Here is the operator:

$$(\textbf{v}\cdot\nabla)\textbf{v}$$

Here is a vector:

$$\textbf{v}=\begin{bmatrix}u(x,y,z)\\v(x,y,z)\\w(x,y,z)\end{bmatrix}$$

Questions:
==========

{1} What is the name of this operator?

{2} How do I apply it to my vector?

{3} Please provide some intuition as to what it is doing. Like the way Divergence is a measure of how much a source or a sink a point in a field is and how Curl is a measure of vorticity.

My attempt to operate
=====================

$$\nabla=\begin{bmatrix} \frac{\partial}{\partial x}\\ \frac{\partial }{\partial y}\\\frac{\partial }{\partial z}\end{bmatrix}$$
$$(\textbf{v}\cdot\nabla)=\begin{bmatrix}u \frac{\partial}{\partial x}\\v \frac{\partial }{\partial y}\\w \frac{\partial }{\partial z}\end{bmatrix}$$
$$(\textbf{v}\cdot\nabla)\textbf{v}=\begin{bmatrix}u \frac{\partial}{\partial x}\\v \frac{\partial }{\partial y}\\w \frac{\partial }{\partial z}\end{bmatrix}\textbf{v}$$

$$(\textbf{v}\cdot\nabla)\textbf{v}=\begin{bmatrix}u \frac{\partial u}{\partial x}\\v \frac{\partial v}{\partial y}\\w \frac{\partial w}{\partial z}\end{bmatrix}$$

Not sure if this is right! As the dot product operator should output a scalar. But this is the most logical interpretation I can come up with.

Context
=======

[![enter image description here](https://i.stack.imgur.com/H1A02.png)](https://i.stack.imgur.com/H1A02.png)