Asymptotic analysis for hydrodynamic force acting on stiff particles

TL;DR

This paper develops explicit asymptotic formulas for the hydrodynamic force on stiff particles in viscous fluids as they approach contact, revealing the dominant squeeze motion effect and providing rigorous justification.

Contribution

It introduces explicit singular functions and asymptotic formulas for hydrodynamic forces in near-contact regimes, with a rigorous variational proof.

Findings

Hydrodynamic force exhibits blow-up as particles approach contact.

Squeeze motion dominates the singular behavior of force.

Explicit formulas accurately describe force asymptotics.

Abstract

A three-dimensional mathematical model of a viscous incompressible fluid with two stiff particles is investigated in the near-contact regime. When one of the particles approaches the other motionless particle with prescribed translational and angular velocities, there always appears blow-up of hydrodynamic force exerted on the moving particle. In this paper, we construct explicit singular functions corresponding to the fluid velocity and pressure to establish precise asymptotic formulas for hydrodynamic force with respect to small interparticle distance, which show that its largest singularity is determined by squeeze motion between two particles. Finally, the primal-dual variational principle is employed to give a complete justification for these asymptotics.