Rectilinear oscillations of two spherical particles embedded in an unbounded viscous fluid

The problem of the rectilinear oscillations of two spherical particles along the line through their centers in an axi-symmetric, viscous, incompressible flow at low Reynolds number is considered. The particles oscillate with the same frequency and with different amplitudes. In addition, the particles may differ in their sizes. In order to solve the Stokes equations for the flow field, a general solution is constructed from the superposition of the basic solutions in the two spherical coordinate systems based at the centers of the particles. A collocation technique is used to satisfy the boundary conditions on the surfaces of the particles. The solution is valid for all values of the frequency parameter subject to the conditions that justify the use of the unsteady Stokes equations. Numerical results displaying the in phase and the out of phase force amplitudes acting on each particle are obtained with good convergence for various values of the physical parameters of the problem. The results are tabulated and represented graphically. Our results agree well with the existing solutions of the steady motion of two spherical particles and with the oscillations of single particle.