Steady-State Bifurcation of a Non-parallel Flow Involving Energy Dissipation over a Hartmann Boundary Layer

A plane non-parallel vortex flow in a square fluid domain is examined. The energy dissipation of the flow is dominated by viscosity and linear friction effect of a Hartmann layer. This is a traditional Navier-Stokes flow when the linear friction effect is not involved, whereas it is a magnetohydrodynamic flow when the energy dissipation is fundamentally dominated by the friction. It is proved that linear critical values of a spectral problem are nonlinear thresholds leading to the onset of secondary steadystate flows, the nonlinear phenomenon observed in laboratory experiments.