Flame Wrinkles From the Zhdanov-Trubnikov Equation

TL;DR

This paper analyzes the Zhdanov-Trubnikov equation for wrinkled flames, deriving analytical solutions for steady flame shapes using polynomial functions and pole density methods, and confirms these with numerical results.

Contribution

It introduces analytical solutions for flame shapes based on polynomial functions and pole densities, extending the understanding of the Zhdanov-Trubnikov equation for different instability regimes.

Findings

Analytical solutions for steady flame shapes using Laguerre and Jacobi polynomials.

Pole density methods relate wrinkle shapes to Meixner-Pollaczek polynomials.

Numerical results confirm the analytical solutions for certain parameter ranges.

Abstract

The Zhdanov-Trubnikov equation describing wrinkled premixed flames is studied, using pole-decompositions as starting points. Its one-parameter (-1< c <1) nonlinearity generalizes the Michelson-Sivashinsky equation (c=0) to a stronger Darrieus-Landau instability. The shapes of steady flame crests (or periodic cells) are deduced from Laguerre (or Jacobi) polynomials when c = -1, which numerical resolutions confirm. Large wrinkles are analysed via a pole density: adapting results of Dunkl relates their shapes to the generating function of Meixner-Pollaczek polynomials, which numerical results confirm for 1<c<0 (reduced stabilization). Although locally ill-behaved if c>0 (over-stabilization) such analytical solutions can yield accurate flame shapes for 0< c <0.6. Open problems are invoked.