Global solution to a three-dimensional spherical piston problem for the relativistic Euler equations

The study of spherically symmetric motion is important for the theory of explosion waves. In this paper, we consider a ‘spherical piston’ problem for the relativistic Euler equations, which describes the wave motion produced by a sphere expanding into an infinite surrounding medium. We use the reflected characteristics method to construct a global piecewise smooth solution with a single shock of this spherical piston problem, provided that the speed of the sphere is a small perturbation of a constant speed.

Approximate Analytic Solution of a Self-Similar Piston Moving in an Inhomogeneous Medium

Self-similar motion for the flow between a piston and strong shock propagating in a non uniform ideal gas at rest has been studied. The solution to the problem is similar to that of hypersonic flows past the power law bodies. The gas ahead of the shock is assumed to be uniform and at rest. This is considered as a particular case of radiative piston problem. The shock is assumed to be very strong and propagating in a medium at rest in which density obeys power laws. This problem with spherical symmetry has got importance in astrophysics. To solve the gas dynamics problem, Chernyii’s expansion techniques have been used in which flow variables are expanded in a series of powers of ε, the density ratio across the strong shock. The approximate analytic solution has been obtained in closed form to the zeroth approximation. The problem discussed belongs to the self-similar motion of the first kind. The resulting analytic solution gives the flow variables distribution for plane, cylindrical, and spherical symmetry for different cases which satisfy the similarity conditions with accurate trend and values.

Poroacoustic Traveling Waves under the Rubin–Rosenau–Gottlieb Theory of Generalized Continua

We investigate linear and nonlinear poroacoustic waveforms under the Rubin–Rosenau– Gottlieb (RRG) theory of generalized continua. Working in the context of the Cauchy problem, on both the real line and the case with periodic boundary conditions, exact and asymptotic expressions are obtained. Numerical simulations are also presented, von Neumann–Richtmyer “artificial” viscosity is used to derive an exact kink-type solution to the poroacoustic piston problem, and possible experimental tests of our findings are noted. The presentation concludes with a discussion of possible follow-on investigations.