# [Plasma physcis] fluid equation

1. Aug 3, 2009

### good_phy

Hi i just learned fluide eqation form chen textbok

The equation is $$mn\left[ \frac{\partial u}{\partial t}+(u\cdot\nabla)u\right]=qn(E+u\times B)-\nabla\cdot P - \frac{mn(u-u_{o})}{\tau}$$. where P is pressure and last term is collision between charged particle and the nutral

I was confusing what is exactly $$\nabla\cdot P$$, force from pressure.

I think in strictly saying, in microsopic view, only electromagnetic force is imposing on each particle and pressure

is also E.M force. and E and B in this equation is Macroscopic field ( external field), not

microscopic field.

Is it right?

2. Aug 3, 2009

### Andy Resnick

That equation looks analogous to ma=F, but written in terms of continuum mechanics. The LHS is ma, or m Dv/Dt, where the D/Dt is called the 'total', or 'material' derivative. The RHS is the force, broken down into electromagnetic (qn(E x uxB)), hydrostatic (divergence of the stress tensor) and some other term which I am unfamiliar with. Normally this is a vector equation, and the term 'P' is a stress tensor, not a (pressure) scalar.

The tensor 'P', based on what is written, is the total stress tensor which can usually be decomposed into an isotropic term (the pressure) and a term relating to the viscosity.

If the book insists that 'P' is the pressure, then it's a typo, and that term should be grad(P), not div(P). Also, the text assumes the fluid is inviscid, since there's no viscosity.

3. Aug 3, 2009

### jasonRF

good_phy,

Yes, the E and B in the equation are macroscopic. The divergence of the pressure tensor represents the force associated with the gas pressure; usually in a plasma the pressure tensor is diagonal to a very good approximation (the off-diagonal terms represent viscosity). It is not a scalar since, in general, the temperatures can be different parallel and perpendicular to a magnetic field. Don't worry about these funny cases at first - get used to working with scalar pressure and you will get a good feel for the physics.

Jason

4. Aug 4, 2009

### good_phy

Thank you for sharing me. Then i'll ask you what is the good plasma physcis textbook

which is good combination with Chen book?

did you get any idea?

5. Aug 4, 2009

### Born2bwire

Ha, I always end up recommending Francis Chen's book in the first place.

6. Aug 4, 2009

### jasonRF

I will agree with Born2bwire: Chen's book is probably the best place to start. It is clearly written, avoids non-essential mathematics that would add no insight, and contains a ton of physics. If you really want another source, check out the e-book by Fitzpatrick:

http://farside.ph.utexas.edu/teaching/plasma/plasma.html

It is more advanced than Chen but is quite well written and is free! I also like his coverage of waves a little better than Chen's.

Good luck,

Jason

7. Feb 8, 2012

Thanks