Problem $2(10\%)$

Recall the volumetric strain rate, defined in the lecture note. In Cartesian coordinates,

$\frac{1}{V}\frac{DV}{Dt}={}_{}+{}_{yy}+{}_{zz}=\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{x}}+\frac{\text{d}\text{e}\text{l}\text{v}}{\text{d}\text{e}\text{l}\text{y}}+\frac{\text{d}\text{e}\text{l}\text{w}}{\text{d}\text{e}\text{l}\text{z}}$

Show that volumetric strain rate is zero for incompressible flow. Discuss the physical interpretation of volumetric strain rate for incompressible and compressible flows.