Non-continuum hydrodynamic interactions between settling inertial droplets

As cloud droplets coalesce, the air squeezed out of the gap gives rise to a continuum lubrication force that diverges with decreasing separation, thus preventing collisions from taking place. For separations comparable to the mean free path of the gas, the non-continuum nature of the force leads to collisions in finite time. This work is a continuation of Sundararajakumar & Koch (J.Fluid M.313:283-308,1996) who analyzed the non-continuum lubrication force for the normal motion of two spheres. We extend that work to include tangential motions over the full range of Knudsen numbers (ratio of the gas mean-free path to the average droplet radius) as a function of the gap thickness. The hydrodynamic resistivity functions are obtained using a uniformly valid approximation that includes non-continuum lubrication forces at small separations and continuum hydrodynamic interactions at larger separations. These hydrodynamic forces are used to calculate the collision rate of inertial droplets settling without a background flow. Using trajectory analysis, the collision efficiency is found for different droplet size ratios, Knudsen numbers and Stokes numbers. Eventually, this analysis will be combined with direct numerical simulations to incorporate the effects of fluid turbulence.