Unsteady motion of two solid spheres in Stokes flow

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 2 a V a ∕ ν and 2 b V b ∕ ν , based on characteristic particles velocities V a and V b , 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.