Growth rates of Bénard-Marangoni convection in a fluid layer in the presence of a magnetic field

In this paper we analyze the effect of a uniform vertical magnetic field on the linear growth (and decay) rates of the steady Bénard-Marangoni instability in a horizontal layer of quiescent, electrically conducting fluid with a uniform vertical temperature gradient subject to a prescribed heat flux at its lower boundary. Explicit analytical expressions for the linear growth rates of long-waves instability modes are derived for the first time. The numerically-calculated linear growth (or decay) rates showing the stabilizing effect of the magnetic field are also presented.