On rotational dynamics of inertial disks in creeping shear flow

• Oblate spheroids rotate in flow-gradient plane irrespective of initial orientation.
• A shape-dependent time scale is proposed for rotation of an oblate spheroid.
• The final stage of spin-up is exponential for disk-like particles.

The rotational motion of an inertial disk-like particle in a creeping linear shear flow is investigated. A disk-like particle in a linear shear flow tends to rotate in the velocity-gradient plane as do rod-like particles. Unlike prolate spheroids, however, oblate spheroids always attain the same steady rotation in the shear plane irrespective of their initial orientation. The drift of the orientation of the rotation axis towards the vorticity vector consists of two qualitatively different stages. First, the wobbling drift towards rotation in the velocity-gradient plane becomes slower with increasing particle inertia, except for the least inertial spheroids. The duration of the second stage, during which the spheroid spins up to match the angular fluid velocity, becomes independent of the aspect ratio for relatively flat particles, provided that a new shape-dependent Stokes number is used.