A cute problem that I spent a bit time on. Please help.
Consider flow domain smooth manifold in $n$ dimension $\Omega$ with boundary and some boundary data. Constant density is assumed.
Assume within the domain the position is defined as $r\in\mathcal{R}^n$ and velocity is prescribed as $u$.
Know that the angular momentum of this fluid is zero. i.e.
$\int_\Omega r\times u dx^n = 0 $
Can you prove the antisymmetric rate of strain tensor $\frac12(\nabla u-\nabla u^T$) is zero?
Thanks.