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 characterizes the electrons and set characterizes the ions. The momentum conservation equation for the electrons is:


The momentum conservation equation for the ions is:


Solve this non linear problem (find solution ) assuming:

  • field is directed along direction and so defines parallel direction and a perpendicular direction (the plane perpendicular to the field).
  • speeds can be written with and .
  • field can be written: .
  • densities can be written . Plasma satisfies the quasi--neutrality condition\index{quasi-neutrality} : .
  • gases are considered perfect: .
  • , have the values they have at equilibrium.