Gas dynamics with relaxation effects

Abstract

T h e way in which gas-phase chemical reactions and the excitation of internal modes of molecular energy storage can be collectively described as relaxation processes is explained. A set of equations which is suitable for a general account of gas flows with relaxation processes is exhibited, and the main part of the discussion of developments over the past decade or so is occupied with situations for which the transport effects of viscosity, heat conduction and mass diffusion are negligible, or sufficiently highly localised as to be so.T h e dissipative-dispersive character of acoustic waves is described with some emphasis on the large effect of absorption on high-frequency waves which travel at the frozen sound speed and on the relatively smaller, bulk-viscosity-like, influence on low-frequency disturbances which travel at the slower equilibrium sound speed. Signalling problems in equilibrium atmospheres show exponential decay at the precursor wave head and diffusive behaviour in those regions which travel at the equilibrium sound speed, as well as the effects of geometric attenuation for other than planar waves. The exploitation of the acoustic approximation in studies of flow problems is described.The effects of a dis-equilibrium atmosphere on acoustic wave propagation, and the consequent possibility of exponential amplification of disturbances, is described with reference to explosion and combustion phenomena, as well as a possible link with upper-atmosphere gravity waves.The analysis and explanation of shock-wave structure is briefly discussed with the aid of Hugoniot curves and Rayleigh lines, and the two main classes, of fully dispersed and partly dispersed waves, are defined. Weak Taylor shocks and fully dispersed waves are compared, with emphasis on the strong limitations on the use of equivalent bulk viscosities to describe relaxation effects. T h e way in which a population inversion can occur in internal molecular modes is illustrated by the case of a partly dispersed shock in a gas with two relaxing modes which have significantly different relaxation times.The way in which disturbances of finite amplitude behave is discussed, with special reference to characteristics theory, for both equilibrium and dis-equilibrium relaxing atmospheres. Modern theories of the non-linear wave propagation of small-amplitude disturbances are described with special reference to their role in the sonic boom problem and to the evolution of fully and partly dispersed shocks.

Decay Rate for Travelling Waves of a Relaxation Model

“…Different kinds of relaxation phenomena occur in water waves, thermo-nonequilibrium gases, rarefied gas dynamics, traffic flow, viscoelasticity with memory, magnetohydrodynamics, etc. [1,7].…”

Nonlinear stability and existence of stationary discrete travelling waves for the relaxing schemes

“…The effect of stiff relaxation is important in a wide range of problems of physical significance. Different kinds of relaxation phenomena occur in water waves, thermo-nonequilibrium gases, rarefied gas dynamics, traffic flow, viscoelasticity with memory and magnetohydrodynamics, etc [1], [5].…”

A linear hyperbolic system with stiff source terms

“…More complicated nonlinear equations of this same general form have been studied for some time in a variety of contexts, and linearized equations with a form similar to (1) are often used to determine stability (for example, see the discussion of flood waves in chapter 3 of Ref. [1] or the discussion of a relaxing gas by Clake [2] . For the system (1), stability requires 1 α ∧ .…”

“…For the system (1), stability requires 1 α ∧ . This is called the sub characteristic condition since this guarantees that the characteristic speed of the equilibrium equation (2) lies between the characteristic speeds of the full system. In this case Chen and Liu in Ref.…”