The wake structure evolution and the drag force decay of an impulsively stopped sphere with a transverse magnetic field

The evolution of the flow structure around an impulsively stopped sphere is investigated in an incompressible viscous fluid under a transverse magnetic field. The study focuses on the wake structure and drag force over the range of Reynolds numbers $60 \leqslant {\textit{Re}}_{\!D} \leqslant 300$ and $ {\textit{Re}}_{\!D}=1000$ , with the interaction parameters $0 \leqslant N \leqslant 10$ , where $N$ characterises the strength of the magnetic field. The wake is fully developed before the impulsive stop, after which it moves downstream and interacts with the sphere under the influence of a transverse magnetic field. The complex flow structures are characterised by skin friction lines on the downstream side of the sphere and categorised into five regimes in the $\{N, {\textit{Re}}_{\!D}\}$ phase diagram based on nearly 200 cases. The drag force generally decays over time following the impulsive stop. A drag decomposition model based on the vorticity diffusion scale is proposed, attributing the drag decay to three components: the original Stokes contribution, an inertia correction at high Reynolds numbers and a magnetohydrodynamic (MHD) correction, where the inertia and MHD effects both contribute a temporal power-law decay with an exponent of $-1/6$ . Temporal scaling laws of the drag decay are derived by coupling these three different effects, considering flow structures at short and long time scales, as well as the dependence on ${\textit{Re}}_{\!D}$ and $N$ . The prediction results are consistent with present simulations. Furthermore, the proposed drag decomposition model is successfully extended to complex vortex flow past a sphere at ${\textit{Re}}_{\!D}=1000$ , to an anisotropic ellipsoidal particle and to different magnetic field orientations.