Gravitational Interactions of Two Small Evaporating Drops

The effect of evaporation on relative trajectories of two spherical drops sedimenting due to gravity in air is investigated. Theoretical analysis and numerical simulation of the interactions are used to obtain results. Several assumptions are made in the model. The drops remain small enough that the Reynolds number, which is linear in the density of the surrounding fluid, is negligible. However, the Stokes number, which is proportional to droplet mass and is a measure of drop inertia, can be much larger than one. Another restriction is that evaporation is dominated by diffusion and that convection of mass is insignificant. In analyzing evaporation when two drops are present, it is noted that the loss of mass is not the same at each point on a droplet surface. That is, evaporation is non-uniform in a spatial sense. In order to maintain the required spherical drop shape, three approaches, involving the isolated droplet and bispherical coordinate solutions, were taken to determine the mass flux due to evaporation and subsequently the drop position at each time step. For a pair of water drops with radii between 2 and 30 \(\upmu\)m, the following conclusions were obtained. In all three methods, evaporation leads to weaker inertial effects and stronger hydrodynamic effects. Most importantly, in comparing critical horizontal offsets, when both attractive molecular forces and Maxwell slip are considered, all three approaches to evaporation lead to similar results, making the choice of method nearly inconsequential.