Three-dimensional dynamics of oblate and prolate capsules in shear flow

We study computationally the dynamics of oblate and prolate spheroidal capsules in simple shear flow with small inertia for a range of dimensionless shear rates. The capsule is modelled as a liquid droplet enclosed by a hyperelastic membrane, and its equatorial plane is initially tilted out of the plane of shear. We find, at low shear rates, the well-accepted tumbling motion is not always stable for both oblate and prolate capsules. For an oblate capsule, the dominant stable modes for increasing dimensionless shear rate are as follows: rolling with the equatorial plane staying in the plane of shear, precessing following Jeffery's orbit [Proc. R. Soc. London A 102, 161 (1922)], and tumbling. Interestingly, the order of modes is reversed for a prolate capsule: tumbling, precessing, and rolling with increasing dimensionless shear rate. At transitional regimes, we find the stable motion of a capsule can depend on its initial titled angle, even at the same shear rate. At high dimensionless shear rates, a spheroidal capsule undergoes a complicated oscillating-swinging motion: Its major axis oscillates about the plane of shear in addition to the swinging about a mean angle with flow direction found previously, and the amplitudes of both oscillations decrease when increasing the dimensionless shear rate towards a steady tank treading motion asymptotically. We summarize the results in phase diagrams and discuss the reorientation of both oblate and prolate capsules in a wide range of dimensionless shear rates.