What does this notation mean where the dot appears before nabla?

Question

I am reading chapter 38 of GPU Gems and the author uses this notation where the dot appears before nabla (sec 38.2.1, eq 1, first term on the right side).

$-(\mathbf u \cdot \nabla)\mathbf u$

The book explains that nabla followed by dot means divergence but they've reversed the order in there. Is it "gradient of u"?

Answer 1

$$(\mathbf u \cdot \nabla)=\sum_{i=1}^{n}u_i\frac{\partial}{\partial x_i}$$

Where $u_i$ is the $i$th component of $\mathbf u$ and $n$ is the number of dimensions.

Answer 2

This is one term in the famous Navier-Stokes equations.

In your 2D context, the boldface symbol $\mathbf{u}=(u,v)$ is a vector; the operator $\mathbf{u}\cdot \nabla$ is simply $u\partial_1+v\partial_2$. Imagine you do inner product symbolically. And the vector $$ (\mathbf{u}\cdot \nabla)\mathbf{u}=(u\partial_1u+v\partial_2u,u\partial_1v+v\partial_2v). $$