Self-Similar Solutions of Compressible Fluids

Abstract

AbstractStrong shock waves are used to compress and heat any matters in the laboratory. The ablation pressure by intense laser is used to compress even solid matters. In plane geometry, it is easier to design multi-shocks to compress the matters, while it is more beneficial to use the spherical compression. No simple solutions are available to know the trajectories of shocks in one-dimensional spherical symmetry. Here we see several analytical solutions with the self-similar method. The method is to find new governing solution of ordinary differential equation from partial differential fluid equations. The self-similar method is known before the birth of computer.The blast wave is the most famous one. Here, we review the basic method to derive several self-similar solutions allowing the spherical implosion, useful to laser driven implosion. The isobaric solution provides uniform pressure and spark-main fuel structure, and isochoric solution gives us uniform density profile at the maximum compression. It is shown that even including thermal conduction, it is possible to find a solution of ablation structure. This is an extended solution more appropriate compared to the steady state solutions shown in the previous chapter.The blast waves are widely used from laser experiments to supernova remnants (SNRs). SNRs are blast waves driven by the matters exploding by supernova explosion. A self-similar solution with forward and reverse shock waves is found to explain many observation data of SNRs. A numerical simulation shows that the solution of ejecta-driven shock changes from Chevalier’s self-similar solution to the other Sedov-Taylor one. The self-similarity is one of the key physics controlling nonlinear hydrodynamic equations.