The interaction between rotationally oscillating spheres and solid boundaries in a Stokes flow

Abstract

We present the results of an experimental and theoretical investigation into the influence of proximate boundaries on the motion of an rotationally oscillating sphere in a viscous fluid. The angular oscillations of the sphere are controlled using the magnetic torque generated by a spatially uniform, oscillatory magnetic field which interacts with a small magnet embedded within the sphere. We study the motion of the sphere in the vicinity of stationary walls that are parallel and perpendicular to the rotational axis of the sphere, and near a second passive sphere that is non-magnetic and free to move. We find that rigid boundaries introduce viscous resistance to motion that acts to suppress the oscillations of the driven sphere. The amount of viscous resistance depends on the orientation of the wall with respect to the axis of rotation of the oscillating sphere. A passive sphere also introduces viscous resistance to motion, but for this case the rotational oscillations of the active sphere establish a standing wave that imparts vorticity to the fluid and induces oscillations of the passive sphere. The standing wave is analogous to the case of an oscillating plate in a viscous fluid; the amplitude of the wave decays exponentially with radial distance from the surface of the oscillating sphere. The standing wave introduces a phase lag between the motion of the active sphere and the response of the passive sphere which increases linearly with separation distance.

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“…The motion of the rotating microrobot was substantially suppressed by the viscous resistance introduced by the rigid wall. The magnitude of viscous resistance was not only related to the orientation of the wall with respect to the axis of rotation (40) but also increased rapidly as the microrobot got closer to the wall.…”

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“…For spheres performing torsional oscillations near stationary boundaries, the viscous resistance introduced by the presence of a wall is negligible provided the separation distance is greater than the thickness of the viscous boundary layer on the wall, independent of the orientation of the rotational axis of the sphere with respect to the surface normal (Box et al 2017). The thickness of the Stokes boundary layer = ( ∕ ) 1∕2 , which gives a measure of the distance over which the amplitude of fluid motion decays to 1/e of the initial value, was estimated to be 2.69a for typical measurements and 4.93a for the PIV measurements.…”