Bifurcations in droplet collisions

Saffman and Turner [P. G. Saffman and J. S. Turner, J. Fluid Mech. 1, 16 (1956)] argued that the collision rate for droplets in turbulence increases as the turbulent strain rate increases. However, the numerical simulations of Dhanasekaran et al. [J. Dhanasekaran et al., J. Fluid Mech. 910, A10 (2021)] in a steady straining flow showed that the Saffman-Turner model is oversimplified because it neglects droplet-droplet interactions. These result in a complex dependence of the collision rate on the strain rate and on the differential settling speed. Here we show that this dependence is explained by a sequence of bifurcations in the collision dynamics. We compute the bifurcation diagram when strain is aligned with gravity and show that it yields important insights into the collision dynamics. First, the steady-state collision rate remains nonzero in the limit Kn -> 0, contrary to the common assumption that the collision rate tends to zero in this limit (Kn is a nondimensional measure of the mean free path of air). Second, the nonmonotonic dependence of the collision rate on the differential settling speed is explained by a grazing bifurcation. Third, the bifurcation analysis explains why so-called closed trajectories appear and disappear. Fourth, our analysis predicts strong spatial clustering near certain saddle points, where the effects of strain and differential settling cancel.