Revisiting Batchelor & Green's study of the hydrodynamic interaction of two spheres in a simple shear flow for aerosol applications

In their pioneering study, Batchelor & Green (J. Fluid Mech. 1972) provided a comprehensive picture of pair trajectories for inertialess non-Brownian spheres in a simple shear flow. They showed that the pair trajectories comprise two families, open and closed, separated by a separatrix. However, all their trajectories are non-colliding. Continuum lubrication forces prevent particles from touching, and collisions occur only with the inclusion of attractive non-hydrodynamic forces (for example, the van der Waals force). In a gaseous medium, the lubrication forces are weaker than their continuum counterpart and thus allow for particle contact in finite time. Our study revisits this classical study and explains the collision dynamics of small solid/liquid spherical particles in gaseous media subjected to a simple shear flow, incorporating various physics that enable collision; the non-continuum lubrication forces being the primary one. We also highlight that when non-continuum effects are weak, the collisional dynamics of two spheres in a simple shear flow significantly differ from other linear flows with purely open pair trajectories. Finally, we discuss the modification to the pair trajectory topology and the collision rate when the role of particle inertia is included in a perturbative manner.