Bifurcations in droplet collisions

Collision rates of droplets in straining and vortical flows exhibit closed trajectories which start and end on the collision spheres. These closed trajectories reduce collision rates by reducing the collision sphere area at which trajectories approaching from large separations can collide. Further, neutral droplet dynamics in the presence of flow gradients, and charged droplet dynamics in a quiescent flow exhibit trajectories where the large droplet overtakes the smaller one before collision. Yet, the mechanisms describing the appearance of such closed trajectories and overtaking trajectories are not understood. We use dynamical systems theory to explain that closed trajectories appear due to the presence of fixed points in the relative droplet dynamics. Finally, we explain the appearance of overtaking trajectories using bifurcation theory.