Hydrodynamic interactions between two spheres at contact

Abstract

This paper is an introductory guide to many-particle hydrodynamic interactions. Basic concepts of the fluid mechanics are assumed to be known. Experience in the Stokes equations is useful but not necessary. The study is estimated to fit five sessions about three hours each.

Collision of multi-particle and general shape objects in a viscous fluid

“…Ekiel-Jezewska et al [4,5] used the roll/slip model of contact for interaction between two spherical particles immersed in a viscous fluid. Comparison of the model with experimental results for the settling motion of a sphere in the vicinity of another fixed sphere showed good agreement.…”

Basic Concepts of Stokes Flows

“…Under the time reversal, the gravitational force is reversed and the sphere centre moves backwards along the same trajectory. However, reversibility of the trajectories is broken when two spheres come so close to each other that their surfaces interact by direct forces, such as van der Waals attraction or mechanical reaction of rough surfaces at the contact [35][36][37][38][39]. The reason is that central direct forces are not symmetric with respect to superposition of the time reversal with reflection in the horizontal plane z = 0.…”

Stokesian dynamics of close particles

“…Such a scenario sometimes involves very close configurations of the spheres, with the distance between the sphere surfaces smaller than 10 −2 radii, for example when a sphere falls under gravity onto the other one held fixed or moving slower almost exactly below. [14,15]. The particles stay close to each other, but for a short time only, and later on the interparticle distances grow indefinitely.…”

“…In a number of systems, motion of several spheres was detected experimentally or evaluated by the Stokesian dynamics, and some very close configurations of the spheres were observed, with the distance between the sphere surfaces smaller than 10 −2 radii [14][15][16]. A lot of examples of 'scattering processes' is known, during which very small gaps between the particle surfaces are reached and then particles again separate -for example, a sphere falling under gravity onto the other one held fixed or moving slower below [14,15]. The question arises if there exist stable systems of a different nature -with at least three spheres staying very close to each other for a long time.…”