A material region currently occupies the cube 0≤ x_1≤ 0.050 m, 0≤ x_2≤ 0.050 m, 0≤ x_3≤…

A material region currently occupies the cube \(0\le x_1\le 0.050\,\mathrm{m}\), \(0\le x_2\le 0.050\,\mathrm{m}\), \(0\le x_3\le 0.050\,\mathrm{m}\). At this instant the spatial mass density is \(\rho=980-400 x_3\,\mathrm{kg/m}^{3}\) (\(x_3\) in m) with \(\partial\rho/\partial t=0\), and the velocity is \(\mathbf{v}=(0.20 x_1-0.10 x_2)\,\mathbf{e}_1+(0.10 x_1+0.20 x_2)\,\mathbf{e}_2-0.40 x_3\,\mathbf{e}_3\) m/s. Using the Reynolds transport theorem, compute the present material time derivative of the mass \(M=\int_{V(t)}\rho\,\mathrm{d}V\). Also compute \(D/Dt\) of the scalar integral \(\int_{V(t)} x_3^{2}\,\mathrm{d}V\) at the same instant.

Convert the material derivatives of the integrals into volume integrals of \(\partial\varphi/\partial t+\mathrm{div}(\varphi\mathbf{v})\) by the Reynolds transport theorem.