Derivation of a curvature-dependent Kuramoto--Sivashinsky equation
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Sep 21, 2024
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Abstract
A curvature-dependent Kuramoto--Sivashinsky equation is derived as a model of flame spreading motion on a cylinder. We start from a three-phases (gas-solid-gas) reaction-diffusion system defined on a cylinder and reduce it to a two-dimensional system by averaging over the thickness. This two-dimensional model includes curvature dependence diffusion coefficients. Matched asymptotic expansion is conducted on the two-dimensional model to construct a traveling wave solution and determine the interface corresponding to the flame front. Finally, a linear stability analysis of the travelling wave solution and geometric arguments to the motion of the front lead to a curvature-dependent Kuramoto--Sivashinsky equation, which is a generalization of a known result obtained by Kagan & Sivashinsky [3].
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Kobayashi, S., & Yazaki, S.
(2024).
Derivation of a curvature-dependent Kuramoto--Sivashinsky equation.
Proceedings Of The Conference Algoritmy, , 189 -198.
Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/2169/1040
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References
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[2] G. I. Sivashinsky, Nonlinear analysis of hydrodynamic instability in laminar flames–I, Acta Astron., 4 (1977), 1177–1206.
[3] L. Kagan and G. I. Sivashinsky, Pattern formation in flame spread over thin solid fuels, Combust. Theor. Model., 12:2 (2008), 269–281.
[4] R. Kimura and H. Torikai, Flame spreading over an accordion folded paper, The Proceedings of the 55th Symposium (Japanese) on Combustion (2017), 456–457.
[5] Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, Dover Publ. (2003).
[6] S. B. Margolis and B. J. Matkowsky, Nonlinear Stability and Bifurcation in the Transition from Laminar to Turbulent Flame Propagation, Combustion Science and Technology, 34 (1983), 44–77.
[7] B. J. Matkowsky and G. I. Sivashinsky, An asymptotic derivation of two models in flame theory associated with the constant density approximation, SIAM. J. Appl. Math., 37 (1979), 686–699.
[8] T. Miyamoto and H. Torikai, Effect of folding width on a flame spreading rate for accordion folded paper, The Proceedings of the Thermal Engineering Conference 2020 (2020), ROMBUNNO.E224.
[9] C. Qian, H. Ishida and K. Saito, Upward Flame Spread along PMMA Vertical Corner Walls Part II: Mechanism of “M” Shape Pyrolysis Front Formation, Combustion and Flame, 99 (1994), 331–338.