Unsteady motion of two solid spheres in Stokes flow

Abstract

This study is concerned with the unsteady motion of two solid spherical particles in an unbounded incompressible Newtonian flow. The background flow is uniform and can be time dependent. In addition, the particle Reynolds numbers 2aVa∕ν and 2bVb∕ν, based on characteristic particles velocities Va and Vb, are assumed to remain small throughout the motion. Here, a and b denote the particle radii and ν is the kinematic viscosity of the fluid. Two approximate methods are employed in order to calculate the unsteady force exerted on each particle. In the first approach, a simplified method of reflections in combination with the point-force method is employed. In the second approach, a simplified method of reflections combined with Burger’s unsteady flow solution is considered. The forces due to the background flow and the disturbed flow created by the presence of particles are treated separately. The equation of motion for each particle is derived and some special cases are presented in detail including the motion with constant acceleration and the motion in a gravitational field. The results indicate that using the Basset force corresponding to the motion of two spheres gives rise to a larger drag force as compared to the solution utilizing the solitary-particle Basset force.

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

“…For dilute suspensions, the unsteady motion of two solid spheres has also been analyzed [9]. However, for intermediate Rey- nolds numbers the use of numerical simulation is generally unavoidable.…”

Two spheres in a free stream of a second-order fluid

“…4 More recently, the unsteady motion of solid spheres and their collisions have been studied by Ardekani and Rangel. 5,6 In this study, the forces acting on two spherical particles in a second-order fluid are investigated.…”

A boundary integral method for computing forces on particles in unsteady Stokes and linear viscoelastic fluids

“…We compare our numerical results with an existing asymptotic solution for motion of two particles in unsteady Stokes flow, which was derived for the case of large separation of the particles [45]. For studies of two-particle microrheology, the two particles need to be sufficiently separated (at least one diameter apart), in contrast to studies of binding kinetics of approaching particles.…”

“…The results are in excellent agreement with the analytical solution, demonstrating that the method is convergent, efficient, and highly accurate. The highly accurate boundary integral result, which serves as a nearly exact solution for forces acting on particle, is used to evaluate the accuracy of an asymptotic solution [45]. The asymptotic solution is derived in the limit as goes to 0 (particles infinitely far from each other), where the nondimensional parameter measures the ratio of the particle size and the particle separation.…”