Relative Trajectories of Contaminated, Spherical Drops in a Temperature Gradient and Gravity at Finite Stokes Numbers

Abstract

This work is a theoretical investigation into the effect of finite droplet inertia on combined gravitational and thermocapillary interactions of spherical drops covered with an incompressible surfactant film. The significance of droplet inertia is indicated by the magnitude of the Stokes number St. Initial calculations are a continuation of previous results from St = 0 to finite Stokes numbers at \(\varvec{O(1)}\) drop-to-medium viscosity and thermal conductivity ratios. Interesting outcomes, such as stable tandem motion and complex relative trajectories, are observed. At more realistic \(\varvec{O(10)}\) ratios, the results tend to be dampened from those at lower values, although unusual behavior still occurs. Finally, interactions are determined for two physical systems, water drops in air and mercury drops in n-pentane. At normal gravity, two limits are usually possible. In order for the thermocapillary and gravitational contributions to be equal, the drops and their corresponding Stokes numbers must be small, while for larger drops at higher St, gravity is the dominant driving force. For water droplets in air with radii less than 10 \(\varvec{\mu }\)m, van der Waals forces control the interactions. However, drop inertia is the most important factor for droplets with radii greater than 25 \(\varvec{\mu }\)m. Even a small thermocapillary effect can have noticeable consequences on the relative trajectories for intermediate-sized drops. Some comments are made on the difficulty in experimentally reproducing the theoretical results, with a recommendation of centi- or milligravity conditions.