Magnetohydrodynamic drag on an oscillating sphere in a rotating spherical cavity

Oscillating-sphere drag is a classical fluid-mechanics problem, yet no theory simultaneously accounts for confinement, rotation, viscosity and magnetic fields. We consider a conducting sphere undergoing translational oscillations inside a rotating spherical cavity, modelling confined magnetohydrodynamic (MHD) flows relevant to planetary interiors and liquid-metal experiments. Planetary applications include polar (rectilinear oscillations) and equatorial (circular trajectories) Slichter modes of solid inner cores or subsurface-ocean interiors relative to outer ice shells. Existing theories are restricted to separate asymptotic regimes, including viscous drag in bounded fluids (Stokes 1851 On the Effect of the Internal Friction of Fluids on the Motion of Pendulums ), rotational effects in inviscid cavities (Busse 1974 J. Geophys. Res. , vol. 79, pp. 753–757), and magnetic coupling through oscillatory boundary layers (Buffett & Goertz 1995 Geophys . J. Intl , vol. 120, pp. 103–110). We derive a unified asymptotic framework for oscillatory drag in confined spherical shells with conductivity and permeability contrasts between the inner sphere, fluid shell and outer solid. Closed-form force laws and dissipation predictions are obtained for polar and equatorial oscillations at weak rotation, with finite-rotation corrections for the polar mode. The framework accounts for viscous and magnetic-pressure contributions, magnetic tension, Alfvén-wave radiation and MHD Stokes–Ekman layers. Recovering classical boundary-layer theories as limiting cases, the analysis yields a local Alfvén-wave drag law for spherical shells, analogous to bounded oscillatory Stokes drag. Beyond the thin boundary-layer approximation, complementary asymptotics yield a closed-form representation of bounded Stokes flow and extend inductionless theory to spherical shells. Direct numerical simulations validate the predictions across a broad parameter range. The framework predicts oscillatory coupling, added mass, drag and dissipation in planetary cores, subsurface oceans and liquid-metal experiments.