The dynamical instabilities of cylindrical shells

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Abstract
Collapsing cylindrical metallic shells have been used to compress magnetic fluxes and generate megagauss magnetic fields. Such shells experience large, rapidly growing accelerations and their symmetry can be completely destroyed by Rayleigh-Taylor instabilities. This paper presents a theoretical study of the Rayleigh-Taylor instability for radially accelerated incompressible cylindrical shells submitted to the pressure of much lighter media. Low-amplitude flute perturbations are considered and Fourier-analyzed in the azimuthal angle. A fourth-order linear differential system with time-dependent coefficients is derived, which determines the two interface-displacements. Stability criteria are discussed. When the perturbation wavelength is much greater or much smaller than the shell thickness, the differential system splits into two independent differential equations and results are greatly simplified; analytical solutions are available for some cases. The case of axial field compression (A.F.C.) is discussed as an application. Numerical solutions give the time behaviour of all possible initially given disturbances. The initial perturbations, which are able to reach the axis during their development and which are consequently dangerous in magnetic field compression experiments, have been calculated. Results are consistent with the few experimental data available. They show that the degree of symmetry of cylindrical devices has to be extremely good in order to get successful compressions. Finally, non-linear and compressibility effects have been taken into account for some A.F.C. cases, solving the full non-linear fluid equations numerically.

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