Hydrodynamic interaction of particles in a viscous fluid

The equation of motion for a cluster of spheres moving in fluid at low Reynolds numbers is developed under the consideration of hydrodynamic interaction among the spheres. The hydrodynamic forces of the spheres in the equation of motion are estimated using the relative velocities. The relative velocities are derived by using the superposition method of Oseen's flow field produced by the translation of each sphere. Experiments for a cluster of equal spheres falling freely in quiescent glycerol at low Reynolds numbers are carried out using a large test tank. The numbers of equal spheres are 3 similar to 12 on a horizontal line and 4x4 similar to 10x10 on a vertical plane. Each spacing between two neighbor spheres is equal to the sphere diameter. The numerical results agree well with the experimental results. It is concluded that the dispersive motions of spheres are dependent on whether the sphere number is odd or even.