Analytical investigation of the combined effect of fluid inertia and unsteadiness on low-Re particle centrifugation

We analyze the explicit contribution of fluid inertia and fluid unsteadiness to the force acting on a solid sphere moving in a vertical solid-body rotation flow, in the limit of small Reynolds and Taylor numbers. This problem can be thought of as a test case where the flow induced by the particle is both unsteady (in the laboratory frame) and convected by the unperturbed flow. Many authors assume that the contributions of these two effects can be approximately superposed, and postulate that the particle motion equation is composed of the classical Boussinesq-Basset-Oseen equation (obtained by neglecting the fluid inertia) plus an additive lift force. In the present paper the simplicity of the unperturbed flow enables one to calculate analytically the explicit contribution of each term appearing in the perturbed flow equation (by using matched asymptotic expansions). Our results show how the convective terms and the unsteady term do contribute to the particle drag and lift coefficients in a very complex and nonadditive manner.