Introduction to Mathematical Physics/Continuous approximation/Exercises

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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.


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



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:


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:


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.