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