Diffusion flames associated with weak explosions

A general formulation for unsteady flows is set forth for which diffusion flames are regarded as discontinuity surfaces. A linearized theory is then developed for weak explosions associated with planar, cylindrical, and spherical symmetries. A simple combustion model for a ternary mixture of fuel, oxidant, and product species is utilized. The one-dimensional linearized shock-tube problem is analyzed in detail. Explicit results are obtained for the flame motion and the flame and flow-field development for arbitrary Prandtl number, Schmidt number, and second coefficient of viscosity. Wave fronts associated with the flame disturbance, initial pressure disturbance, and the value of the Prandtl number are delineated. The motion of a spherical flame associated with weak spherical explosions is analyzed and found ultimately to move toward the origin. The structure of the diffusion flame is analyzed by means of matched asymptotic expansions wherein the details of the flame structure are described by an "inner" expansion that is matched to the "outer" expansion that was obtained, to lowest order, with the flame treated as a discontinuity surface. Thus the variation of the flame structure with time is obtained for reaction broadening.