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A plane non-parallel vortex flow in a square fluid domain is examined. Theenergy dissipation of the flow is dominated by viscosity and linear frictioneffect of a Hartmann layer. This is a traditional Navier-Stokes flow when thelinear friction effect is not involved, whereas it is a magnetohydrodynamicflow when the energy dissipation is fundamentally dominated by the friction. Itis proved that linear critical values of a spectral problem are nonlinearthresholds leading to the onset of secondary steady-state flows, the nonlinearphenomenon observed in laboratory experiments.

Bifurcating steady-state flows involving energy dissipation over a Hartmann boundary layer