Nonlinear hydrodynamic instability of expanding flames: Intrinsic dynamics

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Abstract
In the framework of the weak-thermal-expansion approximation, a potential flow model is employed as an analytical tool to study the dynamics of wrinkled, nearly spherical, expanding premixed flames. An explicitly time-dependent generalization of the nonlinear Michelson-Sivashinsky (MS) equation is found to control the evolution of the flame wrinkles. The new equation qualitatively accounts for the hydrodynamic instability, the stabilizing curvature effects, and the stretch of disturbances induced by flame expansion. Via a linearization and a decomposition of the flame distortion in angular normal modes, it is first shown, in agreement with classical analyses, that the above mechanisms compete at first to make the small disturbances of fixed angular shapes fade out in relative amplitude, and subsequently result in an algebraic growth. Following that, the linear response to small forcings of fixed spatial wave numbers is investigated and exponential growths are obtained. By using a separation of variables, then the pole-decomposition method, the flame evolution is converted into an N-body dynamical system for the complex spatial singularities of the front shape; an infinite number of initial condition dependent, exact solutions to the generalized MS equation are then exhibited. Each of them represents superpositions of locally orthogonal patterns of finite amplitudes which are shown to ultimately evolve into slowly varying ridges positioned at fixed angular locations. The corresponding flame speed histories are determined. Examples of nonlinear wrinkle dynamics are studied, including petal-like patterns that are nearly self-similar asymptotically in time, but in no instance could one observe a spontaneous tendency to repeated cell splitting. Open mathematical and physical problems are also evoked.