Hi,
Assuming the kinetic description is used for electron fluid and the hydrodynamic description for ion fluid. I am wondering what are the physical meaning and limits of such an approach.
Thanks.
Dear Zakaria Bellahsene
For the kinetic description, a fluid can be considered a dense gas, since the liquids are not diluted intermolecular forces are much more complex, and according to several authors the Boltzman approach does not correspond.
Article The kinetic theory of fluids—an introduction
In the case of hydrodynamics if the whole set of the equation is taken into account, still it describes an equilibrium process with fluid following Newton's laws since the main hydrodynamic equations follow in some sense Newton theory, of course, new equation appears such as the one for the entropy but all they can be treated by a more understood formalism, it is my guess.
Best Regards.
First of all, if a system is described kinetically or hydrodynamically is not dependent on the species, i.e. if you look at electrons or ions. The only thing that matters here is the ratio of the collision mean free path and the scale length of your system. If this ratio is small, i.e. you have many collisions in a comparable large volume, you can use hydrodynamics. If this ratio is large you will be better of with kinetics.
The reason for this is that for hydrodynamics you need statistical approaches. Hence there should be a lot of particles in your system. If you have only a few particles you can work with kinetics, meaning that you basically follow all particles individually.
Furthermore, what P. Contreras said is not entirely correct. Using hydrodynamics does not mean that you have more or fewer equations to solve than in kinetics but the structure of the equations is different. Both approaches follow Newton's law of motion and that new equations 'appear' is also not precise. In fact, you could also treat many particles with simple equations of motion, if you have enough time or computer power to solve this. If you have a stream of water, for example, you can still treat each water molecule individually but it is far more efficient if you just use the hydrodynamic approach. Also entropy doesn't have anything to do with how many particles you have (as long as there are more than one), it has more to do with many different microstates they can occupy in the system they are in.
I hope, this lengthy answer helps a bit.
Prof. Johannes Gruenwald thank you for your instructive and clarifying answer.
Yes, both structures of equations are different, for example, the Boltzmann equation in the tao approximation is totally different from the equation for a viscous fluid, but the role of the Newton laws appears from the beginning say in the Navier–Stokes equations were what I meant was the conservation of linear momentum and mass for Newtonian fluids, meanwhile, for ions or charged particles in fluids, we cannot use that math approach directly. If there are charged particles we need the set of magnetohydrodynamic equations.
On the other hand, the kinetic equation always expresses the total evolution of time df /dt, which in one total derivative expression contains different physical phenomena: diffusion, collisions, and newton forces, I only mean that. To solve the kinetic equation we need non-linear integro-differential methods, quite difficult to follow sometimes and yes, we can try for 2 particles, 3 particles, or N particles the collision or scattering term (following for example the Bogoliubov approach
--> which is a method for asymptotic integration of non-linear differential equations), but equilibrium is not present, always the kinetic phenomena express out of equilibrium properties.
In fluids we do not follow that complicated approach, we instead define the fluid from the beginning as something not statistical, I only mean that they are partial differential equations as we introduce them in a beginning fluid mechanics course. Of course, if we use a method such as smoothed particle hydrodynamics (SPH) what I said is not valid, since there is the kernel function that allows a statistical computational approach. And finally, yes, we can follow particle by particle in hydrodynamics as well and even include molecular forces, but it has a very high computational cost, I guess.
Thank you for the clarification.
PD The question about the inclusion of the entropy d S/d t = 0 in magnetohydrodynamics (fluid charged) and not in the kinetic equation approach is very subtle and I prefer not to discuss it here.
Best Regards.
- Badges
- Science topic