A hydrodynamic scaling theory for pair interaction of elastic particles

We propose a fluid lubricated extension of the Hertzian contact model supported by Finite Element Method (FEM) simulations to capture the deformation features of a pair of elastic particles interacting under constant external force, which could be due to an imposed axisymmetric compressional flow. We obtain simple scaling relations that capture the lateral dimension of the deformation (film radius), maximum dynamic pressure in the narrow gap, film thickness along the line joining the centers of the particles, minimum film thickness at the edge of the film, and the particle stresslet. The characteristic length scale chosen to obtain the scaling relations in the fluid film region is time-dependent but universal, independent of the particle stiffness. A comparative study of a pair of interacting elastic particles with the droplets, capsules, and vesicles system shows that the difference in scaling relations can be attributed to the differences in the origin of the lubrication pressure that develops in the narrow gap.