A combustion wave of permanent form in a compressible gas

Abstract

A model is presented for describing the propagation of a one-dimensional wave of permanent form in a compressible gas in a pipe. Energy is lost to the system through the walls of the pipe, but the combustion wave produces heat through an exothermic chemical reaction. The full set of equations for the model is reduced to a phase-plane system, and it is shown that, for small amplitude waves, a weakly non-linear analysis leads to a temperature profile that is a classical solitary wave. A novel shooting method is developed for the full non-linear problem, and this confirms and extends the solitary-wave solution, up to a value of the temperature amplitude at which the wave begins to develop a shock. The jump conditions across the shock are presented, and numerical integration is used to continue the solution for temperature amplitudes at which a shock is present. An example is given of the extreme situation in which the shock is so strong that all the fuel behind the shock is exhausted.

Thermal solitons: travelling waves in combustion

“…Similar temperature profiles were observed by Forbes & Derrick [15]. It is interesting to re-construct the governing differential equation for the second-order temperature perturbation in equation (4.5).…”

“…Similar forms to (3.1) were used by Forbes & Derrick [15]. The approximation technique is based on the method of strained coordinates, an example of which is given in the text by Jordan & Smith [19, p. 148].…”

Co-operation, competition and crowding: a discrete framework linking Allee kinetics, nonlinear diffusion, shocks and sharp-fronted travelling waves

“…An example of a shock-fronted travelling wave solution for the strong Allee effect with both v > 0 and v < 0 is shown in any source terms, that contain shocks have been reported previously [50,51]. Similarly, shockfronted travelling wave solutions arise in other kinds of models, including multispecies models of combustion [54] and haptotactic cell migration [53]. However, the models presented here are very different as our model contains a source term and no advection term, and it is therefore of interest to determine the properties of the reaction-diffusion equation that lead to shock-fronted travelling wave solutions.…”

“…An example of a shock-fronted travelling wave solution for the strong Allee effect with both v > 0 and v < 0 is shown in any source terms, that contain shocks have been reported previously [50,51]. Similarly, shockfronted travelling wave solutions arise in other kinds of models, including multispecies models of combustion [54] and haptotactic cell migration [53]. However, the models presented here are All results are obtained with δx = 0.1, δt = 0.01, = 10 −6 , P g d = 0, (a)-(c) P i m = 0.5, P g m = 0.1, P i p = 0.5, P g p = 0.4, P i d = 0.6, v = 0.009, (d)-(f) P i m = 0.5, P g m = 0.1, P i p = 0.4, P g p = 0.2, P i d = 0.5, v = −0.028.…”

A Note on Travelling Waves in Competitive Reaction Systems

“…The Arrhenius reaction rates (2.4) are somewhat unrealistic, in that they allow reaction to occur at any temperature above absolute zero; this is the famous "cold boundary difficulty", and has been commented on by many authors, including Gray et al [5] and Matkowsky and Sivashinsky [8]. Here, the approach of Forbes and Derrick [4] is adopted, and it is assumed that the reaction only occurs above some critical ("ignition") temperature θ a , taken to be the same for both rates. A further discussion of this concept is given by Zel'dovich et al [13].…”