Some stability and flow problems in hydrodynamics

Abstract

(a) Exploding shell: We see that the disturbances in the plasma pressure and velocity grow with time. They manifest in the form of unbounded oscillations with their amplitudes varying as time and frequencies proportional to ?=2a1/2.\Pi = 2a^{1/2}.?=2a1/2. The perturbations in the plasma magnetic field die as they manifest in the form of damped oscillations. The perturbations in the plasma electric field are harmonic with unbounded amplitudes. There are no perturbations in the outer vacuum magnetic field up to the order of approximation. We have solved the problem. The electric field disturbances in this region of the system appear in the form of unbounded oscillations with frequency equal to N. The disturbances in the magnetic field of the inner vacuum die as time advances. The disturbances in the electric field of this region appear in the form of unbounded oscillations with frequency equal to N?. Thus we conclude that the exploding shell is overstable against long wavelength disturbances. The frequency of these growing oscillations depends on the thickness of the shell through a², the initial magnetic field and the rate of explosion through 2, and on the initial radii of the shell through m.

(b) Imploding shell: Here we note that the disturbances in the plasma pressure, velocity and electric field grow as the shell collapses, the growth rate of the instability being proportional to N=a1/2.N = a^{1/2}.N=a1/2. Similarly, the disturbance in the electric field in the outside vacuum grows as fast ? 1. Thus we conclude that the collapsing shell is unstable, and the growth rate depends on the thickness of the shell through a² and on the rate of collapse through the dimensionless time t.