\[ \int_{V(t)} \frac{\partial}{\partial t} \left( \rho(\mathbf{x}, t) \, \mathbf{u}(\mathbf{x}, t) \right) dV + \int_{A(t)} \rho(\mathbf{x}, t) \, \mathbf{u}(\mathbf{x}, t) \, (\mathbf{u}(\mathbf{x}, t) \cdot \mathbf{n}) \, dA = \int_{V(t)} \rho(\mathbf{x}, t) \, \mathbf{g} \, dV + \int_{A(t)} \mathbf{f}(\mathbf{n}, \mathbf{x}, t) \, dA \]

To develop an integral equation that represents momentum conservation for an arbitrarily moving control volume \( V^*(t) \) with surface \( A^*(t) \), it must be modified to involve integrations over \( V^*(t) \) and \( A^*(t) \). The steps in this process are entirely analogous to those taken for conservation of mass
Set \( \mathbf{F} = \rho \mathbf{u} \) in Reynolds transport theorem \(\boxed{\frac{d}{dt} \int_{V^*(t)} F(\mathbf{x}, t) \, dV = \int_{V^*(t)} \frac{\partial F(\mathbf{x}, t)}{\partial t} \, dV + \int_{A^*(t)} F(\mathbf{x}, t) \, \mathbf{b} \cdot \mathbf{n} \, dA}\) and rearrange it to obtain \[ \int_{V^*(t)} \frac{\partial}{\partial t} \left( \rho(\mathbf{x}, t) \, \mathbf{u}(\mathbf{x}, t) \right) dV = \frac{d}{dt} \int_{V^*(t)} \rho(\mathbf{x}, t) \, \mathbf{u}(\mathbf{x}, t) dV - \int_{A^*(t)} \rho(\mathbf{x}, t) \, \mathbf{u}(\mathbf{x}, t) \, \mathbf{b} \cdot \mathbf{n} \, dA \] Choose \( V^*(t) \) to be instantaneously coincident with \( V(t) \) so that at the moment of interest \( \int_{V(t)} \frac{\partial}{\partial t} \left( \rho(\mathbf{x}, t) \, \mathbf{u}(\mathbf{x}, t) \right) dV = \int_{V^*(t)} \frac{\partial}{\partial t} \left( \rho(\mathbf{x}, t) \, \mathbf{u}(\mathbf{x}, t) \right) dV \quad \textcolor{#7DCDF4}{(a)} \)
\( \int_{A(t)} \rho(\mathbf{x}, t) \, \mathbf{u}(\mathbf{x}, t) \left( \mathbf{u}(\mathbf{x}, t) \cdot \mathbf{n} \right) dA = \int_{A^*(t)} \rho(\mathbf{x}, t) \, \mathbf{u}(\mathbf{x}, t) \left( \mathbf{u}(\mathbf{x}, t) \cdot \mathbf{n} \right) dA \quad \textcolor{#7DCDF4}{(b)} \)
\( \int_{V(t)} \rho(\mathbf{x}, t) \, \mathbf{g} \, dV = \int_{V^*(t)} \rho(\mathbf{x}, t) \, \mathbf{g} \, dV \quad \textcolor{#7DCDF4}{(c)} \)
\( \int_{A(t)} \mathbf{f}(\mathbf{n}, \mathbf{x}, t) \, dA = \int_{A^*(t)} \mathbf{f}(\mathbf{n}, \mathbf{x}, t) \, dA \quad \textcolor{#7DCDF4}{(d)} \)
Substitute \(\textcolor{#7DCDF4}{(a)}\) into \(\boxed{\int_{V^*(t)} \frac{\partial}{\partial t} \left( \rho(\mathbf{x}, t) \, \mathbf{u}(\mathbf{x}, t) \right) dV = \frac{d}{dt} \int_{V^*(t)} \rho(\mathbf{x}, t) \, \mathbf{u}(\mathbf{x}, t) dV - \int_{A^*(t)} \rho(\mathbf{x}, t) \, \mathbf{u}(\mathbf{x}, t) \, \mathbf{b} \cdot \mathbf{n} \, dA}\) and use this result plus \(\textcolor{#7DCDF4}{(b)}\), \(\textcolor{#7DCDF4}{(c)}\), \(\textcolor{#7DCDF4}{(d)}\) to convert to

\[ \frac{d}{dt} \int_{V^*(t)} \rho(\mathbf{x},t) \mathbf{u}(\mathbf{x},t) dV + \int_{A^*(t)} \rho(\mathbf{x},t) \mathbf{u}(\mathbf{x},t) \left( \mathbf{u}(\mathbf{x},t) - \mathbf{b} \right) \cdot \mathbf{n} \, dA = \int_{V^*(t)} \rho(\mathbf{x},t) \mathbf{g} \, dV + \int_{A^*(t)} \mathbf{f}(\mathbf{n}, \mathbf{x}, t) \, dA \]

This is the general integral statement of momentum conservation 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. It merit some additional description that facilitates the derivation of the differential equation representing momentum conservation and allows its simplification under certain circumstances

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