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The wake structure of an incompressible, conducting, viscous fluid past an electrically insulating sphere affected by a transverse magnetic field is investigated numerically over flow regimes including steady and unsteady laminar flows at Reynolds numbers up to 300. For a steady axisymmetric flow affected by a transverse magnetic field, the wake structure is deemed to be a double plane symmetric state. For a periodic flow, unsteady vortex shedding is first suppressed and transitions to a steady plane symmetric state and then to a double plane symmetric pattern. Wake structures in the range 
$210<Re\leqslant 300$ without a magnetic field have a symmetry plane. An angle 
$\unicode[STIX]{x1D703}$ exists between the orientation of this symmetry plane and the imposed transverse magnetic field. For a given transverse magnetic field, the final wake structure is found to be independent of the initial flow configuration with a different angle 
$\unicode[STIX]{x1D703}$. However, the orientation of the symmetry plane tends to be perpendicular to the magnetic field, which implies that the transverse magnetic field can control the orientation of the wake structure of a free-moving sphere and change the direction of its horizontal motion by a field–wake–trajectory control mechanism. An interesting ‘reversion phenomenon’ is found, where the wake structure of the sphere at a higher Reynolds number and a certain magnetic interaction parameter (
$N$) corresponds to a lower Reynolds number with a lower 
$N$ value. Furthermore, the drag coefficient is proportional to 
$N^{2/3}$ for weak magnetic fields or to 
$N^{1/2}$ for strong magnetic fields, where the threshold value between these two regimes is approximately 
$N=4$.
When the Galileo number is below the first bifurcation, the instability and transition of a vertical ascension or the fall of a free sphere affected by a vertical magnetic field are investigated numerically. A compact model is used to explain that the magnetic field can destabilize the fluid–solid system. When the interaction parameter exceeds a critical value, the sphere trajectory is transitioned from a steady vertical trajectory to a steady oblique one. Furthermore, the trajectory will remain vertical at a sufficiently large magnetic field because of a double effect of the magnetic field on the fluid–solid system. Under the influence of an external vertical magnetic field, four wake patterns at the rear of the sphere are found and the physical behaviour of the free sphere is independent of the density ratio. The wake or trajectory of the free sphere is only determined by the Galileo number 
$G$ and the interaction parameter 
$N$. A close relationship between the streamwise vorticity and the sphere motion is found. An interesting ‘agglomeration phenomenon’ is also found, which shows that the vertical velocities are agglomerated into a point for a certain magnetic field regardless of the Galileo number and satisfy a scaling law 
$V_{z}\sim N^{-1/4}$, when 
$N>1$. The principal results of the present work are summarized in a map of regimes in the 
$\{G,N\}$ plane.
The wake structure and transition process of an incompressible viscous fluid flow past a sphere affected by an imposed streamwise magnetic field are investigated numerically over flow regimes that include steady and unsteady laminar flows at Reynolds numbers up to 300. For cases without a magnetic field, a subregion with the existence of a limit cycle is found in the range 
$210<Re<270$. The point of division is between 
$Re=220$ and 
$Re=230$. For cases with a streamwise magnetic field, five wake patterns are the steady axisymmetric wake with an attached separation bubble, the steady plane symmetric wake with a small spiral dismissed, the steady plane symmetric wake with a limit cycle, the steady plane symmetric wake with a small spiral fed by the upstream fluid and the unsteady plane symmetric wake with a wave-like oscillation or vortex shedding. Under the influence of an imposed streamwise magnetic field, the wake will be transitioned to various patterns. An interesting ‘reversion phenomenon’, which describes the topological structure behind a sphere with a higher Reynolds number and a certain interaction parameter which corresponds to a lower Reynolds number case with a certain interaction parameter or a much lower Reynolds number case without a magnetic field, is also found. The principal results of the present work are summarized in a map of regimes in the 
$\{N,Re\}$ plane.
