= Viscous magnetic-relaxation energy identity
{title2=$\dot E=-\rho\nu\int_D|\nabla\times\mathbf v|^2\,dV$}
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, $\int|\nabla\mathbf v|^2=\int|\nabla\times\mathbf v|^2$.
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