When applied to a material volume \( V(t) \) with surface area \( A(t) \), Newton’s second law can be stated directly as
\[
\frac{d}{dt} \int_{V(t)} \rho(\mathbf{x}, t) \, \mathbf{u}(\mathbf{x}, t) \, dV = \int_{V(t)} \rho(\mathbf{x}, t) \, \mathbf{g} \, dV + \int_{A(t)} \mathbf{f}(\mathbf{n}, \mathbf{x}, t) \, dA
\]
where \( \rho \mathbf{u} \) is the momentum per unit volume of the flowing fluid, \( \mathbf{g} \) is the body force per unit mass acting on the fluid within \( V(t) \), \( \mathbf{f} \) is the surface force per unit area acting on \( A(t) \), and \( \mathbf{n} \) is the outward normal on \( A(t) \)
With \( \mathbf{F} = \rho \mathbf{u} \) and \( \mathbf{b} = \mathbf{u} \), the time derivative is expanded using Reynolds transport theorem \(\boxed{\frac{d}{dt} \int_{V^*(t)} F(\mathbf{x}, t) \, dV
= \int_{V^*(t)} \frac{\partial F(\mathbf{x}, t)}{\partial t} \, dV
+ \int_{A^*(t)} F(\mathbf{x}, t) \, \mathbf{b} \cdot \mathbf{n} \, dA}\)
\[
\int_{V(t)} \frac{\partial}{\partial t} \left( \rho(\mathbf{x}, t) \, \mathbf{u}(\mathbf{x}, t) \right) dV
+ \int_{A(t)} \rho(\mathbf{x}, t) \, \mathbf{u}(\mathbf{x}, t) \, (\mathbf{u}(\mathbf{x}, t) \cdot \mathbf{n}) \, dA
= \int_{V(t)} \rho(\mathbf{x}, t) \, \mathbf{g} \, dV
+ \int_{A(t)} \mathbf{f}(\mathbf{n}, \mathbf{x}, t) \, dA
\]
This is a momentum-balance statement between integrated momentum changes within \( V(t) \), integrated momentum contributions from the motion of \( A(t) \), and integrated volume and surface forces.
It is the momentum conservation equivalent of
\(\boxed{\int_{V(t)} \frac{\partial \rho(\mathbf{x}, t)}{\partial t} \, dV
+ \int_{A(t)} \rho(\mathbf{x}, t) \, \mathbf{u}(\mathbf{x}, t) \cdot \mathbf{n} \, dA = 0}\), a mass-balance statement between integrated density changes within \(V(t)\) and integrated motion of its surface \(A(t)\)
◀
▶