Dynamics in a Rod Model of Solid Flame Waves

Abstract

We consider gasless solid fuel combustion in a cylinder of radius $\tilde{R}$, associated with the SHS (self-propagating high-temperature synthesis) process, which employs combustion waves to synthesize materials. A powder mixture of reactants is cold pressed into a sample, typically a cylinder, and ignited at one end. A high-temperature combustion wave then propagates along the sample, converting the unreacted powder into the desired product. It has recently become popular to model systems governed by partial differential equations as an array of interacting oscillators. Here, we extend this approach by considering an array of interacting rods, each of which supports propagating waves. Thus, we employ an array of interacting one-dimensional (1D) rods connected via heat transfer. The heat transfer terms correspond to a discretization of the transverse Laplacian. Both the full 3D model and the rod model allow for a uniformly propagating planar combustion wave. The dispersion relation for this solution is determined for both models and shown to be equivalent when certain parameters in the two models are identified. The rod model is able to describe a number of the features of the 3D model, thus allowing numerical simulations with significantly reduced computational resources. In this paper we consider a rod model consisting of an outer ring of three rods equally spaced along the ring, together with an axial rod. Clearly, this limits the 3D modes of wave propagation which can be described, and the results below have to be considered within this basic limitation. The 3/1 model admits analogues of spin and radial modes which are known to exist experimentally and as solutions of the 3D model. We propose that the new modes of solution behavior that we find are also related to modes of the 3D model. We determine solution behavior as a function of R, the nondimensionalized cylindrical radius. We consider three cases characterized by the Zeldovich number $Z=N(1-\sigma)/2$, where N is a nondimensionalized activation energy and $\sigma$ is the ratio of the unburned to the burned temperature. For Z sufficiently small, the uniformly propagating planar solution is stable to planar, i.e., rod independent, perturbations. In this case, analysis of the dispersion relation predicts that for sufficiently small and sufficiently large R, only the uniformly propagatingsolution is stable. Spin modes, seen in experiments as hot spots spinning periodically around the cylinder as the wave propagates, occur for small R, and radial modes, periodically oscillating solutions independent of the cylindrical angle, occur for larger R. We find analogues of these modes for the rod model and describe the transition from spin to radial modes via a family of quasiperiodic (QP) modes, manifested by periodic variations in the temperature and rotation speed of the spot. For a larger value of Z, the uniformly propagating solution is unstable to planar perturbations.In this case, we find that for sufficiently small values of R, the singly periodic pulsating planar (PP) solution is the only stable solution. For larger values of R, there is bistability between PP solutions and spin solutions. As R increases further, both solutions lose stability, and the only stable solution is a QP mode. This mode is a combination of spin and radial behavior. Unlike the smaller Z case, stable radial solutions are not found. As R increases, the two generator frequencies of the QP solutions converge to the dominant frequency of the PP solution. In the time domain the solution in each rod is approximately the PP solution on intermediate time scales, with a long time envelope corresponding to the small difference between the two generator frequencies. There is a phase difference between adjacent outer rods which is essentially constant over intermediate time scales, but which varies over longer time scales. The intermediate time scale increases as R increases. Thus, we find QP spin-like behavior which on intermediate time scales appears as a spinning manifestation of the PP solution. We refer to this behavior as spinning PP (SPP) behavior. We also consider a yet larger value of Z. We find a PP solution which is now period doubled. In addition, we find spin and QP spin solutions as before. We also find an interval in R where spinning-type solutions reverse direction in a periodic or QP fashion.