Material volume ($V(t)$)

A material volume is a region containing the same fluid particles at every time: its boundary moves with the fluid velocity field. Its mass is fixed by mass conservation, while its volume can change in compressible motion. Applying the Reynolds transport theorem to its volume gives $d|V(t)|/dt={\int }_{V(t)}\nabla \cdot \mathbf{u}dV$. A material volume element is the infinitesimal version.