Nonlinear Dynamics and Pattern Formation in Combustion

马特考斯基9705670我们从事的是燃烧中的非线性动力学和花纹形成的研究项目。该程序包括燃烧和火焰传播问题的分析和数值研究的协同作用，它已经成功地阐明了我们所研究的高度非线性的PDE系统的溶液行为。特别是，人们的兴趣集中在气体燃烧和固体燃料燃烧中表现出复杂的时空动力学的问题上。在气体燃烧中，我们考虑了四个不同的问题：(A)具有连续反应的火焰，(B)拉伸流动中的火焰，(C)燃烧器上稳定的火焰，以及(D)充气管道中的火焰。我们还考虑了固体燃料燃烧中的问题，在固体燃料燃烧中，燃烧波被用于合成先进材料。在这一相对较新和创新的工艺过程中，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品，似乎比传统技术具有许多优势。我们的分析研究基于分叉和非线性稳定性理论，在分叉或其他过渡点的邻域内使用渐近分析和奇异摄动理论，在适当的临界点的邻域内得到解的局部描述。然后，将作者和A.Bayliss提出的自适应伪谱方法应用于大规模科学计算，利用该方法对局部描述进行全局扩展。所考虑的问题的解表现出层状行为，即解在其中变化非常迅速的局部区域。如何准确有效地解决这类问题，是数值方法面临的一个挑战。自适应伪谱方法已被证明成功地应对了这一挑战。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转变。我们的目标是描述这种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，例如参数相关性外，还可作为后续数值计算的基准。因此，代码不仅针对简单的解决方案进行验证，还针对复杂的时空行为进行验证。除了研究特定的燃烧问题外，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，将与相关的实验观测结果进行比较。我们研究描述燃烧过程的数学问题。这些研究的主要关注点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前提。特别是，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使用燃烧波来合成先进材料。在传统技术中，将各种成分的粉末混合物放入炉子中并进行烘烤，直到它变成做得好。在燃烧合成中，混合物被压成固体，并在一端点燃。燃烧波随后在固体中传播，将其转化为所需的产品。这一过程比传统技术快得多，成本也低得多。产品的性质由燃烧波的传播方式决定，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.