Question 4.11: Reynolds Transport Theorem in Terms of Relative Velocity Beg...

Equation 4–44 is to be proven.
Analysis   The general three-dimensional version of the Leibniz theorem, Eq. 4–50, applies to any volume. We choose to apply it to the control volume of interest, which can be moving and/or deforming differently than the material volume (Fig. 4–64). Setting G to 𝜌b, Eq. 4–50 becomes

\frac{d}{dt}\int\limits_{V(t)}{G(x, y, z, t)}  dV = \int\limits_{V(t)}{\frac{∂G}{∂t} }  dV + \int\limits_{A(t)}{G\vec{V}_A.\vec{n}  dA }             (4.50)

\frac{d}{dt} \int\limits_{CV}{\rho b  dV} = \int\limits_{CV}{\frac{∂}{∂t}(\rho b)  dV } + \int\limits_{CS}{\rho b\vec{V}_{CS}.\vec{n}  dA }         (1)

We solve Eq. 4–53 for the control volume integral,

\int\limits_{CV}{\frac{∂}{∂t}(\rho b)  dV } = \frac{dB_{sys}}{dt} – \int\limits_{CS}{\rho b\vec{V}.\vec{n}  dA }         (2)

Substituting Eq. 2 into Eq. 1, we get

\frac{d}{dt} \int\limits_{CV}{\rho b  dV} = \frac{dB_{sys}}{dt} – \int\limits_{CS}{\rho b\vec{V}.\vec{n}  dA } + \int\limits_{CS}{\rho b\vec{V}_{CS}.\vec{n}  dA }             (3)

Combining the last two terms and rearranging,

\frac{dB_{sys}}{dt} = \frac{d}{dt} \int\limits_{CV}{\rho b  dV} + \int\limits_{CS}{\rho b(\vec{V} – \vec{V}_{CS} ).\vec{n}  dA}        (4)

But recall that the relative velocity is defined by Eq. 4–43. Thus,

\vec{V}_r = \vec{V} – \vec{V}_{CS}           (4.43)

RTT in terms of relative velocity:    \frac{dB_{sys}}{dt} = \frac{d}{dt}\int\limits_{CV}{\rho b  dV} + \int\limits_{CS}{\rho b\vec{V}_{r}.\vec{n}  dA }      (5)

Discussion Equation 5 is indeed identical to Eq. 4–44, and the power and elegance of the Leibniz theorem are demonstrated.