Flame wrinkles from the Zhdanov–Trubnikov equation

The Zhdanov–Trubnikov equation describing wrinkled premixed flames is studied, using pole decompositions as starting points. Its one-parameter (−1⩽c⩽+1−1⩽c⩽+1) nonlinearity generalises the Michelson–Sivashinsky equation (c=0c=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≈−1c≈−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 −10c>0 (over-stabilisation) such analytical solutions can yield accurate flame shapes for 0⩽c⩽0.60⩽c⩽0.6. Open problems are invoked.

► We study a 1-parameter (c  ) nonlinear integral equation and get flame-wrinkle shapes.
► Pole decompositions of the front slope (periodic or not) are used as a basis.
► In limiting cases we relate the flame shapes to Laguerre or Jacobi polynomials.
► Linear integral equations for pole densities give accurate large-wrinkle shapes if c<0c<0.
► Though locally singular the shapes so obtained for c>0c>0 can be fairly accurate.