Vol.21, Special Issue, 2021, pp. S55–S63
UDC:

The problem of Stokes flow of a viscous fluid past a pair of separated solid spheres solved by Payne and Pell is revisited in this paper. Payne and Pell worked on the peripolar coordinate system, whereas we consider a bipolar system in this work. One impressive result of this study is that we derived an expression for the drag experienced by the system of two-spheres by modifying the expression that Payne and Pell gave for a general axisymmetric body. Further, this study gave rise to some interesting observations. Though one sphere's presence affects the other, the drag on the system is found equal to the sum of the drag on individual spheres. For spheres of equal radius, we computed the drag on each sphere using the formulae given by Stimson and Jeffery and found that it is precisely half the drag computed on the system. If the spheres are of unequal radius, we arrive at an empirical formula to compute bounds for each sphere's drag. These bounds include values calculated by Jeffery and Stimson in their work on the motion of two spheres in a viscous fluid. We also observe that the drag on the sphere facing the fluid flow first gets saturated at a value that equals the drag on the system with decreasing radius of the other (latter) sphere. Another remarkable feature of our work is that, as a limiting case, we derive the individual spheres' drag, and the values are in excellent agreement with those computed by Stokes formula for drag on a single sphere. Further to these, we have also carried out numerical evaluations for flow visualization and plots of pressure. 

Keywords: pair of separated spheres, bipolar coordinates, Gegenbaur functions, stream function, drag