# Plasma Fluid Mechanics - Convective Derivatives

• Tomishiyo
In summary: Substituting this into our equation, we get:\frac{\partial}{\partial t}(\rho \vec{v})+(\vec{v}\cdot\nabla)\rho \vec{v}+(\rho \vec{v}\cdot\nabla)\vec{v}+(\vec{v}\cdot\nabla)(\rho \vec{v})-(\rho \
Tomishiyo

## Homework Statement

Use the continuity and momentum conservation equations for a single species to construct the following "convective derivative" equation for the fluid velocity:

$$\frac{\partial\vec{v}}{\partial t}+\vec{v}\cdot\nabla\vec{v}=\vec{g}-\frac{1}{\rho}\nabla p +\frac{q}{m}(\vec{E}+\vec{v}\times\vec{B}),$$
where it was assumed that the stress tensor is approximately given by $\Pi_{ij}=p\delta_{ij}$, with $p=nkT$ beign the pressure for that species.

## Homework Equations

1.Continuity equation:
$$\frac{\partial \rho}{\partial t}+\nabla\cdot(\rho\vec{v})$$
2.Momentum conservation equation
$$\frac{\partial}{\partial t}(\rho\vec{v})+\nabla\cdot(\rho \vec{v} \vec{v} )+\nabla \cdot\Pi=\rho\vec{g}+\rho_{e}\vec{E}+\vec{J}\times \vec{B}$$

## The Attempt at a Solution

Ok, first problem I have: those equations I wrote were derived in class, and I got no textbook to study the subject. So there are some terms that are quite unclear to me. I would appreciate some book indications (plasma books, specially the ones good in the fluids part).

My attempt at the solution so far is to write the lhs of the momentum conservation equation as:

$$\frac{\partial}{\partial t}(\rho \vec{v})+\nabla \cdot (\vec{v} \vec{v})+\nabla \cdot \Pi = \vec{v}\frac{\partial \rho}{\partial t}+\rho\frac{\partial \vec{v}}{\partial t}+\vec{v}\nabla\cdot(\rho\vec{v})+\rho\vec{v} \nabla \cdot \vec{v}+\nabla\cdot\Pi$$
But by the continuity equation, the third term in the rhs is $\vec{v}\nabla \cdot (\rho\vec{v})=-\vec{v}\frac{\partial \rho}{\partial t}$ and thus the equation becames:

$$\frac{\partial}{\partial t}(\rho \vec{v})+\nabla \cdot (\vec{v} \vec{v})+\nabla \cdot \Pi=\rho\frac{\partial \vec{v}}{\partial t}+\rho\vec{v} \nabla \cdot \vec{v} + \nabla \cdot \Pi.$$

And that is pretty much all I could do so far. I know $\rho_{e}\vec{E}$ is a very small and usually negligible, and I think second term in the rhs of the equation above can be transformed using some kind of vectorial identity like:

$$\nabla \dot (f\vec{A})=f \nabla \cdot \vec{A}+\vec{A}\cdot \nabla f,$$
if I only could grante that the relation holds for tensorial quantities like the vectors in the equation (and also, I would need one of the terms to vanish in the identity displayed). I have absolutely no idea on how to get rid of $\vec{J}$ or $\nabla \cdot \Pi$.

Thank you for your time!

Thank you for your post. Let me try to help you with your problem. First of all, I would recommend looking into the book "Introduction to Plasma Physics" by Francis F. Chen. It covers the basics of plasma physics including fluid equations and is a great resource for beginners.

Now, let's get to your problem. The first thing you need to do is to write the momentum conservation equation in terms of the fluid velocity instead of the momentum density. This can be done by using the definition of momentum density, which is \rho \vec{v}. So, the equation becomes:

\frac{\partial}{\partial t}(\rho \vec{v})+\nabla \cdot (\rho \vec{v} \vec{v})+\nabla \cdot \Pi=\rho \vec{g}+\rho_{e}\vec{E}+\vec{J}\times\vec{B}
Now, using the continuity equation, we can simplify the second term in the lhs as you did before. But, instead of using the vector identity, we can use the product rule for divergence, which states that:

\nabla \cdot(\vec{A}\vec{B})=(\vec{A}\cdot\nabla)\vec{B}+(\vec{B}\cdot\nabla)\vec{A}+\vec{A}\times\nabla\times\vec{B}+\vec{B}\times\nabla\times\vec{A}
Applying this to our problem, we get:

\frac{\partial}{\partial t}(\rho \vec{v})+(\vec{v}\cdot\nabla)\rho \vec{v}+(\rho \vec{v}\cdot\nabla)\vec{v}+\vec{v}\times\nabla\times(\rho \vec{v})+\nabla\cdot\Pi=\rho \vec{g}+\rho_{e}\vec{E}+\vec{J}\times\vec{B}
Now, using the vector identity \nabla\times(\vec{a}\times\vec{b})=\vec{a}(\nabla\cdot\vec{b})-\vec{b}(\nabla\cdot\vec{a})+(\vec{b}\cdot\nabla)\vec{a}-(\vec{a}\cdot\nab

## 1. What is plasma fluid mechanics?

Plasma fluid mechanics is a branch of physics that studies the behavior of ionized gases, or plasmas, and their interactions with electric and magnetic fields.

## 2. What are convective derivatives in plasma fluid mechanics?

Convective derivatives refer to the terms in the equations of motion that describe the transport of plasma properties, such as mass, momentum, and energy, due to fluid motion.

## 3. How do convective derivatives affect plasma behavior?

Convective derivatives play a crucial role in determining the dynamics of plasmas, as they govern the rate at which plasma properties are transported and mixed by fluid motion.

## 4. What are some applications of plasma fluid mechanics?

Plasma fluid mechanics has many practical applications, including plasma propulsion for spacecraft, plasma processing in semiconductor manufacturing, and plasma confinement in fusion energy research.

## 5. Are there any current challenges or unanswered questions in plasma fluid mechanics?

Yes, there are still many open questions and challenges in plasma fluid mechanics, such as understanding turbulence and instabilities in plasmas, developing accurate models for plasma behavior, and improving plasma confinement for fusion energy applications.

Replies
1
Views
477
Replies
3
Views
645
Replies
17
Views
914
Replies
29
Views
2K
Replies
3
Views
1K
Replies
1
Views
1K
Replies
5
Views
2K
Replies
19
Views
2K
Replies
6
Views
3K
Replies
32
Views
874