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:
$$y+\frac{dy}{dx}$$

 

[blue_raver22]

The notation \left( \textbf{v} \cdot \nabla \right) \textbf{v} in Navier-Stokes represents the convective term in the Navier-Stokes equation, which describes the transport of momentum by the fluid flow. The gradient operator, denoted by \nabla, is a vector operator that represents the rate of change of a scalar field in each direction. In this context, it is used to calculate the local change in velocity in the direction of the flow.

Expanding the notation, we have:

\left( \textbf{v} \cdot \nabla \right) \textbf{v} = \left( v_x \frac{\partial}{\partial x} + v_y \frac{\partial}{\partial y} + v_z \frac{\partial}{\partial z} \right) \left( v_x \hat{\textbf{i}} + v_y \hat{\textbf{j}} + v_z \hat{\textbf{k}} \right)

= v_x \frac{\partial v_x}{\partial x} + v_y \frac{\partial v_x}{\partial y} + v_z \frac{\partial v_x}{\partial z} \hat{\textbf{i}} + v_x \frac{\partial v_y}{\partial x} + v_y \frac{\partial v_y}{\partial y} + v_z \frac{\partial v_y}{\partial z} \hat{\textbf{j}} + v_x \frac{\partial v_z}{\partial x} + v_y \frac{\partial v_z}{\partial y} + v_z \frac{\partial v_z}{\partial z} \hat{\textbf{k}}

This notation is used to represent the convective acceleration of the fluid, which is an important factor in the Navier-Stokes equation as it takes into account the nonlinear effects of the fluid flow. It is often used in fluid dynamics to study the behavior of fluids in motion and is a crucial component in understanding complex flow patterns.