Analysing instability of combustion waves using the Evans function

Jason Sharples, Harvi Sidhu, Vladimir Gubernov

Abstract


We consider travelling wave solutions of a reaction-diffusion system corresponding to a single step, homogeneous, premixed combustion scheme with Newtonian heat loss and general Lewis number. Particular attention is paid to unstable combustion wave regimes, especially those associated with oscillatory behaviour. The instability analysis is conducted with the use of Evans function techniques, which we use to derive eigenvalues of the linear stability problem via the argument principle and Nyquist plots. These techniques permit the study of transitions to different modes of unstable behaviour in great detail. Threshold values of the parameters corresponding to Hopf and Bogdanov--Takens bifurcation are established and it is shown that for certain parameter values the system exhibits a period doubling route to chaotic behaviour.

References
  • Afendikov, A. L. and Bridges, T. J. Instability of the Hocking--Stewartson pulse and its implications for three-dimensional Poiseuille flow. P. R. Soc. A, 457:1--16, 2001. doi:10.1098/rspa.2000.0665
  • Alexander, J., Gardner, R. and Jones, C. A topological invariant arising in the stability analysis of travelling waves. J. Reine Angew. Math., 410:167--212, 1990.
  • Dold, J. W. Premixed flames modelled with thermally sensitive intermediate branching kinetics. Combust. Theor. Model., 11:909--948, 2007. doi:10.1080/13647830701294599
  • Dold, J. W., and Zinoviev, A. Fire eruption through intensity and spread rate interaction mediated by flow attachment. Combust. Theor. Model., 13:763--793, 2009. doi:10.1080/13647830902977570
  • FlexPDE. http://www.pdesolutions.com
  • Gubernov, V. V., Mercer, G. N., Sidhu, H. S. and Weber, R. O. Evans function stability of combustion waves. SIAM J. Appl. Math., 63:1259--1275, 2003. http://www.siam.org/journals/siap/63-4/40024.html
  • Gubernov, V. V., Mercer, G. N., Sidhu, H. S. and Weber, R. O. Evans function stability of non-adiabatic combustion waves. P. R. Soc. A, 460:2415--2435, 2004. doi:10.1098/rspa.2004.1285
  • Gubernov, V. V., Sidhu, H. S. and Mercer, G. N.. Generalized compound matrix method. Appl. Math. Lett., 19:458--463, 2006. doi:10.1016/j.aml.2005.07.002
  • Gubernov, V. V, Kolobov, A. V., Polezhaev, A. A., Sidhu, H. S. and Mercer, G. N. Period doubling and chaotic transient in a model of chain-branching combustion wave propagation. Proc. Roy. Soc. Lond. A, 466:2747--2769, 2010. doi:10.1098/rspa.2009.0668
  • Makino, A. Fundamental aspects of the heterogeneous flame in the self-propagating high-temperature synthesis (SHS) process. Prog. Energ. Combust., 27:1--74, 2001.
  • Merzhanov, A. G., and Rumanov, E. N. Physics of reaction waves. Rev. Mod. Phys., 71:1173--1211, 1999.
  • Pego, R. L., Smereka, P. and Weinstein, M. I. Oscillatory instability of travelling waves for a KdV-Burgers equation. Physica D, 67:45--65, 1993. http://www.sciencedirect.com/science/journal/01672789
  • Sandstede, B. Stability of travelling waves. In B. Fiedler, editor, Handbook of Dynamical Systems II, pp. 983--1055. Elsevier, 2002.
  • Sharples, J. J., Sidhu, H. S. and Gubernov, V. V. Properties of nonadiabatic premixed combustion fronts arising in single-step reaction schemes. In Proceedings of Chemeca 2010, Paper No. 357, 26--29 September 2010, Adelaide. ISBN 978 085 825 9713.
  • Thomas, P. H., Bullen, M. L., Quintiere, J. G. and McCaffrey, B. J. Flashover and instabilities in fire behaviour. Combust. Flame, 38:159--171, 1980. http://www.sciencedirect.com/science/article/pii/0010218080900486
  • Weber, R. O., G. N. Mercer, H. S. Sidhu, and Gray, B. F. Combustion waves for gases (Le$=1$) and solids (Le$=\infty $). Proc. Roy. Soc. Lond. A, 453:1105--1118, 1997. doi:10.1098/rspa.1997.0062

Keywords


combustion wave; stability; Evans function; argument principle

Full Text:

PDF BibTeX


DOI: http://dx.doi.org/10.21914/anziamj.v52i0.3944



Remember, for most actions you have to record/upload into this online system
and then inform the editor/author via clicking on an email icon or Completion button.
ANZIAM Journal, ISSN 1446-8735, copyright Australian Mathematical Society.