A body of continuous matter with volume V(t) and surface area A(t) (Figure 2) has, according to the principle of conservation of mass, the same mass at any time. The mass \( m \) may be expressed by the integral: mathematical terms may be expressed by: $$ \begin{equation} m = \Int_{V(t)} \rho(\mathbf{r},t) \, dV \tag{2.35} \end{equation} $$ and consequently due to the above mentioned principle: $$ \begin{equation} \frac{d m}{dt} =\dot{m} = \frac{d}{dt} \Int_{V(t)} \rho(\mathbf{r},t) \, dV = 0 \tag{2.36} \end{equation} $$ The conservation of mass equation is obtained by setting \( b=\rho \) in Reynolds transport theorem (2.30) $$ \begin{equation} \dot{m}= \frac{d}{dt} \Int_{V(t)} \rho(\mathbf{r},t) \, dV = \Int_V \partd{\rho}{t} \, dV + \Int_A \rho \, (\mathbf{v \cdot n}) \, dA = 0 \tag{2.37} \end{equation} $$
Note that \( V(t) \) denote the time-varying volume of the body at a given time \( t \), whereas \( V \) and \( A \) are the fixed volume and surface area of a control volume, respectively. The control volume \( V=V(t) \) at time \( t \).
By using the Gauss theorem the surface integral in (2.37) is transformed to a volume integral and the two integrals in (2.37) collapse to one: $$ \begin{equation} \dot{m} = \Int_V \partd{\rho}{t}+ \nabla \cdot (\rho \mathbf{v}) \, dV = 0 \tag{2.38} \end{equation} $$
As (2.38) must be valid for arbitrary volumes \( V \) the integrand must be zero and a differential form for mass conservation is obtained: $$ \begin{equation} \partd{\rho}{t}+ \nabla \cdot (\rho \mathbf{v}) = 0 \tag{2.39} \end{equation} $$ This is an differential equation which represents mass conservation in an Eulerian reference frame. The importance of the reference frame will become clearer in the chapter 3 Deformation, where the concepts of deformation are introduced.
An equivalent presentation of the mass conservation on component form reads: $$ \begin{equation} \partd{\rho}{t}+ (\rho v_i)_{,i} = 0 \tag{2.40} \end{equation} $$ For incompressible fluid the mass density \( \rho \) is constant and (2.40) reduces to: $$ \begin{equation} \tag{2.41} \nabla \cdot \mathbf{v} = 0 \quad \Leftrightarrow \quad v_{i,i}=0 \end{equation} $$