Asymptotic Methods in Magnetoconvection

The effects of Lorentz force and non-uniform temperature gradient on the onset of magnetoconvection in an electrically conducting horizontal Boussinesq fluid layer permeated by a uniform transverse magnetic field are studied analytically using linear stability analysis by specifying constant temperature or constant heat flux at the boundaries. It is shown that when the Chandrasekhar number Q → ∞ the correct asymptotic value of the critical Rayleigh number, Rc, can be obtained from the non-viscous MHD equations using a single-term Galerkin expansion. The criterion for the onset of magnetoconvection is determined using a regular perturbation technique with wave-number as perturbation parameter. The method of matched asymptotics is used to predict explicitly the effect of the Hartmann boundary layer (that exists at the rigid boundary for large values of Q) on the onset of magnetoconvection. It is shown that the effect of the Hartmann boundary layer is to increase the asymptotic value of Rc by an amount proportional to the value of the Hartmann number M. We find that the ratios Rci/Rcl where Rci (i = 1 to 6) are the asymptotic values of RC for different nonlinear temperature profiles, are independent of Q but dependent on thermal depth e. It is also shown that the power law for asymptotic values of Rci depends crucially on the nature of heating and not on the nature of the boundaries.