Dynamics of a driven spheroid in a slow oscillating creeping shear flow

Abstract

We report the orientation dynamics of a sinusoidally driven spheroid suspended in a slow and weak/strong oscillatory shear flow without Brownian and inertial forces, derive the governing equations, find the classical Jeffery orbits, and then solve them numerically. These equations describe Jeffery's orbits for no external force and no flow oscillations. When the external forces are small, and there are no oscillations, they can be seen as perturbations of the equations that result in Jeffery's orbits. The small perturbations disturb the Jeffery orbits. We also analyze the chaotic and regular dynamics regimes in nearly quiescent, simple shear, and weak/strong and slow oscillating shear flows. We observe quantitative and qualitative differences in the particle dynamics for an oscillating shear flow compared to simple shear flow, as seen from the Poincaré sections, attractors, phase diagrams, time series, and Lyapunov exponents. The analysis indicates that the slow oscillations reduce the complexity of the dynamics of the particle compared to simple shear flow. The steady-state solutions for both prolate and oblate spheroids remain in the flow gradient plane in the case of strong oscillatory shear. At the same time, there is some disturbance from the flow gradient plane for weak oscillations due to the external force instead of inertial forces reported earlier in the literature. In addition, we propose a mechanism to improve particle separation based on shape using a combination of simple and oscillating shear flows, offering significant advantages in separating particles from a colloidal mixture that would otherwise be impossible.