what is divergence of $\text{div} (nu)$?

Question

Let $n$ be the density of the fluid and $u\in \mathbb R^3$ its velocity field.

My Question is: what is $\text{div} (nu)$?

Motivation: I'm trying to understand the equation: $\partial_t n + \text{div} (nu)=0, x \in \mathbb T^3, t>0$ (Navier-stack equation on 3 dimensional torus).

Answer 1

In Euclidian coordinates, we have that $$\operatorname{div}(nu)=\partial_i (nu_i)=(\partial_i n)u_i+n \partial_iu_i=\operatorname{grad}(n)\cdot u+ n \operatorname{div}(u)$$