Notation (v*gradient operator)v in Navier-Stokes

[TylerH]

Notation "(v*gradient operator)v" in Navier-Stokes

What does $\left( \textbf{v} \cdot \nabla \right) \textbf{v}$ mean, assuming knowledge of the gradient operator? And, specifically, how would that be expanded? In general, I'm ignorant of the notation $\left( f \left( y, \frac{d}{dx}, \frac{d^2}{dx^2}, \ldots \right) y \right)$ where, for example, $f \left( y, \frac{d}{dx}, \frac{d^2}{dx^2}, \ldots \right)$ could be $y + \frac{d}{dx}$.

 

[HallsofIvy]

That is, by the way, the "del" or "nabla" operator. It is specifically only $\nabla f(x,y,z$, with f a scalar valued function, that is the "gradient". $\nabla\cdot \vec{v}$ is the "gradient" or grad $\vec{v}$ and $\nabla\times\vec{v}$ is the curl $\vec{v}$.

In any case
$$\vec{v}\cdot\nabla= v_x\frac{\partial }{\partial x}+ v_y\frac{\partial }{\partial y}+ v_z\frac{\partial }{\partial z}$$
so that
$$\vec{v}\cdot\nabla \vec{v}= \left(v_x\frac{\partial }{\partial x}+ v_y\frac{\partial }{\partial y}+ v_z\frac{\partial }{\partial z}\right)v_x\vec{i}$$$$+ \left(v_x\frac{\partial }{\partial x}+ v_y\frac{\partial }{\partial y}+ v_z\frac{\partial }{\partial z}\right)v_y\vec{j}$$$$+ \left(v_x\frac{\partial }{\partial x}+ v_y\frac{\partial }{\partial y}+ v_z\frac{\partial }{\partial z}\right)v_z\vec{k}$$
$$= \left(v_x\frac{\partial v_x}{\partial x}+ v_y\frac{\partial v_x}{\partial y}+ v_z\frac{\partial v_x}{\partial z}\right)\vec{i}$$$$+ \left(v_x\frac{\partial v_y}{\partial x}+ v_y\frac{\partial v_y}{\partial y}+ v_z\frac{\partial v_y}{\partial z}\right)\vec{j}$$$$+ \left(v_x\frac{\partial v_z}{\partial x}+ v_y\frac{\partial v_z}{\partial y}+ v_z\frac{\partial v_z}{\partial z}\right)\vec{k}$$

 

[TylerH]

Oh, I didn't understand it as well as I thought I did. Thanks.

Just wondering, is $\left( y + \frac{d}{dx} \right)y$ meaningless, or can it be expanded? If it can, what is it?

Thanks for your time.

 

[hunt_mat]

it can be expanded as:
$$y+\frac{dy}{dx}$$