# 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 ${\displaystyle n_{e},v_{e},p_{e}}$ characterizes the electrons and set ${\displaystyle n_{i},v_{i},p_{i}}$ characterizes the ions. The momentum conservation equation for the electrons is:

me

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

The momentum conservation equation for the ions is:

mi

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

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

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