The hydrodynamic interaction of a particle pair sedimenting in a horizontal shear flow

Collisional dynamics of particles suspended in air have significance in many industrial and natural processes. We calculate the collision rate of a particle pair sedimenting in a simple shear flow where gravity acts in the gradient direction of the simple shear flow. For particles approaching each other in a gaseous medium, continuum lubrication approximation becomes no longer valid, and non-continuum lubrication interactions become important. The Knudsen number, defined as the ratio of the mean free path of the medium to the mean radius of the two interacting spheres, measures the strength of non-continuum effects. Typical pair trajectories for simple shear flow alone or gravity alone are well-studied. We report the non-trivial topology of the pair trajectories for this coupled problem. We evaluate the ideal collision rate as a function of the relative strength of gravity and uniaxial compressional flow and find that it deviates significantly from a linear superposition of these driving terms. By performing trajectory analysis, we show how collision efficiencies vary with the size ratio, the Knudsen number, and the dimensionless parameters capturing the strength of gravity and simple shear flow.