The second step is to choose the arbitrary control volume \( V^*(t) \) to be instantaneously coincident with the material volume \( V(t) \) so that at the moment of interest \( V(t) = V^*(t) \) and \( A(t) = A^*(t) \)
The \(\boxed{ \left( \frac{d}{dt} \right) \int \rho \, dV }\)-terms are not equal
in \(\frac{d}{dt} \int_{V(t)} \rho(\mathbf{x}, t) \, dV = 0\) and \(
\frac{d}{dt} \int_{V^*(t)} \rho(\mathbf{x}, t) dV
- \int_{V^*(t)} \frac{\partial \rho(\mathbf{x}, t)}{\partial t} \, dV
- \int_{A^*(t)} \rho(\mathbf{x}, t) \mathbf{b} \cdot \mathbf{n} \, dA = 0\)
However, the volume integration
of \( \partial \rho / \partial t \) in
\(\int_{V(t)} \frac{\partial \rho(\mathbf{x}, t)}{\partial t} \, dV + \int_{A(t)} \rho(\mathbf{x}, t) \, \mathbf{u}(\mathbf{x}, t) \cdot \mathbf{n} \, dA = 0\)
is equal to that in \(
\frac{d}{dt} \int_{V^*(t)} \rho(\mathbf{x}, t) dV
- \int_{V^*(t)} \frac{\partial \rho(\mathbf{x}, t)}{\partial t} \, dV
- \int_{A^*(t)} \rho(\mathbf{x}, t) \mathbf{b} \cdot \mathbf{n} \, dA = 0\) and the surface integral
of \( \rho \mathbf{u} \cdot \mathbf{n} \) over \( A(t) \) is equal to that over \( A^*(t) \):
\[
\int_{V^*(t)} \frac{\partial \rho(\mathbf{x}, t)}{\partial t} \, dV
= \int_{V(t)} \frac{\partial \rho(\mathbf{x}, t)}{\partial t} \, dV
= - \int_{A(t)} \rho(\mathbf{x}, t) \mathbf{u}(\mathbf{x}, t) \cdot \mathbf{n} \, dA
= - \int_{A^*(t)} \rho(\mathbf{x}, t) \mathbf{u}(\mathbf{x}, t) \cdot \mathbf{n} \, dA
\]
The two ends allow the central volume-integral
term in
\(\boxed{
\frac{d}{dt} \int_{V^*(t)} \rho(\mathbf{x}, t) dV
- \boxed{\int_{V^*(t)} \frac{\partial \rho(\mathbf{x}, t)}{\partial t} \, dV}
- \int_{A^*(t)} \rho(\mathbf{x}, t) \mathbf{b} \cdot \mathbf{n} \, dA = 0}
\)
to be replaced by a surface integral to find
\( \mathbf{u} \) and \( \mathbf{b} \) must both be observed in the same frame of reference. This is the general integral statement of conservation of mass for an arbitrarily moving control volume. It can be specialized to stationary, steadily moving, accelerating, or deforming control volumes by appropriate choice of \( \mathbf{b} \). When \( \mathbf{b} = \mathbf{u} \), the arbitrary control volume becomes a material volume and \( \frac{d}{dt} \int_{V^*(t)} \rho(\mathbf{x}, t) \, dV + \int_{A^*(t)} \rho(\mathbf{x}, t) \left( \mathbf{u}(\mathbf{x}, t) - \mathbf{b} \right) \cdot \mathbf{n} \, dA = 0\) reduces to \(\frac{d}{dt} \int_{V(t)} \rho(\mathbf{x}, t) \, dV = 0\)