Lubrication theory and boundary element hybrid method for calculating hydrodynamic forces between particles in near contact

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Suspensions of small particles are ubiquitous; examples include aqueous dispersions of solid particles, vesicles, capsules, cells, swimming microorganisms and artificial microswimmers. The rheological and diffusion properties of small particle suspensions are strongly influenced by near-field interactions between particles. Lubrication theory (LT) is a classical analytical technique used to obtain asymptotic solutions of forces and torques acting in the lubrication region. The boundary element method (BEM), on the other hand, is a computational method for accurately calculating Stokes flow around particles. To dramatically improve the near-field accuracy of the BEM, a novel hybrid method combining LT and BEM (LT-BEM) is proposed in this study, in which the inner solution is obtained by LT, whereas the outer solution is obtained by BEM. The validity of the LT-BEM was confirmed based on the shearing, rotational, and squeezing motions of two spheres. The asymptotic nature of the forces and torques exerted on the two spheres were efficiently captured by the LT-BEM. Especially, the squeezing force of the BEM was improved dramatically by the LT-BEM, which is important to prevent particle overlap. The obtained results can be expanded to the arbitrary motions of many spheres without loss of generality. Moreover, the LT-BEM can accommodate particles with arbitrary shapes. The advantages of the LT-BEM are demonstrated for two kinds of applications; (i) dynamic interactions of pairwise microswimmers with surface velocities, and (ii) pairwise interactions of bacteria. The obtained knowledge should be useful for computing various suspensions of small particles in nature and industrial applications.

Original languageEnglish
Article number110913
JournalJournal of Computational Physics
Publication statusPublished - 2022 Mar 1


  • Boundary element method
  • Hydrodynamic interactions
  • Lubrication
  • Microswimmer
  • Stokes flow
  • Suspension


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