The collection kernel for the coalescence of water droplets

Dimensional analysis is used to determine the general functional behaviour of the collection kernel, which is the rate of increase in the swept-out volume of air by one water droplet relative to another falling through still air. By considering the symmetry and analyticity of the kernel with respect to the gravitational acceleration, the detailed behaviour of the kernel for small droplets is found. The analysis is extended to the case of droplets falling through a uniform vertical shear flow which is shown to increase the collection kernel particularly for droplets of comparable size.