Fluid Dynamics – Conservation of Mass from Material Derivative

conservation-lawsdifferentiationfluid dynamicsmassnavier-stokes;

Let the mass be $m=\rho \text{Vol}$, where $\text{Vol}$ is the volume of the domain and the velocity is $u$.

Applying the material derivative, then

$$\frac{Dm}{Dt}=\frac{\partial (\rho \text{Vol})}{\partial t}+ u \cdot \nabla (\rho \text{Vol})=0$$

Since the volume is constant, then it reads

$$\frac{\partial \rho}{\partial t}+ u \cdot \nabla \rho=0$$

However, the conservation of mass (the correct one as far as I know)

$$\frac{\partial \rho}{\partial t}+ \nabla \cdot (\rho u)=0$$

It is not clear What's wrong I am doing here.

Best Answer

What is wrong is your assumption that Vol is constant. What is constant is the mass within the volume "Vol". This is $M=\rho\, {\rm Vol}$. So as $\rho$ increases ${\rm Vol}$ decreases.