Chain rule on gradient of two vectors (Momentum Equation) I am confused about the application of the chain rule with the following equation: 
$$\boldsymbol{\nabla} \cdot (\rho \mathbf V \mathbf V)$$ where: 
$\mathbf V  = u \hat {\mathbf i} +v \hat{\mathbf j} + w \hat{\mathbf k}$ and $\rho $ is denoting density, a scalar quantity.
I am confused with the expansion of this once I have applied the chain rule and what the end result would look like. 
Thanks for your help! 

Consider the problem with distinct vectors.
$$\eqalign{
\nabla\cdot(\rho\,ab)
 &= (\nabla\rho)\cdot ab + \rho(\nabla\cdot a)b + \rho\,a\cdot(\nabla b) \\
}$$
Each term in the product is differentiated according the usual rule. The only vector consideration is to keep the dot product between the $\nabla$-operator and the $a$-vector.
When $a=b=v,\,$ the result can be written (with fewer parentheses) as
$$\eqalign{
\nabla\cdot(\rho\,vv)
 &= vv\cdot\nabla\rho + \rho v\,\nabla\cdot v + \rho\,v\cdot\nabla v \\
}$$
