Differential continuity equation for Fluid (conservation of mass)

Question

I know the equation can be written in this way:

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

How can i write this equation to $$\frac{\partial \rho}{\partial t} + \rho \nabla \cdot ( \vec{v})=0$$

Answer 1

In general, $\nabla \cdot (\rho \vec v)=\nabla (\rho) \cdot \vec v+\rho \nabla \cdot (\vec v)$. If $\nabla \rho =0$, then

$$\nabla \cdot (\rho \vec v)=\rho \nabla \cdot (\vec v)$$

Note that $\nabla \rho =0$ implies that $\rho$ does not depend on the spatial coordinates.