films on a cylinder and sphere, J. Fluid Mech. 647, 221 (2010).
  32. G. Balestra, David Minh-Phuc Nguyen, and F. Gallaire, Rayleigh-Taylor instability under a spherical substrate, Phys. Rev. Fluids 3, 084005 (2018).
  33. D. Kang, A. Nadim, and M. Chugunova, Dynamics and equilibria of thin viscous coating films on a rotating sphere, J. Fluid Mech. 791, 495 (2016).
  34. D. Kang, A. Nadim, and M. Chugunova, Marangoni effects on a thin liquid film coating a sphere with axial or radial thermal gradients, Phys. Fluids 29, 072106 (2017).
  35. A. Grigor'ev, M. Munichev, and S. Shiryaeva, Influence of disjoining pressure upon stability in the electric field of a charged liquid film on the surface of a hard core, J. Colloid Interface Sci. 166, 267 (1994).
  36. B. Li, H. Zhao, and X. Feng, Spontaneous instability of soft thin films on curved substrates due to van der Waals interaction, J. Mech. Phys. Solids 59, 610 (2011).
  37. J. Colin, Morphological instability of two stressed spherical shells, Int. J. Solids Struct. 44, 3218 (2007).
  38. D. Moreno-Boza, A. Martínez-Calvo, and A. Sevilla, Stokes theory of thin-film rupture, Phys. Rev. Fluids 5, 014002 (2020).
  39. I. Napari and A. Laaksonen, Disjoining pressure of thin films on spherical core particles, J Chem. Phys. 119, 10363 (2003).
  40. J. Israelachvili, Intermolecular and Surface Forces (Academic Press, San Diego, CA, 1985).
  41. A. Scheludko, Sur certaines particularités des lames mousseuses. II. Stabilité cinétique, épaisseur critique et épaisseur d'équilibre, Proc. Koninkl Nederl Akad Wetenschap 65, 87 (1962).
  42. A. Vrij, Possible mechanism for the spontaneous rupture of thin, free liquid films, Discuss. Faraday Soc. 42, 23 (1966).
  43. R. Jain and E. Ruckenstein, Stability of stagnant viscous films on a solid surface, J. Colloid Interface Sci. 54, 108 (1976).
  44. E. Ruckenstein and R. K. Jain, Spontaneous rupture of thin liquid films, J. Chem. Soc., Faraday Trans. 2 70, 132 (1974).
  45. Wolfram Inc., Mathematica, Version 12.3.1 (Wolfram, Champaign, IL, 2021).
  46. A. Vrij and J. Overbeek, Rupture of thin liquid films due to spontaneous fluctuations in thickness, J. Am. Chem. Soc. 90, 3074 (1968).
  47. R. Barrera, G. Estevez, and J. Giraldo, Vector spherical harmonics and their application to magnetostatics, Eur. J. Phys. 6, 287 (1985).
  48. S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability (Clarendon Press, Oxford, UK, 1961).
  49. A. Prosperetti, Viscous effects on perturbed spherical flows, Q. Appl. Math. 34, 339 (1977).

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Information

Phys. Rev. Fluids 9, 014004– Published 24 January, 2024
  • Received 11 July 2023
  • Accepted 14 November 2023
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