Reynolds transport theorem: \[ \frac{d}{dt} \int_{V^*(t)} F(\mathbf{x}, t) \, dV = \int_{V^*(t)} \frac{\partial F(\mathbf{x}, t)}{\partial t} \, dV + \int_{A^*(t)} F(\mathbf{x}, t) \, \mathbf{b} \cdot \mathbf{n} \, dA \]
The implications of \(\frac{d}{dt} \int_{V(t)} \rho(\mathbf{x}, t) \, dV = 0\) for the fluid velocity field may be better displayed by using Reynolds transport theorem (RTT) with \( F = \rho \) and \( \mathbf{b} = \mathbf{u} \) to expand the time derivative in \(\boxed{\frac{d}{dt} \int_{V(t)} \rho(\mathbf{x}, t) \, dV} = 0\): \[ \int_{V(t)} \frac{\partial \rho(\mathbf{x}, t)}{\partial t} \, dV + \int_{A(t)} \rho(\mathbf{x}, t) \, \mathbf{u}(\mathbf{x}, t) \cdot \mathbf{n} \, dA = 0 \]
Total mass in a material volume changes with time (mass is conserved):
\(\frac{d}{dt} \int_{V(t)} \rho(\mathbf{x}, t) \, dV = 0
\)
Using RTT on this left-hand side: \(
\frac{d}{dt} \int_{V(t)} \rho(\mathbf{x}, t) \, dV
= \int_{V(t)} \frac{\partial \rho(\mathbf{x}, t)}{\partial t} \, dV
+ \int_{A(t)} \rho(\mathbf{x}, t) \, \mathbf{u}(\mathbf{x}, t) \cdot \mathbf{n} \, dA \)
Substituting this back into the conservation statement:
\(
\boxed{
0 = \int_{V(t)} \frac{\partial \rho(\mathbf{x}, t)}{\partial t} \, dV
+ \int_{A(t)} \rho(\mathbf{x}, t) \, \mathbf{u}(\mathbf{x}, t) \cdot \mathbf{n} \, dA
}
\)
This is a mass-balance statement between integrated density changes within \(V(t)\) and integrated motion of its surface \(A(t)\)