Chaotic dynamics in solid fuel combustion. (English) Zbl 0781.58029
The free-boundary problem for models describing the dynamics of condensed fuel gasless combustion is studied numerically. The solution of the propagating combustion wave type is considered. The effective one-dimensional boundary problem which determines the flame wave in the framework of the model is formulated in the following way
\[
{\partial \theta\over \partial \tau} - V(\tau){\partial \theta\over \partial \xi} = {\partial^ 2 \theta\over \theta\xi^ 2} + V(\tau)\delta(\xi), \;\theta(+\infty,\tau) = 0,\quad \theta(-\infty,\tau) < \infty,
\]
where \(\theta = \theta(\xi,\tau)\) is the dimensionless temperature and \(\xi\), \(\tau\) are the dimensionless spatial coordinate and time, respectively. The function \(V(\tau)\) is defined by
\[
V(\tau) = \Omega[\theta(0,\tau)], \;\Omega = \exp(\alpha(\theta-1)/(\sigma + (1-\sigma)\theta)),
\]
where \(\sigma\) and \(\alpha\) (Zeldovich number) are the dimensionless parameters. The problem admits a basic state solution
\[
\theta^ b = \exp(-\xi),\quad \xi > 0;\quad \theta^ b = 1,\quad \xi < 0; \quad V^ b = 1,
\]
corresponding to a steady planar combustion wave. This solution loses its stability at \(\alpha > \alpha_ c = 2+5^{1/2}\) by Hopf bifurcation and consequently a sequence of the period doublings which leads to a chaotic behaviour should be realized. The authors try to observe these phenomena numerically.
Reviewer: Y.P.Virchenko (Khar’kov)
MSC:
37N20 | Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics) |
80A25 | Combustion |
65P20 | Numerical chaos |
65P30 | Numerical bifurcation problems |
80M25 | Other numerical methods (thermodynamics) (MSC2010) |
37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |
80A22 | Stefan problems, phase changes, etc. |
References:
[1] | Merzhanov, A. G., Arch. Combust., 23-48 (1981) |
[2] | Holt, J. B., Mater. Res. Soc. Bull., 12, 60-64 (1982) |
[3] | Shkadinsky, K. G.; Khaikin, B. I.; Merzhanov, A. G., Combust. Explos. Shock Waves, 7, 15-22 (1971) |
[4] | Matkowsky, B. J.; Sivashinsky, G. I., SIAM J. Appl. Math., 33, 465-478 (1978) · Zbl 0404.76074 |
[5] | Matkowsky, B. J.; Sivashinsky, G. I., SIAM J. Appl. Math., 37, 617-697 (1979) |
[6] | Sivashinsky, G. I., SIAM J. Appl. Math., 40, 432-438 (1981) · Zbl 0479.76086 |
[7] | Frankel, M. L., Physica D, 40, 403-414 (1989) · Zbl 0706.35066 |
[8] | Bayliss, A.; Matkowsky, B. J., J. Comput. Phys., 81, 421-443 (1989) · Zbl 0668.65092 |
[9] | Dimitriou, P.; Puszynsky, J.; Hlavacek, V., Combust. Sci. Tech., 68, 101-111 (1989) |
[10] | Bayliss, A.; Matkowsky, B. J., SIAM J. Appl. Math., 50, 437-459 (1990) · Zbl 0696.76082 |
[11] | Brailovsky, I.; Sivashinsky, G. I., Combust. Sci. Tech., 87, 389-393 (1992) |
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