Droplet motion in a predefined line vortex undergoing stretching is governed by viscosity and gravity forces only, so Stokes equation to examine droplet trajectory is used. Both analytical and numerical solutions of Stokes equation allow to identify features such as stationary points, stationary orbits and limit cycles, which may influence preferential concentration and/or collisions.
Stationary orbits exist in vortices aligned with gravity only and depend on all system parameters: droplet mass, vortex stretching strength and circulation. In contrast, such orbits do not exist in oblique vortices, but it is concluded from numerical simulations that under certain conditions droplets are attracted by limit cycles and/or by stationary points. Condition of existence of stationary point can be obtained analytically. For a given vortex circulation and stretching rate, stationary point can exist for the deflection from vertical exceeding certain critical value and for droplets exceeding certain critical radius. Stationary behavior is observed for two components of motion in a plane perpendicular to vortex axis only.
Simulations of motion of numerous droplets of the same size, distributed homogeneously in space were conducted in an oblique vortex. For different sets of parameters these simulations showed appearance of stationary points, limit cycles or both. Figure shows droplets initiated in a a plane stable with respect to the motion along the axis (in such a case motion remains two-dimensional) and in unstable plane, where droplets move in third dimension along the vortex axis.
Bajer, K., K.M. Markowicz, S.P. Malinowski, 2000: Influence of the small-scale turbulence structure on the concentration of cloud droplets. Proceedings of 13th Conference on Clouds and Precipitation, 159-162, IAMAP, Reno, Nevada, USA, 14-18 August 2000.
Hill., R.J., 2005: Geometric collision rates and trajectories of cloud droplets falling into a Burgers vortex. Phys. Fluids, 17, 037103, 1-23.
Figure. Trajectories of 36 droplets of radius R=10 Î¼m distributed uniformly at the t=0 on a rectangle of a size 8cm a) in an unstable stationary plane b) below the plane.