Two-liquid annular viscous flows driven by slippery concentric spherical surfaces undergoing time-periodic torsional oscillations

To elucidate the underlying physics and subtle intricacies of the fundamental flow fields commonly found in ocular fluid mechanics and bacterial local motion research, we present a comprehensive theoretical investigation on the annular viscous flows of two concentrically arranged, immiscible liquids between a pair of concentric spherical surfaces undergoing time-periodic torsional/rotational oscillations while subject to asymmetric hydrodynamic slip conditions. Systematic parametric analyses on the analytical solutions are conducted for the configurations of two-liquid flows between concentric spherical surfaces, a solid sphere encapsulated within two concentrically arranged liquid layers, as well as single-liquid flows between concentric spherical surfaces serving as a baseline for comparison. One common observation on the results of the three different configurations examined is that the torque magnitudes are found to achieve local extrema whenever the inner (or single) liquid viscous penetration depth approximately matches the inner (single) liquid layer thickness as the inner (single) liquid layer thickness is increased. Various scaling relationships are derived, likely for the first time, to explain the reason for the existence of such local extrema in the torque magnitudes, and to describe how the torque magnitudes vary with respect to the Womersley number, liquid layer thickness, and hydrodynamic slip lengths for the different matching scenarios between the viscous penetration depths and liquid layer thicknesses. We also show that the inner/outer slip lengths or kinematic viscosity ratio of the two liquids may, counterintuitively, have negligible influence over the torque responses depending on the respective oscillatory boundary layer developments in the two liquid layers.