Let,
V → Control Volume
b → Intensive value of B in any small element of the fluid
ρ → Density of the flow
\(\vec{v}\) → Velocity of fluid entering or leaving the control volume
After applying Gauss divergence theorem, how does the term representing ‘net flow of B into and out of the control volume’ look like?
a) \(\int_v \nabla.(\rho \vec{v}b)dV\)
b) \(\int_s \nabla.(\rho \vec{v}b)dS\)
c) \(\int_v(\rho \vec{v}b)dV\)
d) \(\int_s(\rho \vec{v}b)dS\)

Answer: a
Explanation: The term representing ‘net flow of B into and out of the control volume’ is
\(\int_s b \rho \vec{v}.\vec{n}dS\)
Applying Gauss divergence theorem,
\(\int_s b \rho \vec{v}.\vec{n}dS = \int_v \nabla.(\rho \vec{v}b)dV\).