Derived with \( b=\rho \) in Reynolds Transport Theorem Eq. (2.33) $$ \begin{equation} \dot{m}= \frac{d}{dt} \Int_{V(t)} \rho(\boldsymbol{r},t) \, dV = \Int_V \partd{\rho}{t} \, dV + \Int_A \rho \, (\boldsymbol{v \cdot n}) \, dA = 0 \tag{5.7} \end{equation} $$
Field equation $$ \begin{equation} \partd{\rho}{t}+ \nabla \cdot (\rho \boldsymbol{v}) = 0 \tag{5.8} \end{equation} $$ $$ \begin{equation} \partd{\rho}{t}+ (\rho v_i)_{,i} = 0 \tag{5.9} \end{equation} $$
Equivalent presentations of conservation of mass.
Conservative formulation: $$ \begin{equation} \partd{\rho}{t} + (\rho v_i)_{,i} = 0 \quad \Leftrightarrow \quad \partd{\rho}{t} + \nabla \cdot (\rho \boldsymbol{v}) = 0 \tag{5.10} \end{equation} $$ Non-conservative formulation $$ \begin{equation} \partd{\rho}{t}+ \rho_{,i} v_i + \rho v_{i,i} = 0 \quad \Leftrightarrow \quad \dot{\rho} + \rho \nabla \cdot \boldsymbol{v}= 0 \tag{5.11} \end{equation} $$ where we have introduced the material derivative of \( \rho \) is defined in Eq. (2.20).
Mass conservation for incompressible flow (\( \rho= \) constant) $$ \begin{equation} \partd{\rho}{t}+ \nabla \cdot (\rho \boldsymbol{v}) = 0 \tag{5.12} \end{equation} $$ $$ \begin{equation} \nabla \cdot \boldsymbol{v} = 0 \quad \Leftrightarrow \quad v_{i,i}=0 \tag{5.13} \end{equation} $$