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. 660 K Madhukar et alKeywords. Low Reynolds number; quiescent fluid; periodically forced prolate spheroids; aspect ratio; inertial effects and particle separation.

We study the effects of both convective and unsteady inertia on the dynamics and rheology of a dilute suspension of periodically forced neutrally buoyant spherical particles, at low Reynolds numbers, in a quiescent Newtonian fluid. The inclusion of inertia results in additional terms in the equation governing the dynamics of the particle that represent a fading memory of the entire history of the particle motion. The inclusion of convective inertia in the low Reynolds number limit makes the memory term nonlinear. Several tests were performed to show that the results presented in this paper are physically reasonable and correct. A perturbation analysis of the problem yields strong evidence that the results of our simulations are correct. It is observed that there is a preferred direction in this system which manifests itself in the properties of the solution. This preferred direction is identified as the direction of the initial motion of the particle. We present here results on the behavior of various parameters with respect to Reynolds numbers and the amplitude of the periodic force. These include phase-space plots between particle displacement and particle velocity and the variation of a rheological parameter, namely a 'normal stress' with respect to Reynolds number and the amplitude of the periodic force. We believe that our results may be technologically important since the rheological parameter depends strongly on controllable parameters such as the Reynolds number and the amplitude of the periodic force. Further, this system is one of the simplest systems whose rheology shows non-Newtonian behavior, such as the presence of a normal stress. In addition, this system represents a physically realizable system for experimentally testing the frameworks developed to calculate the collective behavior of systems of oscillators with memory.

We make use of the formulation developed by Lovalenti and Brady [1] for the hydrodynamic force acting upon a spherical particle undergoing arbitrary time dependent motion in an arbitrary time dependent uniform flow field at low Reynolds numbers, to derive an expression for the effects of a constant bias force acting on a periodically forced rigid spherical particle in a Newtonian fluid. We use Newton's second law to relate the total force acting on the particle to the motion of the particle. The total force is given by: Total force = F ext +F H , where, F ext is the external force inclusive of both the periodic force and the constant bias force. F H is the hydrodynamic force derived by Lovalenti and Brady [1] including both unsteady and convective inertia. The equation derived contains a nonlinear history term and is nonlinear. This equation is solved numerically using an adaptive step size Runge -Kutta scheme. We obtain several phase plots (plots between particle displacement and particle velocity), which show the effects of low Reynolds numbers, the periodic force and the effects of the constant bias force on the particle motion. It is observed that at low magnitudes of the periodic forcing the external constant force dominates and the particle moves along the direction of the external constant force. As we increase the magnitude of the periodic forcing, the periodic force is seen to dominate and the particle is seen to oscillate along a mean position with a slight drift along the direction of the periodic force and the external constant force, when they are imposed in the same direction. However the motion of the particle becomes more complicated when the directions of the periodic forcing and external constant force are opposite to each other. We also observe a reflection in phase space when the directions of both the forces are reversed. The phase plots typically are of a half sinusoidal, sinusoidal and a coiled (solenoidal) pattern. These plots include the effects of both periodic force and the constant bias force. As the Reynolds numbers increases the drift of the particle reduces, which indicates the effects of inertia. We present a preliminary analysis of the dynamics in this paper.