 Solution

This is the velocity field of solid-body rotation about the axis parallel to $\mathbf{\omega }$ through the origin. A fluid-following body's centre moves around that axis at angular speed $|\mathbf{\omega }|$, keeping its axial coordinate and perpendicular distance fixed. Its orientation rotates at the same angular velocity. The symmetric part of the velocity-gradient tensor is zero, so the prescribed flow produces no local strain or change of shape. Indeed, the separation $\mathbf{d}$ between two fluid points obeys $d|\mathbf{d}{|}^2/dt=2\mathbf{d}\cdot (\mathbf{\omega }\times \mathbf{d})=0$.

This is a kinematic prediction for an entrained body or material object. A finite inertial body initially at rest cannot instantaneously acquire the prescribed velocity and rotation without forces and torques; its transient after placement would also require density, drag and coupling data. Neglecting gravity alone does not specify that transient. On the rotation axis its centre can remain fixed while its orientation still rotates.