Applied Mathematics and Mechanics

, Volume 37, Issue 12, pp 1597–1606 | Cite as

Exact solution for capillary interactions between two particles with fixed liquid volume

  • Fengxi ZhouEmail author
  • Qiang Ma


The capillary interactions, including the capillary force and capillary suction, between two unequal-sized particles with a fixed liquid volume are investigated. The capillary interaction model is used within the Young-Laplace framework. With the profile of the meridian of the liquid bridge, the capillary suction, and the liquid volume as state variables, the governing equations with two-fixed-point boundary are first derived using a variable substitution technique, in which the gravity effects are neglected. The capillary suction and geometry of the liquid bridge with a fixed volume are solved with a shooting method. In modeling the capillary force, the Gorge method is applied. The effects of various parameters including the distance between two particles, the ratio of particle radii, and the liquid-solid contact angles are discussed.

Key words

liquid bridge Young-Laplace equation fixed liquid volume shooting method 

Chinese Library Classification


2010 Mathematics Subject Classification

76B45 76D45 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Kralchevsky, P. and Nagayama, K. Particles at Fluid Interfaces and Membranes, Elsevier, Amsterdam (2001)Google Scholar
  2. [2]
    Payam, A. F. and Fathipour, M. A capillary force model for interactions between two spheres. Particuology, 9, 381–386 (2011)CrossRefGoogle Scholar
  3. [3]
    Haines, W. B. Studies in the physical properties of soils II: a note on the cohesion developed by capillary forces in an ideal soil. The Journal of Agricultural Science, 15, 529–535 (1925)CrossRefGoogle Scholar
  4. [4]
    Haines, W. B. Studies in the physical properties of soils IV: a further contribution to the theory of capillary phenomena in soil. The Journal of Agricultural Science, 17, 264–290 (1927)CrossRefGoogle Scholar
  5. [5]
    Fisher, R. A. On the capillary forces in an ideal soil: correction of formulae given by WBHaines. The Journal of Agricultural Science, 16, 492–505 (1926)CrossRefGoogle Scholar
  6. [6]
    Fisher, R. A. Further note on the capillary forces in an ideal soil. The Journal of Agricultural Science, 18, 406–410 (1928)CrossRefGoogle Scholar
  7. [7]
    Orr, F. M., Scriven, L. E., and Rivas, A. P. Pendular rings between solids: meniscus properties and capillary force. Journal of Fluid Mechanics, 67, 723–742 (1975)CrossRefzbMATHGoogle Scholar
  8. [8]
    Lechman, J. and Lu, N. Capillary force and water retention between two uneven-sized particles. Journal of Engineering Mechanics, 134, 374–384 (2008)CrossRefGoogle Scholar
  9. [9]
    Chen, Y. C., Zhao, Y. Z., Gao, H. L., and Zheng, J. Y. Liquid bridge force between two unequalsized spheres or a sphere and a plane. Particuology, 9, 374–380 (2011)CrossRefGoogle Scholar
  10. [10]
    Molenkamp, F. and Nazemi, A. H. Interactions between two rough spheres, water bridge and water vapour. Geotechnique, 53, 255–264 (2003)CrossRefGoogle Scholar
  11. [11]
    Mu, F. and Su, X. Analysis of liquid bridge between spherical particles. China Particuology, 5, 420–424 (2007)CrossRefGoogle Scholar
  12. [12]
    Soulié, F., Cherblanc, F., El-Youssoufi, M. S., and Saix, C. Influence of liquid bridges on the mechanical behaviour of polydisperse granular materials. International Journal for Numerical and Analytical Methods in Geomechanics, 30, 213–228 (2006)CrossRefzbMATHGoogle Scholar
  13. [13]
    Dormann, M. and Schmid, H. J. Simulation of capillary bridges between nanoscale particles. Langmuir, 30, 1055–1062 (2014)CrossRefGoogle Scholar
  14. [14]
    Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed., Academic Press, San Diego (1992)Google Scholar
  15. [15]
    Rabinovich, Y. I, Esayanur, M. S., and Moudgil, B. M. Capillary forces between two spheres with a fixed volume liquid bridge: theory and experiment. Langmuir, 21, 10992–10997 (2005)CrossRefGoogle Scholar
  16. [16]
    Hotta, K., Takeda, K., and Iinoya, K. The capillary binding force of a liquid bridge. Powder Technology, 10, 231–242 (1974)CrossRefGoogle Scholar
  17. [17]
    Lian, G., Thornton, C., and Adams, M. J. A theoretical study of the liquid bridge forces between two rigid spherical bodies. Journal of Colloid and Interface Science, 161, 138–147 (1993)CrossRefGoogle Scholar
  18. [18]
    Keller, H. B. Numerical Solution of Two Point Boundary Value Problems, SIAM, Philadelphia (1976)CrossRefGoogle Scholar
  19. [19]
    Zhou, F. X., Li, S. R., and Lai, Y. M. Three-dimensional analysis for transient coupled thermoelastic response of a functionally graded rectangular plate. Journal of Sound and Vibration, 330, 3990–4001 (2011)CrossRefGoogle Scholar

Copyright information

© Shanghai University and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.School of Civil EngineeringLanzhou University of TechnologyLanzhouChina

Personalised recommendations