Time-dependent force acting on a particle moving arbitrarily in a rotating flow, at small Reynolds and Taylor numbers

Abstract

The arbitrary motion of a solid sphere released in a solid-body rotating fluid is investigated theoretically in the limit of small Reynolds and Taylor numbers. The angular velocity of the fluid is assumed to be constant and under the premise that Ta1/2 ≫ Re, the simplicity of the unperturbed flow enables us to calculate analytically the force acting on a particle moving with a harmonic slip velocity (by means of matched asymptotic expansions), when both inertia and unsteady effects are taken into account. Subsequently, these single-frequency results are used in order to determine the temporal expression of the force acting on an arbitrarily moving sphere, since the problem under study is linear. This force is first determined in a co-rotating reference frame and takes the form of two convolution products involving the particle acceleration and the particle velocity. For convenience, the corresponding expression of this force is also derived in the laboratory reference frame, and the particle motion equation obtained is thereafter illustrated by dealing with two practical situations, where unsteady and inertia effects must be taken into account to predict the particle dynamics accurately.