Journal of Mathematical Fluid Mechanics

, Volume 20, Issue 4, pp 1681–1699 | Cite as

The Laplace Parallel Plates Problem in Capillarity Theory

  • Rajat Bhatnagar
  • Robert FinnEmail author


When two parallel plates of perhaps different materials are immersed vertically into a fluid bath in a vertically downward gravity field, they may experience an attracting or repelling force. Following a framework introduced by Laplace, we seek explication based on surface tension theory, which we use to correlate the forces with estimates for the height of the liquid surface. In this context, we provide (as special cases of more general estimates), an exact formula for the mean rise height in the channel formed by the plates, together with a strict bound for the variation of rise height within the channel. We base our procedures directly on the nonlinear formulation of the governing equations as formulated by Young and by Laplace, and introduce no further structural hypotheses. As support for this decision we present an explicit example based on our Theorem 5, displaying errors of unlimited magnitude that can arise from customary linearization procedures, and which can also obscure physical phenomena of central interest. As corollary of the development, we obtain also a new characterization of the classes of attracting and repelling solutions. We establish further an asymptotically exact form of the Laplace discovery, that aside from an isolated exceptional case of repelling forces bounded from infinity above (and, as we show, bounded also from zero below), the force between the plates always becomes attracting and grows as the inverse square of the separation distance, as the plates are brought together with fixed contact angles. We continue our efforts, initiated by us in earlier publications, toward clarifying this exotic behavior in quantitative terms.


Surface tension Capillarity Laplace parallel plates configuration Attracting and repelling parallel plates 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The second author is indebted to the Mathematische Abteilung der Universität, in Leipzig, for its hospitality and for excellent working conditions, and also to the Max–Planck–Institut MIS for additional support. He is indebted to the Simons Foundation for a very helpful grant facilitating his research. He thanks Erich Miersemann for incisive comments on some conceptual points related to the problem considered, and Stephan Luckhaus for his interest and for help in many ways. Both authors thank Thomas Vogel for suggestions leading to improvements in exposition.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Young, T.: An essay on the cohesion of fluids. Philos. Trans. R. Soc. Lond. 95, 65–87 (1805)ADSCrossRefGoogle Scholar
  2. 2.
    Laplace, P.S.: Traité de mécanique céleste, Oeuvres complète, Vol. 4. Gauthier-Villars, Paris, 1805, Supplément 1, livre X, pp. 771–777; and Traité de mécanique céleste, Oeuvres compléte, Vol. 4 Gauthier-Villars, Paris, 1806, Supplément 2, Livre X pp. 909–945. See also the annotated English translation by N. Bowditch (1839), reprinted by Chelsea, New York (1966)Google Scholar
  3. 3.
    Bakker, G.: Handbuch der Experimentalphysik. Akademische, Leipzig (1928)Google Scholar
  4. 4.
    Poincaré, H.: Capillarité. Leçons professées pendant le deuxième semester 1888–1889, rédigées par J. Blondin Gauthier-Villars, Paris (1895)Google Scholar
  5. 5.
    Poynting, J.H., Thomson, J.J.: A Text-book of Physics: Properties of Matter, 4th edn. C. Griffin, Chicago (1907)Google Scholar
  6. 6.
    Steinberg, M.: Reconstruction of tissues by dissociated cells. Science 141, 401–408 (1963)ADSCrossRefGoogle Scholar
  7. 7.
    Bowden, N., Choi, I.S., Grzybowski, B.A., Whitesides, G.M.: Meso-scale self-assembly of hexagonal plates using Lateral capillary forces: synthesis using the ‘Capillary Bond’. J. Am. Chem. Soc. 121, 5373–5391 (1999)CrossRefGoogle Scholar
  8. 8.
    Bowden, N., Oliver, S., Whitesides, G.M.: Mesocale self-assembly: capillary bonds and negative menisci. J. Phys. Chem. B 104, 2714–2724 (2000)CrossRefGoogle Scholar
  9. 9.
    Wolfe, D.B., Snead, A., Mao, C., Bowden, N.B., Whitesides, G.M.: Mesoscale self- assembly: capillary interactions when positive and negitive menisci have similar amplitudes. Langmuir 19, 2206–2214 (2003)CrossRefGoogle Scholar
  10. 10.
    Vella, D., Mahadevan, L.: The “Cheerios effect”. Am. J. Phys. 73(9), 817–825 (2005)ADSCrossRefGoogle Scholar
  11. 11.
    Bhatnagar, R., Finn, R.: Attractions and repulsions of parallel plates partially immersed in a liquid bath: III. Bound. Value Probl. 2013, 1 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Bhatnagar, R., Finn, R.: The force singularity for partially immersed parallel plates. J. Math. Fluid Mech. 18, 739 (2016)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Bhatnagar, R., Li, W., Cho, S.K., Finn, R.: Adjoin Same Issue JGoogle Scholar
  14. 14.
    Finn, R.: Capillary graph-interfaces “in the Large”. Vietnam Math. J. (in press)Google Scholar
  15. 15.
    Finn, R.: On Young’s paradox, and the attractions of immersed parallel plates. Phys. Fluids 22, 017103 (2010)ADSCrossRefGoogle Scholar
  16. 16.
    Gyemant, A.: Kapillarität. In: Handbuch der Physik, Springer, Berlin (1927)CrossRefGoogle Scholar
  17. 17.
    Aspley, A., He, C., McCuan, J.: Force profiles for parallel plates partially immersed in a liquid bath. J. Math. Fluid Mech. 17, 87–102 (2015)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Poisson, S.D.: Nouvelle théorie de l’action capillaire. Bachelier, Paris (1831)Google Scholar
  19. 19.
    Finn, R., Lu, D.: Mutual attractions of partially immersed parallel plates. J. Math. Fluid Mech. 15(2), 273–301 (2013)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Bhatnagar, R., Finn, R.: Addenda to the preceding paper. J. Math. Fluid Mech. 18, 757 (2016)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Verge GenomicsSan FranciscoUSA
  2. 2.Mathematics DepartmentStanford UniversityStanfordUSA

Personalised recommendations