The structure and stability of nonadiabatic flame balls: II. Effects of far-field losses

A theory of spherical premixed flames in which the only transport processes are diffusion and radiation (“flame balls”) is extended to include the effects of heat loss from the far-field (unburned gas). Using matched asymptotic expansions for large activation energy, stationary solutions are constructed and an evolution equation for the radial motion of the flame is derived. Linear stability analyses are performed for both one-dimensional and three-dimensional perturbations. It is shown that when far-fieldlosses are included, the stability properties are qualitatively changed. A smaller range of conditions produces flames that are stable to one-dimensional disturbances and, in some cases, the linear growth rates are found to have imaginary components. Numerical intergation of the evolution equation reveals oscillations (but not limit cycles), in agreement with a bifurcation analysis. The minimum flame-ball radius for which three-dimensional instabilities will occur is shown to depend only on the near-field losses, and becomes arbitrarily large as these losses are reduced.