Time-dependent two-dimensional translation of a freely rotating sphere in a viscoelastic fluid

This study examines the movement of a small freely rotating spherical particle in a two-dimensional trajectory through a viscoelastic fluid described by the Giesekus model. The fluid equations of motion in the inertialess limit and the Giesekus constitutive equation are expanded as a power series in the Weissenberg number, for which analytical solutions for velocity and pressure profiles at low order can be determined for the case of a steady-state flow. These steady solutions are then related to Fourier-transformed variables in frequency space through the use of correspondence relationships, allowing the analysis of time-dependent particle trajectories. The relative unsteadiness and nonlinearity of these time-dependent flows are quantified through a Deborah and Weissenberg number, respectively. The impact of changing these dimensionless parameters on the characteristics of the flow is discussed at length. We calculate the predicted rate of rotation of a small particle undergoing an arbitrary two-dimensional translation through a viscoelastic fluid, as well as the predicted correction to the force exerted on the particle arising from the interaction of particle rotation and translation. Finally, we calculate the angular velocity and total force including second-order corrections for particles executing a few specific trajectories that have been studied experimentally, as well as the predicted trajectory for a particle being directed by a known time-dependent forcing protocol.