Viscous magnetic-relaxation energy identity ($\dot{E}=-\rho \nu {\int }_D|\nabla \times \mathbf{v}{|}^{2}dV$)

For constant mass density $\rho$, kinematic viscosity $\nu >0$, ideal induction, zero boundary velocity and tangent magnetic field, put $E={\int }_D[\rho |\mathbf{v}{|}^2/2+|\mathbf{B}{|}^2/(2{\mu }_0)]dV$. Multiplying the momentum equation by $\mathbf{v}$ and integrating cancels pressure work and advective flux. The magnetic-energy derivative is minus the integral of Lorentz force density dotted with velocity, canceling the same work in the kinetic-energy equation. Integration by parts then gives the displayed identity. For incompressible flow with zero boundary velocity, $\int |\nabla \mathbf{v}{|}^2=\int |\nabla \times \mathbf{v}{|}^2$.