The derivation of the advection-diffusion equation uses $\nabla\cdot(c\vec{v})=(\vec{v}\cdot\nabla)c$. Why doesn't the order of the derivative matter?

Question

In a derivation of the advection-diffusion equation, it is exploited that $\vec{\nabla} \cdot (c \vec{v}) = ( \vec{v}\cdot \vec{\nabla})c$, where $\vec{v}$ and c respectively are the velocity and the concentration. How can the order of the gradient not matter?

Answer 1

If $c$ and $\vec v$ is an arbitrary pair of functions, then the identity you wrote is false; instead it must read $$ \nabla \cdot (c\vec v) = (\vec v\cdot \nabla) c + c (\nabla \cdot \vec v), $$ which is easy to prove component-wise.

If your text is disregarding the second term, then presumably they're working under conditions where $\nabla \cdot \vec v = 0$. That's a natural assumption to have if $\vec v$ is the velocity field of an incompressible static flow, but you'll need to check your text for exactly what reasoning underlies that assumption.