How do I apply the weird vector operator (in the Navier-Stokes Convection Term) to a vector?

Question

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

Answer 1

$$\nabla = \frac{\partial}{\partial x}\hat i + \frac{\partial}{\partial y}\hat j + \frac{\partial}{\partial z}\hat k$$

If $u = u_1\hat i + u_2\hat j + u_3\hat k$, then $$u\cdot \nabla = u_1 \frac{\partial}{\partial x} + u_2\frac{\partial}{\partial y} + u_3\frac{\partial}{\partial z}$$

Note that while $\nabla$ carries a real-valued function to a vector field, $u\cdot \nabla$ carries a real-valued function to another real-valued function. Thus you can apply it component-wise to another vector field $v$:

$$(u\cdot \nabla) v = \begin{bmatrix}u_1 \frac{\partial v_1}{\partial x} + u_2\frac{\partial v_1}{\partial y} + u_3\frac{\partial v_1}{\partial z}\\u_1 \frac{\partial v_2}{\partial x} + u_2\frac{\partial v_2}{\partial y} + u_3\frac{\partial v_2}{\partial z}\\u_1 \frac{\partial v_3}{\partial x} + u_2\frac{\partial v_3}{\partial y} + u_3\frac{\partial v_3}{\partial z}\end{bmatrix}$$

It measures how fast $v$ is changing in the direction of $u$.

Answer 2

Dear confused future self...

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

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

which is a scalar

scalar multiplication is component-wise

$$\lambda\textbf{v}=\lambda \begin{bmatrix} u\\ v\\w\end{bmatrix}=\begin{bmatrix} \lambda u\\ \lambda v\\\lambda w\end{bmatrix}$$

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

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