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arXiv · 1207.5416

Resolvent methods for steady premixed flame shapes governed by the Zhdanov-Trubnikov equation

Gaëtan Borot·Bruno Denet·Guy Joulin
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Abstract

Using pole decompositions as starting points, the one parameter (-1 =< c < 1) nonlocal and nonlinear Zhdanov-Trubnikov (ZT) equation for the steady shapes of premixed gaseous flames is studied in the large-wrinkle limit. The singular integral equations for pole densities are closely related to those satisfied by the spectral density in the O(n) matrix model, with n = -2(1 + c)/(1 - c). They can be solved via the introduction of complex resolvents and the use of complex analysis. We retrieve results obtained recently for -1 =< c =< 0, and we explain and cure their pathologies when they are continued naively to 0 < c < 1. Moreover, for any -1 =< c < 1, we derive closed-form expressions for the shapes of steady isolated flame crests, and then bicoalesced periodic fronts. These theoretical results fully agree with numerical resolutions. Open problems are evoked.

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Gaëtan Borot, Bruno Denet, Guy Joulin. 2013-04-11. Resolvent methods for steady premixed flame shapes governed by the Zhdanov-Trubnikov equation. https://doi.org/10.1088/1742-5468%2F2012%2F10%2Fp10023

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