Introduction to Mathematical Physics/Continuous approximation/Exercises

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