Dynamics and ‘normal stress’ evaluation of dilute suspensions of periodically forced prolate spheroids in a quiescent Newtonian fluid at low Reynolds numbers

Abstract

The problem of determining the force acting on a particle in a fluid where the motion of the fluid and the particle is given has been considered in some detail in the literature. In this work, we propose an example of a new class of problems where, the fluid is quiescent and the effect of an external periodic force on the motion of the particle is determined at low non-zero Reynolds numbers. We present an analysis of the dynamics of dilute suspensions of periodically forced prolate spheroids in a quiescent Newtonian fluid at low Reynolds numbers including the effects of both convective and unsteady inertia. The inclusion of both forms of inertia leads to a nonlinear integro — differential equation which is solved numerically for the velocity and displacement of the individual particle. We show that a ‘normal stress’ like parameter can be evaluated using standard techniques of Batchelor. Hence this system allows for an experimentally accessible measurable macroscopic parameter, analogous to the ‘normal stress’, which can be related to the dynamics of individual particles. We note that this ‘normal stress’ arises from the internal fluctuations induced by the periodic force. In addition, a preliminary analysis leading to a possible application of separating particles by shape is presented. We feel that our results show possibilities of being technologically important since the ‘normal stress’ depends strongly on the controllable parameters and our results may lead to insights in the development of active dampeners and smart fluids. Since we see complex behaviour even in this simple system, it is expected that the macroscopic behaviour of such suspensions may be much more complex in more complex flows.