Magnetohydrodynamic shear work ($S=a(B_x{B}_y/{\mu }_0-\rho v_x{v}_y)$)

The energy of the residual flow and magnetic field in a maintained shear flow has source $S=-a{T}_{xy}$, where $T_{xy}=\rho v_x{v}_y-B_x{B}_y/{\mu }_0$. The kinetic and magnetic stresses can exchange energy with the imposed background. The source vanishes when their off-diagonal contributions cancel.