Assessment of the effects of azimuthal mode number perturbations upon the implosion processes of fluids in cylinders

Highlights: • Implosion instabilities are studied by linearizing about a symmetric implosion. • This suggests azimuthal instabilities decrease with time and mode number. • Numerics capture the delta functions from linearized solutions of conservation laws. • The mass of these delta functions is used to estimate perturbations in shock fronts. • The linear Klein–Gordon equation in one dimension is solved via formal asymptotics. Fluid instabilities arise in a variety of contexts and are often unwanted results of engineering imperfections. In one particular model for a magnetized target fusion reactor, a pressure wave is propagated in a cylindrical annulus comprised of a dense fluid before impinging upon a plasma and imploding it. Part of the success of the apparatus is a function of how axially-symmetric the final pressure pulse is upon impacting the plasma. We study a simple model for the implosion of the system to study how imperfections in the pressure imparted on the outer circumference grow due to geometric focusing. Our methodology entails linearizing the compressible Euler equations for mass and momentum conservation about a cylindrically symmetric problem and analysing the perturbed profiles at different mode numbers. The linearized system gives rise to singular shocks and through analysing the perturbation profiles at various times, we infer that high mode numbers are dampened through the propagation. We also study the Linear Klein–Gordon equation in the context of stability of linear cylindrical wave formation whereby highly oscillatory, bounded behaviour is observed in a far field solution.