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

TL;DR

This paper develops resolvent methods to analyze steady premixed flame shapes governed by the Zhdanov-Trubnikov equation, providing explicit solutions and resolving previous pathologies in the model.

Contribution

It introduces complex resolvent techniques to solve the Zhdanov-Trubnikov equation for all parameter values, extending and refining previous results.

Findings

Derived closed-form solutions for flame shapes.

Resolved issues in previous models for certain parameter ranges.

Confirmed theoretical results with numerical simulations.

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.