The effects of a constant magnetic field on electrically conducting liquid-metal flows in a parallelepiped cavity are investigated using a spectral numerical method involving direct numerical solution of the Navier–Stokes and Ohm equations for three-dimensional flows. Three horizontal Bridgman configurations are studied: buoyancy-driven convection in a confined cavity and in a cavity where the top boundary is a stress-free surface and thirdly, thermocapillary-driven flow in a cavity where the upper boundary is subjected to effects of surface tension. The results of varying the Hartmann number (Ha) are described for a cavity with Ax = L/H = 4 and Ay = W/H = 1, where L is the length, W is the width and H is the height of the cavity. In general, an increase in the strength of the applied magnetic field leads to several fundamental changes in the properties of thermal convection. The convective circulation progressively loses its intensity and when Ha reaches a certain critical value, which is found to depend on the direction (longitudinal or vertical) of the applied magnetic field, decrease of the flow intensity takes on an asymptotic form with important changes in the structure of the flow circulation. The flow structure may be separated into three regions: the core flow, Hartmann layers which develop in the immediate vicinity of the rigid horizontal boundaries or at the endwalls, and parallel layers appearing in the vicinity of the sidewalls. The behaviour of the maxima of velocity and of the overall flow circulation is found to depend on both the boundary conditions used and the direction of the applied magnetic field. Furthermore, the interaction of the electric current density with the applied magnetic field which leads to the structural reorganization described above can also create more subtle flow modifications, such as flow inversions which are observed mainly in the central region of the cavity.