Three-dimensional steady combined free and forced convective magnetohydrodynamic (MHD) flow are simulated in a cylindrical cavity filled with a liquid metal and submitted to a vertical temperature gradient and an axial magnetic field. The forcing corresponds to a swirling flow produced by counter-rotation of the top and bottom disks. The governing Navier-Stokes, energy, and potential equations along with appropriate boundary conditions are solved by using the finite-volume method. Comparisons with previous results were performed and found to be in excellent agreement. The effects of magnetic field on flow and temperature fields are analysed. When the Reynolds number is increased, the axisymmetric basic state loses stability and different complex flow appear. Axisymmetric (m = 0) and asymmetric m = 1 and m = 2 azimuthal modes are observed. Azimuthal mode m = 3 are found when the Hartmann number, Ha is sufficiently large. In the mixed convection case the m = 1 becomes the dominant mode in place of m = 0. The stability diagram (Rea Ha) corresponding to the transition from axisymmetric to non-axisymmetric flow by increasing values of the Richardson number is obtained. (C) 2015 Elsevier Ltd. All rights reserved.