Translation of two rigid spheres perpendicular to their line-of-centers and normal to a plate

The translation of two identical rigid spheres perpendicular to their line-of-centers normal to a rigid plate is analyzed theoretically for Reynolds number ranges from 0.1 to 40. The geometry considered allows us to examine simultaneously the effect of the presence of a boundary and that of nearby particles on the translation of a particle. We show that the presence of the plate has a significant influence on the flow field near the spheres, especially when Reynolds number is low. Due to the competition between the nozzle effect and the sphere–sphere interaction, the degree of the boundary effect on the drag coefficients of the spheres has a local minimum as the separation distance between two particles varies. In addition, the deviation of the ln(drag coefficient)–ln(Reynolds number) curve from a Stokesʹ-law-like relation may also have a local minimum as the separation distance between two spheres varies. An empirical relation is proposed to correlate the drag coefficient with the key parameters of the present problem for the case where Reynolds number is smaller than unity.