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

TL;DR

This paper investigates steady-state vortex flows in a square domain with energy dissipation influenced by viscosity, linear friction, and magnetic effects, revealing critical thresholds for flow transitions observed experimentally.

Contribution

It demonstrates the nonlinear thresholds for secondary steady flows in magnetohydrodynamic and viscous flows with friction, linking spectral analysis to experimental phenomena.

Findings

Identification of critical values leading to secondary flows

Connection between spectral thresholds and experimental observations

Analysis of energy dissipation mechanisms in Hartmann layers

Abstract

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 steady-state flows, the nonlinear phenomenon observed in laboratory experiments.