For time varying domains the Reynolds transport theorem states $${\frac{d}{dt}}\int_{\Omega(t)}\mathbf{f}\,dV{\displaystyle =\int_{\Omega(t)}{\frac{\partial\mathbf{f}}{\partial t}}\,dV+\int_{\partial\Omega(t)}\left(\mathbf{v}^{b}\cdot\hat{\mathbf{n}}\right)\mathbf{f}\,dA}$$
I want to integrate the mass conservation in fluid with time varying domain; this is $$\int_{\Omega(t)}\left(\frac{d\rho}{dt}+\nabla\cdot\rho\mathbf{v}\right)\thinspace dV=0$$
The first term can be integrated straighrforwardly using Reynolds theorem; however, for the second term $$\int_{\Omega(t)}\left(\nabla\cdot\rho\mathbf{v}\right)\thinspace dV=0$$ I am not sure. By intuition I would say that I can use Gauss theorem to give
$$\int_{\partial\Omega(t)}\left(\rho\mathbf{v}_{r}\cdot\hat{\mathbf{n}}\right)\thinspace dA$$
Now being $\mathbf{v}_{r}$ a velocity relative to the moving boundary.
My question is: is there a way to integrate without defining a relative velocity?