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

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 so-called 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.

Abstract loop equations, topological recursion and new applications

“…For ρ " ´1, it contains for instance the Ising model on faces of random triangulations [Eyn06]. In the context of convergent integrals, such a model has also appeared in the context of quantum entanglement [BN12], and in relation with dynamics of fluid interfaces [BDJ12].…”

Shapes and speeds of steady forced premixed flames

“…For example, the strategy of Sec. II can the two pole-densities belonging to such bi-coalesced [or two-crested] fronts obey a pair of coupled integral equations that are easy to write [15,36] but are still awaiting solutions.…”

“…  entails adaptation of the complex-variable techniques used in [36]. Although cotangent base-slopes [Sec.III] and their sinusoidal limit [Sec.IV] seem within reach, and the key property of linearity [Secs.…”

Premixed-flame shapes and polynomials

“…Equation (1) exhibits a number of remarkable features, most notably the existence of poledecompositions whereby the search for ( , ) t x  is converted to a 2N -body problem for the complex poles of the front slope [10,11]; see §2. Thanks to this property one can: (i) Explain the formation of front arches joined by sharper crests whose mergers ultimately produce the widest admissible steady cell; (ii) Access the latter's arc-length vs. wavelength curve [12], which yields the effective flame speed; (iii) Solve stability issues [13,14] without the effect of spurious noises hampering the non-self-adjoint linearized dynamics [15]; (iv) Compute pole density and front shapes for isolated crests, and then for periodic cells [11,16], if 1 N  ; (v) Study stretched crests [17]; (vi) Set up tools to study extensions of (1) that incorporate higher orders of the 1   expansion, at least in the large-N limit ( [18,19] and Refs. therein).…”

“…which yields the effective flame speed; (iii) Solve stability issues [13,14] without the effect of spurious noises hampering the non-self-adjoint linearized dynamics [15]; (iv) Compute pole density and front shapes for isolated crests, and then for periodic cells [11,16], if 1 N  ; (v) Study stretched crests [17]; (vi) Set up tools to study extensions of (1) that incorporate higher orders of the 1   expansion, at least in the large-N limit ( [18,19] and Refs. therein).…”