| Abstract: | The time-dependent Stokes equations were solved in the vicinity of two spheres colliding in a viscous fluid with viscosity ν to determine the rate of change of the hydrodynamic forces during large accelerations associated with Hertzian mechanical contact of small duration \({\tau_{\rm c}}\). It was assumed that the gap clearance remains finite during contact and is approximately equal to the height σ of surface micro-asperities. The initial condition corresponds to the steady-state axisymmetric solution of Cooley and O’Neill (Mathematika 16:37–49, 1969), and the initial value problem for the time-dependent Stokes streamfunction was solved using Laplace transform methods. Assuming that σ is small compared to the sphere radius a, we used singular perturbation expansions and tangent-sphere coordinates to obtain an asymptotic solution for the viscous flow in the gap and around the moving sphere. The solution provides the dependence of the resistance, added mass and history forces on σ, the sphere velocity and acceleration, and the ratio of the sphere diameters. We found that the relative importance of viscous and mechanical forces during contact depends on a new Stokes number \({St_{\rm c}=\sigma^2/\nu \tau_{\rm c}}\). Integration of Newton’s equation for the motion of the sphere during mechanical contact showed that there is a critical \({St_{\rm c}=O(\sigma/a)}\) for which there is no rebound at the end of contact. |
