Improved formulations of the superposition method

Two formulations of an improved superposition method are proposed for studying droplet-droplet hydrodynamic interactions. The formulations make explicit use of the boundary conditions on the surface of the two interacting droplets. The improved formulations are described through a consistent and rigorous consideration of the relationship between the drag force and representation of disturbance flows. It is demonstrated that the improved formulations are much more accurate than the original implementation of the superposition method. Specifically, for the case of Stokes disturbance flows, the relative errors on the drag force can be reduced by one order of magnitude using the improved formulations, when compared with the original formulation, for situations when the lubrication effect is not dominant. Using the improved superposition method, collision efficiencies of small cloud droplets falling in calm air are also computed and compared with previously published results.