The Effects of a Constant Bias Force on the Dynamics of a Periodically Forced Spherical Particle in a Newtonian Fluid at Low Reynolds Numbers

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: Totalforce=F ext+ FH, 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. © 2010 American Institute of Physics