The Forces on Two Spheres in Steady Stokes Flow.

An analysis of the hydrodynamic forces on cloud droplets with radii less than 30 microns near each other in a viscous fluid. Previous work has solved the problem for velocities parallel to the line of centers of the droplets; the present study considers the far more complex case of motion perpendicular to that line. The Stokes linearized hydrodynamic theory is used, extending Hocking's analysis of nonrotating spheres to spheres that are assumed to rotate freely enough to nullify the torques that would be produced by fluid shear. Velocity expressions were constructed in terms of spherical harmonics centered on each of the spheres. The linear system of an infinite number of algebraic equations in an infinite number of unknowns was truncated to finite size. The forces and torques were evaluated, and the forces were superposed, according to the Stokes procedure, to obtain the force under a condition of rotation such that the torque on each sphere was zero. Reasonably reliable force calculations are feasible at separations down to 0.1 of the radius of the larger droplet; beyond that, the results are numerically unstable. Above 0.5 separation, results are the same as Hocking's approximations. 25 pp.