= Magnetized-fluid tensor virial theorem
{title2=$\ddot I_{ij}/2=2K_{ij}-\int T_{ij}\,dV$}
For an isolated self-gravitating fluid obeying <ideal magnetohydrodynamics>, let $I_{ij}=\int\rho x_ix_jdV$ and $K_{ij}=\frac12\int\rho u_iu_jdV$. Use the force-stress convention $\rho D_tu_i=\partial_jT_{ij}$, with <pressure>, magnetic and negative <Newtonian gravitational stress tensor> contributions. Two time derivatives of $I$ give $4K_{ij}$ plus $\int(x_i\partial_kT_{jk}+x_j\partial_kT_{ik})dV$. Integration by parts and symmetry yield the theorem when surface terms vanish. This fluid version complements the collisionless formulation of the <tensor virial theorem>.
Back to article page
