Non-equilibrium effects on the breakdown of weak solutions in non-linear wave propagation

Abstract

The weak solutions of the first order quasi-linear system governing a non-equilibrium flow of a gas are studied in the characteristic plane. It is shown that a linear solution in the characteristic plane can exhibit non-linear behaviour in the physical plane. As an application of the theory the law of propagation and the growth equation of weak discontinuities are obtained and the breakdown of weak solutions is discussed. It is also shown that non-equilibrium effects play an important role in the process of steepening and flattening of weak non-linear waves with planar, cylindrical and spherical symmetry. The critical timet c for the breakdown of weak solutions is obtained. The critical amplitudea c of the initial compressive disturbance has been determined such that any compressive weak wave with an initial amplitude greater thana c terminates into a shock wave, while an initial amplitude less than the critical one results in a decay of the disturbance. A non-linear steepening and non-equilibrium effects provide a particular answer to the substantial question as for when a shock wave will be formed on the breakdown of weak solutions, when the characteristics will pile up on the wave front. The non-equilibrium effects have a stabilizing influence on the wave propagation in the sense that not all compressive disturbances will grow into shock waves, whereas in ordinary gas flows all compressive waves will grow into shock waves.