Collision efficiency of rapidly settling particles in a turbulent flow

We calculate the collision rate constant of hydrodynamically interacting low inertia spherical particle pairs sedimenting in a homogeneous isotropic turbulent flow in the rapid settling limit where the settling time across a Kolmogorov eddy is short compared with Kolmogorov time scale. Due to the sub-Kolmogorov particle sizes, we approximate the fluid motion in the vicinity of the particle pair locally as a linear flow with a fluctuating velocity gradient that appears from the background turbulent flow. The response of the relative particle position is small over the correlation time of the flow and therefore, a diffusive process characterizes the relative motion with a diffusivity $D_{ij}^H$ and the hydrodynamic interactions lead to a net drift, $V_i^H$, toward small inter-particle separations. The drift-diffusion fluxes are expressed in terms of the velocity gradient auto-correlation function along the settling trajectory and in this particular problem due to rapid settling assumption we are able to relate it with the turbulence energy spectrum. A convection-diffusion equation for pair probability density function, $P(\textbf{r}, t)$, is derived in terms of the hydrodynamic turbulent pair diffusion and drift and then solved numerically to calculate the collision rate constant.