Introduction to Mathematical Physics/Continuous approximation/Exercises

Exercice:

Give the equations governing the dynamics of a plate (negligible thickness) from powers taking into account the gradient of the speeds (first gradient theory). Compare with a approach starting form conservation laws.

Exercice:

Same question as previous problem, but with a rope clamped between two walls.

Exercice:

exoplasmapert

A plasma\index{plasma} is a set of charged particles, electrons and ions. A classical model of plasma is the "two fluid model": the system is described by two sets of functions density, speed, and pressure, one for each type of particles, electrons and ions: set $n_{e},v_{e},p_{e}$ characterizes the electrons and set $n_{i},v_{i},p_{i}$ characterizes the ions. The momentum conservation equation for the electrons is:

me

$n_{e}m_{e}({\frac {\partial v_{e}}{\partial t}}+({v}_{e}.\nabla v_{e}))=en_{e}\nabla \phi -en_{e}v_{e}\wedge B-\nabla p_{e}$

The momentum conservation equation for the ions is:

mi

$n_{i}m_{i}({\frac {\partial v_{i}}{\partial t}}+({v}_{i}.\nabla v_{i}))=-en_{i}\nabla \phi +en_{i}v_{i}\wedge B-\nabla p_{i}.$

Solve this non linear problem (find solution $n_{i}({\vec {r}},t)$) assuming:

• $B$ field is directed along direction $z$ and so defines parallel direction and a perpendicular direction (the plane perpendicular to the $B$ field).
• speeds can be written $v_{a}={\tilde {v}}_{a}+v_{a}^{0}$ with $v_{i}^{0}=0$ and $v_{e}^{0}=-{\frac {k_{B}T_{e}\nabla _{x}n^{0}}{eBn^{0}}}e_{\theta }$.
• $E_{\perp }$ field can be written: $E_{\perp }=0+{\tilde {E}}_{\perp }$.
• densities can be written $n_{a}=n_{0}+{\tilde {n}}_{a}$. Plasma satisfies the quasi--neutrality condition\index{quasi-neutrality} : ${\tilde {n}}_{e}={\tilde {n}}_{i}={\tilde {n}}$.
• gases are considered perfect: $p_{a}=n_{a}k_{B}T_{a}$.
• $T_{e}$, $T_{i}$ have the values they have at equilibrium.