Let,
V → Control Volume
B → Flow property
b → Intensive value of B in any small element of the fluid
ρ → Density of the flow
t → Instantaneous time
Which of these terms represent the ‘instantaneous total change of the flow property within the control volume’ after Leibniz rule is applied?
a) \( \frac{d}{dt}(\int_vb \rho dV)\)
b) \( \int_v \frac{\partial}{\partial T}(b \rho)dV\)
c) \(\rho \int_v \frac{\partial b}{\partial T} dV \)
d) \(\rho \int_v \frac{\partial \rho}{\partial b} dV\)

Answer: b
Explanation: According to Leibniz rule, if the variation of f(x, t) is independent of t,
\( \frac{d}{dx} \int f(x,t)dt = \int \frac{\partial}{\partial x}f(x,t)dt\)
Instantaneous total change of the flow property within the control volume is given by,
\( \frac{d}{dt}(\int_vb \rho dV)\)
Applying Leibniz rule,
\( \frac{d}{dt}(\int_vb \rho dV) = \rho \int_v \frac{\partial b}{\partial T}dV\).