Nonlinear Dynamics and Pattern Formation in Combustion

我们的研究方向是燃烧过程中的非线性动力学和模式形成。该程序涉及燃烧和火焰传播问题的分析和数值研究的协同作用，已成功地阐明了我们所研究的高度非线性偏微分方程系统的解行为。特别是，兴趣集中在气体燃烧和固体燃料燃烧中表现出复杂时空动态的问题。在气体燃烧中，我们考虑四个不同的问题领域：(a)连续反应的火焰，(b)拉伸流中的火焰，(c)在燃烧器上稳定的火焰，以及(d)充满气体的管道中的火焰。我们还考虑了固体燃料燃烧中的问题，其中燃烧波被用来合成先进材料。在这种相对较新的创新技术过程中，燃烧波在样品中传播，将未反应的固体粉末混合物转化为固体产物，似乎比传统技术具有许多优点。我们的分析研究基于分岔和非线性稳定性理论，在分岔或其他过渡点的邻域中采用渐近分析和奇异摄动理论，得到解在适当临界点的邻域中的局部描述。在此基础上，提出了一种适用于大规模科学计算的自适应伪谱方法，将局部描述扩展到全局。所考虑的问题的解表现出层型行为，即解变化非常快的局部区域。如何准确、高效地求解这种行为对数值方法是一个挑战。自适应伪谱方法成功地解决了这一难题。当问题的关键参数被超过时，就会发生向具有更大时空复杂性的状态的过渡。我们的目标是描述这些转变以及由此产生的时空模式。分析研究，除了阐明系统行为，例如，参数依赖性，也作为随后的数值计算的基准。因此，代码不仅适用于简单的解决方案，也适用于复杂的时空行为。除了研究特定的燃烧问题，我们还为研究这些问题和其他问题的分析和数值方法的发展做出了贡献。最后，在可能的情况下，将与相关的实验观察结果进行比较。我们研究描述燃烧过程的数学问题。这些研究主要关注的是了解影响燃烧的基本机制，包括因果关系，这是任何试图控制燃烧过程的必要先决条件。特别地，我们研究了燃烧波的结构、传播速度、稳定性等特性。所考虑的问题的一个例子是利用燃烧波合成先进材料。在传统技术中，将各种成分的粉末混合物放在炉子里“烘烤”直到完全熟透。在燃烧合成中，混合物被压成固体并在一端点燃。然后，燃烧波通过固体传播，将其转化为所需的产物。这一过程比传统技术快得多，也便宜得多。产物的性质是由燃烧波的传播方式决定的，这是我们研究的主题。

Matkowsky 9705670 We pursue a research program in nonlinear dynamics and pattern formation in combustion. The program involves a synergism of analytical and numerical studies of problems in combustion and flame propagation, which has been successful in elucidating solution behavior of the highly nonlinear systems of PDEs which we study. In particular, interest centers on problems exhibiting complex spatiotemporal dynamics in gaseous combustion as well as in solid fuel combustion. In gaseous combustion, we consider four different problem areas: (a) flames with sequential reactions, (b) flames in stretched flows, (c) flames stabilized on a burner, and (d) flames in gas filled tubes. We also consider problems in solid fuel combustion, in which combustion waves are used to synthesize advanced materials. In this relatively new and innovative technological process, which appears to enjoy many advantages over conventional technology, the combustion wave propagates through the sample, converting unreacted solid powder mixture to solid product. Our analytical studies are based on bifurcation and nonlinear stability theories, which employ asymptotic analysis and singular perturbation theory in the neighborhood of bifurcation or other transition points, resulting in a local description of the solution in a neighborhood of the appropriate critical point. Then an adaptive pseudo-spectral method, developed by the proposer and A. Bayliss, is employed for large scale scientific computations, with which the local description is globally extended. The solutions of the problems considered exhibit layer type behavior, i.e., localized regions in which the solution varies very rapidly. It is a challenge to numerical methods to accurately and efficiently resolve such behavior. The adaptive pseudo-spectral method has been shown to successfully meet this challenge. As critical parameters of the problem are exceeded, transitions to states with successively greater degrees of spatio-temporal complexity occur. Our goal is to describe the transitions as well as the resulting spatio-temporal patterns. The analytical studies, in addition to elucidating system behavior, e.g., parameter dependencies, also serve as benchmarks for the ensuing numerical computations. Codes are thus validated not only for simple solutions, but for complex spatio-temporal behavior as well. In addition to studying specific combustion problems, we contribute to the development of both analytical and numerical methodology for the investigation of these and other problems. Finally, where possible, comparisons to relevant experimental observations will be made. We investigate mathematical problems describing combustion processes. The primary concern in these studies is the understanding of basic mechanisms, including cause and effect, influencing combustion, which is a necessary prerequisite to any attempt to control the combustion process. In particular, we study the characteristics, e.g., structure, propagation speed, stability, etc., of combustion waves. An example of the problems considered is the use of combustion waves to synthesize advanced materials. In the conventional technology, a powder mixture of components is placed in a furnace and "baked" until it is well done. In combustion synthesis the mixture is pressed into a solid and ignited at one end. A combustion wave then propagates through the solid converting it to desired product. The process is significantly faster and cheaper than the conventional technology. The nature of the product is determined by the manner of propagation of the combustion wave, which is the subject of our investigation.