NONLINEAR HYDRODYNAMIC INSTABILITY OF EXPANDING FLAMES - INTRINSIC DYNAMICS

In the framework of the weak-thermal-expansion approximation, a potent ial flow model is employed as a analytical tool to study the dynamics of wrinkled, nearly spherical, expanding premixed flames. An explicitl y 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 instabi lity, the stabilizing curvature effects, and the stretch of disturbanc es induced by flame expansion. Via a linearization and a decomposition of the flame distortion in angular normal modes, it is first shown, i n 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 gro wth. Following that, the linear response to small forcings of fixed sp atial wave numbers is investigated and exponential growths are obtaine d. By using a separation of variables, then the pole-decomposition met hod, the flame evolution is converted into a N-body dynamical system f or the complex spatial singularities of the front shape; an infinite n umber of initial condition dependent, exact solutions to the generaliz ed MS equation are then exhibited. Each of them represents superpositi ons of locally orthogonal patterns of finite amplitudes which are show n to ultimately evolve into slowly varying ridges positioned at fixed angular locations. The corresponding flame speed histories are determi ned. Examples of nonlinear wrinkle dynamics are studied, including pet al-like patterns that are nearly self-similar asymptotically in time, but in no instance could one observe a spontaneous tendency to repeate d cell splitting. Open mathematical and physical problems are also evo ked.