Continuity Equation

Another basic equation comes from the mass conservation. This is often called the continuity equation, which relates the change of the volume to its density. Consider a fluid element whose volume is equal to $\Delta V$. The mass contained in the volume is constant. Thus

$$\frac{d \rho \Delta V}{d t}=\frac{d \rho}{d t}\Delta V +\frac{d \Delta V}{d t}\rho=0.$$ (A.8)

The variation of the volume $\frac{d \Delta V}{d t}$ is rewritten as

$$\frac{d \Delta V}{d t}=\int_{\partial \Delta V}{\bf v}\cdot d{\bf S}=\int_{\Delta V}{\rm div}{\bf v} dV,$$ (A.9)

where $\partial \Delta V$ represents the surface of the fluid element $\Delta V$. From equations (A.8) and (A.9), we obtain the mass continuity equation for Lagrangian time derivative as

$$\frac{d \rho}{d t}+\rho{\rm div} {\bf v}=0.$$ (A.10)

Using equation (A.6) this is rewritten to Eulerian form as

$$\frac{\partial \rho}{\partial t}+{\rm div} (\rho{\bf v})=0.$$ (A.11)

Basic equations using the Lagrangian derivative are equations (A.3) and (A.10), while those of the Euler derivative are equations (A.7) and (A.11).