Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-25T10:01:27.308Z Has data issue: false hasContentIssue false
  • English
  • Français

TRAVELLING WAVES AND OSCILLATIONS IN SAL’NIKOV’S COMBUSTION REACTION IN A COMPRESSIBLE GAS

Part of: Compressible fluids and gas dynamics, general

Published online by Cambridge University Press:  19 January 2015

RHYS A. PAUL  and
LAWRENCE K. FORBES
RHYS A. PAUL*
Affiliation:
School of Mathematics and Physics, University of Tasmania, Tasmania 7001, Australia email [email protected], [email protected]
LAWRENCE K. FORBES
Affiliation:
School of Mathematics and Physics, University of Tasmania, Tasmania 7001, Australia email [email protected], [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider a two-step Sal’nikov reaction scheme occurring within a compressible viscous gas. The first step of the reaction may be either endothermic or exothermic, while the second step is strictly exothermic. Energy may also be lost from the system due to Newtonian cooling. An asymptotic solution for temperature perturbations of small amplitude is presented using the methods of strained coordinates and multiple scales, and a travelling wave solution with a sech-squared profile is derived. The method of lines is then used to approximate the full system with a set of ordinary differential equations, which are integrated numerically to track accurately the evolution of the reaction front. This numerical method is used to verify the asymptotic solution and investigate behaviours under different conditions. Using this method, temperature waves progressing as pulsatile fronts are detected at appropriate parameter values.

Keywords

Sal’nikov reactioncombustionsolitonspulsatile wavesstrained coordinatesmultiple scalesmethod of lines

MSC classification

Primary: 76N15: Gas dynamics, general
Type
Research Article
Information
The ANZIAM Journal , Volume 56 , Issue 3 , January 2015 , pp. 233 - 247
Copyright
© 2015 Australian Mathematical Society 

References

Ames, W. F., Numerical methods for partial differential equations (Academic, New York, 1977).Google Scholar
Barenblatt, G., Zeldovich, Ya. B., Librovich, V. B. and Makhviladze, G., The mathematical theory of combustion and explosions (Consultants Bureau, New York, 1985).Google Scholar
Bayliss, A. and Matkowsky, B. J., “Fronts, relaxation oscillations, and period doubling in solid fuel combustion”, J. Comput. Phys. 71 (1987) 147–168; doi:10.1016/0021-9991(87)90024-6.CrossRefGoogle Scholar
Coppersthwaite, D. P., Griffiths, J. F. and Gray, B. F., “Oscillations in the hydrogen $+$ chlorine reaction: experimental measurements and numerical simulation”, J. Phys. Chem. 7 (1991) 6961–6967; doi:10.1021/j100171a043.CrossRefGoogle Scholar
Fisher, R. A., “The wave of advance of advantageous genes”, Ann. Eugen. 7 (1937) 1–26; doi:10.1111/j.1469-1809.1937.tb02153.x.CrossRefGoogle Scholar
Forbes, L. K., “Limit-cycle behaviour in a model chemical reaction: the Sal’nikov thermokinetic oscillator”, Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 430 (1990) 641–651;doi:10.1098/rspa.1990.0110.Google Scholar
Forbes, L. K., “Thermal solitons: travelling waves in combustion”, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 469 (2013) 20120587; doi:10.1098/rspa.2012.0587.Google Scholar
Forbes, L. K. and Derrick, W., “A combustion wave of permanent form in a compressible gas”, ANZIAM J. 43 (2001) 35–58; doi:10.1017/S144618110001141X.CrossRefGoogle Scholar
Forbes, L. K. and Gray, B. F., “Forced oscillations in an exothermic chemical reaction”, Dyn. Stab. Syst. 9 (1994) 253–269; doi:10.1080/02681119408806181.Google Scholar
Forbes, L. K., Myerscough, M. R. and Gray, B. F., “On the presence of limit-cycles in a model exothermic chemical reaction: Sal’nikov’s oscillator with two temperature-dependent reaction rates”, Proc. Math. Phys. Sci. 435 (1991) 591–604; doi:10.2307/52094.Google Scholar
Gray, B. F. and Roberts, M. J., “Analysis of chemical kinetic systems over the entire parameter space I. The Sal’nikov thermokinetic oscillator”, Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 416 (1988) 391–402; doi:10.1098/rspa.1988.0040.Google Scholar
Kolmogorov, A. N., Petrovskii, I. G. and Piskunov, N. S., “A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem”, Byul. Moskovskogo Gos. Univ. 1 (1937) 1–26; http://books.google.com/books?id=ikN59GkYJKIC&lpg=PP1&dq=A.N.Kolmogorov3ASelectedWorks&client=firefox-a&pg=PA242#v=onepage&q=&f=false.Google Scholar
Liepmann, H. W. and Roshko, A., Elements of gasdynamics (Wiley, New York, 1957).CrossRefGoogle Scholar
Matkowsky, B. J. and Sivashinsky, G. I., “Propagation of a pulsating reaction front in solid fuel combustion”, SIAM J. Appl. Math. 35 (1978) 465–478; doi:10.2307/2100631.CrossRefGoogle Scholar
MATLAB version 8.0.0.783 (R2012b) (The MathWorks Inc., Natick, USA, 2012).Google Scholar
Nelson, M. I. and Sidhu, H. S., “Bifurcation phenomena for an oxidation reaction in a continuously stirred tank reactor I. Adiabatic operation”, J. Math. Chem. 31 (2002) 155–186; doi:10.1021/j100171a043.CrossRefGoogle Scholar
Sal’nikov, I. Ye., “Contribution to the theory of the periodic homogeneous chemical reactions”, Zh. Fiz. Khim. 23 (1949) 258–272.Google Scholar
Scott, S. K., Chemical chaos (Oxford University Press, London, 1993).Google Scholar
Stocker, T., Introduction to climate modelling (Springer, Dordrecht, 2011).CrossRefGoogle Scholar
Weber, R. O., Mercer, G. N., Sidhu, H. S. and Gray, B. F., “Combustion waves for gases (Le $=$ 1) and solids ($\text{Le}\rightarrow \infty$)”, SIAM J. Appl. Math. 453 (1997) 1105–1118; doi:10.1098/rspa.1997.0062.Google Scholar
Whitham, G. B., Linear and nonlinear waves (Wiley-Interscience, New York, 1993).Google Scholar
Williams, F. A., Combustion theory: the fundamental theory of chemically reacting flow systems (Addison-Wesley, Reading, MA, 1965).Google Scholar