Nonlinear analysis on the natural convection between vertical plates in the presence of a horizontal magnetic field

Abstract

Finite-amplitude secondary motions of an electrically conducting fluid between two vertical parallel plates heated differntially in the presence of a horizontal magnetic field are obtained numerically in the limit of a small Prandtl number and a small magnetic Prandtl number. We find that, as in the purely hydrodynamic case with the Hartmann number H = 0 examined by Nagata and Busse (1983), the bifurcation of the secondary flows is supercritical at the critical Grashof number G = G c when H is small. Subcritical bifurcations occur at higher wavenumbers. As H is increased, the occurrence of the subcritical bifurcations moves gradually towards a smaller wavenumber region along the neutral curve. For H = 10 subcritical motions dominate across the whole range of wavenumbers. The stability analysis shows that the stability of the supercritical motions for small H is bounded by an onset of monotonically growing instability at a higher Grashof number G = G 2. c . The stability region ( G c < G < G 2. c ) becomes smaller with G 2. c approaching G c from above as H is increased. The region vanishes completely at H ⋍ 7.