Nonlinear Dynamics and Pattern Formation in Combustion

数学科学：燃烧中的非线性动力学和模式形成[Abstract] matkowsky。这项研究涉及燃烧和火焰传播问题的分析和数值研究的协同作用。这种方法非常成功地解释了所研究的高度非线性偏微分方程组解的行为。兴趣集中在气体燃烧以及固体燃料燃烧和非均相燃烧（固体加气体）中表现出复杂时空动态的问题，这些问题出现在先进材料的自传播高温合成过程中。本研究的目的是描述向复杂解行为的过渡和结构。该项目解决了气体燃烧问题的火焰与顺序反应和火焰在充满气体的管道。随着参数的变化，这些系统表现出向具有更大程度时空复杂性的状态的过渡。该项目还分析了固体燃料燃烧中的问题，其中使用高温燃烧波（称为固体火焰）来合成先进材料。研究集中在固体火焰和过滤燃烧领域，它涉及非均相燃烧，气固两相之间发生反应。解析研究基于分岔理论和非线性稳定性理论，利用分岔或其他过渡点附近的渐近分析和奇异摄动理论得到解的局部描述。在大规模科学计算中采用自适应伪谱方法，将局部描述扩展到全局。溶液呈现出层状，在局部区域中溶液变化迅速。自适应伪谱法很好地解决了准确、高效地求解这类层型行为的难题。这个项目关注的是理解发生在能量转换和材料合成中的过程，这两个领域具有国家重要性。能量转换的研究主要集中在火焰的结构和动力学方面，涉及多种物理机制的相互作用。为了确定各种机制之间的因果关系，本工作探讨了参数依赖性。材料合成的研究是利用燃烧波来生产具有所需性能的材料，例如，极高的硬度或对极端温度的不渗透性。在这种创新的技术过程中，燃烧波通过样品传播，将未反应的固体粉末混合物转化为固体产物，这似乎比传统技术具有优势。当这样合成时，材料可以具有优越的性能。

NSF Award Abstract - DMS-0072491Mathematical Sciences: Nonlinear Dynamics and Pattern Formation in CombustionAbstract0072491 MatkowskyThis research involves a synergism of analytical and numerical studies of problems in combustion and flame propagation. This approach is highly successful in elucidating behavior of solutions to the highly nonlinear systems of partial differential equations under study. Interest centers on problems exhibiting complex spatio-temporal dynamics in gaseous combustion as well as in solid fuel combustion and in heterogeneous combustion (solid plus gas), which arise in the self-propagating high-temperature synthesis process for advanced materials. The goal of the research is to describe transitions to, and the structure of, complex solution behavior. The project addresses gaseous combustion problems in the areas of flames with sequential reactions and flames in gas filled tubes. These systems exhibit transitions to states with greater degrees of spatio-temporal complexity as parameters are varied. The project also analyzes problems in solid fuel combustion, in which high temperature combustion waves, referred to as solid flames, are used to synthesize advanced materials. Research centers on the areas of solid flames and filtration combustion, which involves heterogeneous combustion, with reaction between gas and solid phases. The 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 to yield a local description of the solution. An adaptive pseudo-spectral method is employed for large scale scientific computations, with which the local description is globally extended. The solutions exhibit layers, localized regions in which the solution varies rapidly. The adaptive pseudo-spectral method successfully meets the challenge of accurately and efficiently resolving such layer type behavior. This project is concerned with understanding processes that occur in energy conversion and materials synthesis, two areas of national importance. The research in energy conversion is focused on flame structure and dynamics, which involve interactions among many physical mechanisms. To determine cause and effect relations among the various mechanisms, this work explores parameter dependencies. The research in materials synthesis studies the use of combustion waves to produce materials having desired properties, for example, extreme hardness or imperviousness to temperature extremes. In this innovative technological process, which appears to enjoy advantages over conventional technology, the combustion wave propagates through the sample, converting an unreacted solid powder mixture to solid product. When so synthesized, materials can have superior properties.