Fluid Conservation of Momentum

This is an application of the Reynolds Transport Theorem where the property $b$ we are interested in is velocity $\mathbf{V}$, such that

$$b=\mathbf{V}⟶B=m\mathbf{V}$$

which then gives

$$\begin{array}{rl}{B}_{\text{sys}}& ={\int }_{\text{sys}}\mathbf{V}\rho dd\forall ={\mathbf{M}}_{\text{sys}}\\ B_{\text{C.V.}}& ={\int }_{\text{C.V.}}\mathbf{V}\rho d\forall ={\mathbf{M}}_{\text{C.V.}}\end{array}$$

This gives us the linear momentum equation, which basically says that the time rate of change of linear momentum of the system is equal to the sum of external forces acting on the system.

$$\frac{D{\mathbf{M}}_{\text{sys}}}{Dt}=\frac{\partial }{\partial t}{\int }_{\text{C.V.}}\mathbf{V}\rho d\forall +{\int }_{\text{C.S.}}\mathbf{V}\rho \mathbf{V}\cdot \mathbf{n}dA=\sum {\mathbf{F}}_{\text{content of C.V.}}$$

where:

• $\frac{D{\mathbf{M}}_{\text{sys}}}{Dt}$ is the momentum change in the system
• $\frac{\partial }{\partial t}{\int }_{\text{C.V.}}\mathbf{V}\rho d\forall$ is the momentum change in the C.V.
• ${\int }_{\text{C.S.}}\mathbf{V}\rho \mathbf{V}\cdot \mathbf{n}dA$ is the net momentum flow rate through the C.S.

Note that we can also write

$$\frac{D{\mathbf{M}}_{\text{sys}}}{Dt}=\frac{D(m_{\text{sys}}\mathbf{V})}{Dt}=m_{\text{sys}}\frac{D\mathbf{V}}{Dt}=m_{\text{sys}}\mathbf{a}=\sum {\mathbf{F}}_{\text{sys}}$$

For moving C.V.:

$$\frac{\partial }{\partial t}{\int }_{\text{C.V.}}\mathbf{V}\rho d\forall +{\int }_{\text{C.S.}}\mathbf{V}\rho \mathbf{W}\cdot \mathbf{n}dA=\sum {\mathbf{F}}_{\text{content of C.V.}}$$