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

• TylerH
In summary, the notation "(v*gradient operator)v" in Navier-Stokes refers to the dot product of the vector v and the gradient operator, which can be expanded as the sum of the partial derivatives of the vector components with respect to the coordinates. The del or nabla operator is specifically used for scalar valued functions, and \nabla\cdot\vec{v} and \nabla\times\vec{v} represent the gradient and curl of the vector v, respectively. Additionally, the notation \left( f \left( y, \frac{d}{dx}, \frac{d^2}{dx^2}, \ldots \right) y \right) can be expanded as y
TylerH

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 $y + \frac{d}{dx}$.

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

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?

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

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.

## What is the "Notation (v*gradient operator)v in Navier-Stokes"?

The notation (v*gradient operator)v in Navier-Stokes refers to a mathematical expression used in the Navier-Stokes equations to describe the rate of change of a fluid's velocity over time and space.

## How is the (v*gradient operator)v term used in the Navier-Stokes equations?

In the Navier-Stokes equations, the (v*gradient operator)v term is used to represent the convection of momentum in a fluid. It describes the transfer of momentum from one point in the fluid to another due to the velocity gradient.

The gradient operator (∇) in (v*gradient operator)v represents the change in velocity over a small distance in space. It is commonly used in mathematical expressions to represent the rate of change of a function with respect to its variables.

## Why is the (v*gradient operator)v term important in the Navier-Stokes equations?

The (v*gradient operator)v term is important in the Navier-Stokes equations because it accounts for the effects of fluid motion and turbulence on the overall behavior of the fluid. It helps to describe the complex interactions between different regions of the fluid and how they influence the fluid's overall motion.

## Are there any other notations used to represent the (v*gradient operator)v term in the Navier-Stokes equations?

Yes, there are other notations that can be used to represent the (v*gradient operator)v term in the Navier-Stokes equations, such as ∇⋅(v⊗v) or v⋅∇v. These notations are equivalent and are used interchangeably depending on the context and personal preference.

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