Solid Flame Waves

We consider the gasless combustion model of the SHS (Self-Propagating High Temperature Synthesis) process in which combustion waves are employed to synthesize desired materials. In this case the combustion phenomenon is referred to as a “solid flame”. Specifically, we consider the combustion of a solid sample in which combustion occurs on the surface of a cylinder of radius R. In addition to uniformly propagating planar waves there are many other types of waves. The study of different waves is important since the nature of the wave determines the structure of the product material. For the fixed value of the Zeldovich number Z which we employ the uniformly propagating planar wave is unstable. We describe stable waves only. We consider solution behavior as R is increased. For R sufficiently small, slowly propagating planar pulsating flames are the only modes observed. As R is increased, transitions to more complex modes of combustion occur, including (i) spin modes in which one or several symmetrically spaced hot spots rotate around the cylinder as the flame propagates along the cylindrical axis, thus following a helical path, (ii) counterpropagating (CP) modes, in which spots propagate in opposite angular directions around the cylinder, executing various types of dynamics. These include spots which pass through each other essentially unchanged, much the same as solitons, and spots which appear to be annihilated, only to be regenerated further along the sample in one of a variety of ways, (iii) alternating spin CP modes (ASCP), where rotation of a spot around the cylinder is interrupted by periodic events in which a new spot is spontaneously created ahead of the rotating spot. The new spot splits into counterpropagating daughter spots, one of which collides with the original spot leading to their mutual annihilation, while the other continues to spin, (iv) modulated spin waves consisting of either one or two symmetrically located rotating spots which exhibit a periodic modulation in speed and temperature as they rotate, (v) asymmetric spin waves in which two spots of unequal strength and not separated by angle π, rotate together as a bound state, (vi) modulated asymmetric spin waves in which the two asymmetric spots oscillate in a periodic manner as they rotate, alternately approaching each other and then moving apart periodically in time, (vii) asymmetric ASCP modes in which a slowly varying bound state of two spots rotates around the cylinder with the leading spot, and subsequently the trailing spot, exhibiting episodes of ASCP behavior, and (viii) 3-headed spins in which three spots rotate around the cylinder in a nonuniform fashion so that each cell alternately approaches one of its neighbors and then the other. In one case the motion is quasiperiodic, with neighboring spots approaching and departing from each other periodically in time as they rotate. In another case the motion is apparently chaotic. Two neighboring spots nearly collide, after which one spot is rapidly propelled away from the other as they rotate. Finally, for a slightly higher value of R, two neighboring spots collide, leading to annihilation of one spot and collapse of the 3-headed spin to a 2-headed spin mode.