Abstract
Dynamics of temperature distribution and interfacial front propagation in a generalized solid combustion model are studied through both asymptotic and numerical analyses. For asymptotic analysis, we focus on the weakly nonlinear case where a small perturbation of a neutrally stable parameter is taken so that the linearized problem is marginally unstable. Multiple scale expansion method is used to obtain an asymptotic solution for large time by modulating the most linearly unstable mode. On the other hand, we integrate numerically the exact problem by the Crank-Nicolson method. Since the numerical solutions are very sensitive to the derivative interfacial jump condition, we integrate the partial differential equation to obtain an integral-differential equation as an alternative condition. The result system of nonlinear algebraic equations is then solved by the Newton’s method, taking advantage of the sparse structure of the Jacobian matrix. Finally, we show that our asymptotic solution captures the marginally unstable behaviors of the solution for a range of model parameters.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Merzhanov, A.G.: SHS processes: combustion theory and practice. Arch. Combust. 1, 2–48 (1981)
Munir, Z.A., Anselmi-Tamburini, U.: Self-propagating exothermic reactions: the synthesis of high-temperature materials by combustion. Mat. Sci. Rep. 3, 277–365 (1989)
Varma, A., Rogachev, A.S., Mukasyan, A.S., Huang, S.: Combustion synthesis of advanced materials: principles and applications. Adv. Chem. Eng. 24, 7–226 (1998)
Yang, Y., Gross, L.K., Yu, J.: Comparison study of dynamics in one-sided and two-sided solid-combustion models. SIAM J. Appl. Math. 70(8), 3022–3038 (2010)
Matkowsky, B.J., Sivashinsky, G.I.: Propagation of a pulsating reaction front in solid fuel combustion. SIAM J. Appl. Math. 35(93), 465–478 (1978)
Chen, K., Gross, L.K., Yu J., Yang, Y.: On a generalized free-interface model of solid combustion. J. Eng. Math. (submitted)
Bayliss, A., Matkowsky, B.J.: Two routes to chaos in condensed phase combustion. SIAM J. Appl. Math. 50(2), 437–459 (1990)
Brailovsky, I., Sivashinsky, G.: Chaotic dynamics in solid fuel combustion. Physica D 65, 191–198 (1993)
Frankel, M.L., Roytburd, V., Sivashinsky, G.: Complex dynamics generated by a sharp interface model of self-propagating high-temperature synthesis. Combust. Theory Model. 2, 1–18 (1998)
Frankel, M.L., Roytburd, V.: Dynamical portrait of a model of thermal instability: cascades, chaos, reversed cascades and infinite period bifurcations, Internat. J. Bifur. Chaos Appl. Sci. Eng. 4(3), 579–593 (1994)
Chen, K.: Mathematical analysis of some partial differential equations with applications. Ph.D. thesis, The University of Vermont (in progress)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Yu, J., Chen, K., Gross, L.K. (2019). Asymptotic and Numerical Analysis of Dynamics in a Generalized Free-Interfacial Combustion Model. In: Dimov, I., Faragó, I., Vulkov, L. (eds) Finite Difference Methods. Theory and Applications. FDM 2018. Lecture Notes in Computer Science(), vol 11386. Springer, Cham. https://doi.org/10.1007/978-3-030-11539-5_76
Download citation
DOI: https://doi.org/10.1007/978-3-030-11539-5_76
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-11538-8
Online ISBN: 978-3-030-11539-5
eBook Packages: Computer ScienceComputer Science (R0)