Bounds on dissipation in magnetohydrodynamic Couette and Hartmann shear flows

Shear flow with an applied cross-stream magnetic field is studied using dissipative incompressible magnetohydrodynamics. The study incorporates exact solutions, the energy stability method, and exact bounds on the total energy dissipation rate. Two physical configurations are examined: magnetic Couette flow and Hartmann flow, the latter being Poiseuille flow with the existence of a perpendicular magnetic field. Explicit expressions are derived for energy stability regions in the parameter space and these expressions are compared with numerically obtained results. For large enough Reynolds numbers the energy dissipation rate is shown to be bounded by a function of the magnetic Prandtl number. The bounds obtained on the dissipation rate are compared with experimental results