Buoyancy-driven attraction of active droplets

For dissolving active oil droplets in an ambient liquid, it is generally assumed that the Marangoni effect results in repulsive interactions, while the buoyancy effects caused by the density difference between the droplets, diffusing product and the ambient fluid are usually neglected. However, it has been observed in recent experiments that active droplets can form clusters due to buoyancy-driven convection (Krüger et al. Eur. Phys. J. E, vol. 39, 2016, pp. 1-9). In this study, we numerically analyze the buoyancy effect, in addition to the propulsion caused by Marangoni flow (with its strength characterized by Péclet number Pe). The buoyancy effects have their origin in (i) the density difference between the droplet and the ambient liquid, which is characterized by Galileo number Ga, and (ii) the density difference between the diffusing product (i.e. filled micelles) and the ambient liquid, which can be quantified by a solutal Rayleigh number Ra. We analyze how the attracting and repulsing behaviour of neighbouring droplets depends on the control parameters Pe, Ga, and Ra. We find that while the Marangoni effect leads to the well-known repulsion between the interacting droplets, the buoyancy effect of the reaction product leads to buoyancy-driven attraction. At sufficiently large Ra, even collisions between the droplets can take place. Our study on the effect of Ga further shows that with increasing Ga, the collision becomes delayed. Moreover, we derive that the attracting velocity of the droplets, which is characterized by a Reynolds number Re_d, is proportional to Ra^1/4/(ℓ/R), where ℓ/R is the distance between the neighbouring droplets normalized by the droplet radius. Finally, we numerically obtain the repulsive velocity of the droplets, characterized by a Reynolds number Re_rep, which is proportional to PeRa^-0.38. The balance of attractive and repulsive effect leads to Pe ~ Ra^0.63, which agrees well with the transition curve between the regimes with and without collision.