# 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

$n_e m_e(\frac{\partial v_e}{\partial t}+({v}_e.\nabla v_e))=en_e\nabla \phi - e n_e v_e \wedge B -\nabla p_e$

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 $v_e^0=-\frac{k_B T_e \nabla_x n^0}{e B n^0} e_{\theta}$.
• $E_\perp$ 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: $p_a=n_ak_BT_a$.
• $T_e$, $T_i$ have the values they have at equilibrium.