Conservation of mass for a system: $ \frac{Dm_{sys}}{Dt}=0 $ Rate of change of mass within an enclosed system is equal to zero $ \underbrace{...}_{\substack{\text{Some long text that} \\ \text{should be multiline}}} $ For a control volume: $ \underbrace{\frac{\partial}{\partial t}\int_{CV}\rho dV}_{\substack{\text{rate of change of mass} \\ \text{in control volume}}}+\underbrace{\int_{CS}\rho\vec{V}\cdot\hat{n}\,dA}_{\substack{\text{mass flux across} \\ \text{control surface}}}=0 $ Consider an infinitesimal fluid element with sides $dx$, $dy$, and $dz$, with a density $\rho$ located at the element centroid. Per unit volume the amount of mass leaving the element in the $x$ direction is: $ \left[\rho u+\frac{\partial (\rho u)}{\partial x}\frac{dx}{2}\right]dydz $ Similarly, the amount of mass entering the element in the $x$ direction is: $ \left[\rho u-\frac{\partial(\rho u)}{\partial x}\frac{dx}{2}\right]dydz $ Therefore the net mass flux is: $ \frac{\partial(\rho u)}{\partial x}dxdydz $ This is extended to the $y$ and $z$ directions: $ \frac{\partial(\rho v)}{\partial y}dxdydz $ $ \frac{\partial(\rho w)}{\partial z}dxdydz $