Gravitational Interactions of Small Contaminated Drops ina Temperature Gradient at Finite Stokes Numbers

Relative trajectories are calculated for two sedimenting spherical drops in the presence of a vertical temperature gradient with exact methods for determining the hydrodynamic forces at finite Stokes number and low Reynolds number. The drops are covered with incompressible surfactant, and thermal convection and Brownian motion are negligible. When the Reynolds number is small, fluid inertia is negligible, and the hydrodynamic forces are linear functions of the translational velocities of the drops. However, at nonzero Stokes numbers, drop inertia must be taken into account, and the hydrodynamic forces do not balance the applied forces. For drops in close approach, lubrication forces and attractive molecular forces are considered. In the absence of attractive molecular forces, inertia leads to asymmetry in the drops’ relative trajectories. Interesting behavior, such as two drops moving in stable, tandem motion with a finite gap, is possible in the dimensionless parameter space. Moreover, retrograde motion is observed in the numerical results, depending on the relative strength of the thermocapillary and gravitational driving forces. An important application is to raindrop growth, where the model is applicable to water droplets with radii between 10 and 30 μm.