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In this study, two-dimensional incompressible steady and time-dependent magnetohydrodynamic (MHD) flows around a cylinder were simulated by applying a finite volume method. The method was first validated through fair agreement of its results with the analytical solution for MHD flows inside the channels. It was then applied to simulate the MHD flows around a cylinder at Reynolds (Re) numbers varying from 100 to 1000 and Hartmann (Ha) numbers varying from 10 to 100. According to the results, although the increase in Re number strengthens the flow instability, the magnetic field suppresses this effect. The evidence for this claim is that the increase in Ha number leads to reduce in the separation angle for each Re number. This was in such a way that no flow separation was observed at Ha number equal to 50 and above for any considered Re number. Instability and therefore unsteady flows were seen only for the cases where Ha number was equal to 10 (the minimum considered value) and Re number was 500 or above. The drag coefficient (as a constant value for steady flows) or its average value (for unsteady flows) was enhanced by increasing the Ha number. The amplitude of the drag and lift coefficient fluctuations for unsteady cases were grown up by increasing the Re number. The Strouhal number was also found as a parameter that is dependent on both Re and Ha numbers. [ABSTRACT FROM AUTHOR]