Gravitational collision efficiencies of small viscous drops at finite Stokes numbers and low Reynolds numbers

Abstract

Collision efficiencies are calculated for two sedimenting spherical drops of different radii with internal circulation under conditions such that drop inertia can be significant while the surrounding fluid inertia remains negligible. A trajectory analysis is employed with exact methods for determining the hydrodynamic forces. When the Reynolds number is small, fluid inertia is negligible, and the hydrodynamic forces are linear functions of the 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, slip and attractive molecular forces are considered. Comparison is made between the effects of unretarded and retarded van der Waals forces and Maxwell slip on collision efficiencies. Slip is calculated exactly for viscous drops using bispherical coordinates for motion along the drops’ line of centers. For water droplets in air, at drop radii between 10 and 30μm, drop inertia is important while the Reynolds based on the surrounding air is still small. The collision efficiency goes through a minimum and then approaches the Smoluchowski limit of no hydrodynamic interactions as the drop size and Stokes number become increasingly large. Viscous effects are most important at drop radii of about 15μm. For drops with a radius less than 10μm, the most significant factors are unretarded van der Waals forces, Maxwell slip, and retarded van der Waals forces, respectively. When van der Waals forces and slip are combined, the collision efficiencies for drops with radii of 10μm are significantly greater than the results of previous researchers. At an intermediate drop radius of 20μm, surrounding fluid inertia, as considered in earlier work, appears to lead to greater collision efficiencies at intermediate size ratios, while van der Waals forces, incorporated in the present study, cause larger values at smaller and larger size ratios. At a drop radius of 30μm, drop inertia dominates interactions, and similar results occur for both the current and prior investigations. Since temperature affects both density and viscosity, collision efficiencies can differ by a factor of two or more between 0 °C and 100 °C at certain drop radii. Change in the viscosity ratio itself leads to about a 7% difference in the collision efficiency between drops with internal circulation and solid spheres at room temperature.