Plasma Fluid Mechanics - Convective Derivatives

[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!

 

[anathema]

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