Mathematical Sciences: Mathematical Topics in Combustion

[600103 . Volpert]提出了两个有关燃烧波数学研究的研究课题。第一个主题是确定一些燃烧问题的燃烧波的传播速度。有大量的工作使用各种方法来解决这个问题，包括匹配渐近展开方法和基于有关燃烧波结构的先验物理假设的方法。这些工作得出了许多关于传播速度的近似公式。然而，很少有工作涉及对其准确性或适用性的仔细评估。因此，不同文献对同一燃烧波的传播速度推导的不同公式不一定一致。基于极大极小原理的测定燃烧速度的新方法在我们以前的工作中得到了发展。这些方法使我们不仅可以推导出许多燃烧问题中传播速度的近似公式，而且可以确定该公式的准确性。许多气体燃烧的问题，包括简单的和复杂的化学，提出了使用这些新方法来研究。第二个主题是研究具有宽反应区的燃烧合成波，这种波在实验中经常观察到。所有以前的理论，在材料生产中使用的过程，是基于一个狭窄的反应区假设。这是由于研究具有宽反应区的合成波的数学困难，以及缺乏完善的分析方法。发展宽反应区燃烧波理论是本项目的主要目标。初步结果表明，虽然具有宽反应区的波确实具有非常复杂的结构，但新的渐近方法使人们能够成功地处理这个问题。具体讨论的问题包括具有宽反应区的非绝热燃烧、具有复杂化学和相变的合成波以及传播波的稳定性。在这些问题中，将研究燃烧波的结构，其传播速度，燃烧温度，反应物的转化程度等，以了解具有宽反应区的合成过程的基本机制。材料的燃烧合成是利用燃烧波进行材料生产的过程。在这个过程的最简单的表现中，在一端点燃由反应物的粉末混合物组成的样品。然后，具有正面结构的高温热波在样品中传播，将反应物转化为所需的产物。该工艺与传统工艺相比具有一定的优势，传统工艺将混合物放入熔炉中。这包括(i)更短的合成时间，（ii）更少的费用，因为使用反应物的内部化学能而不是使用烘箱的外部能量，（iii）使用更简单的设备，（iv）更纯净的产品，因为高温波燃烧掉挥发性杂质。提出了一项研究计划，旨在进一步了解燃烧合成所涉及的机制。所提出的研究的主要对象是涉及广泛反应区的燃烧合成波，尽管由于化学转化的非均质性而经常发生，但由于该问题的数学复杂性，尚未对其进行理论研究。该项目的目的是了解具有广泛反应区的合成过程的基本机制，从而能够有效地控制产品的组成和质量。＊＊＊

9600103 Volpert Two research topics concerned with mathematical studies of combustion waves are proposed. The first topic is the determination of the propagation velocity of combustion waves for a number of combustion problems. There are a large number of works which use various approaches to this problem, including the method of matched asymptotic expansions and those based on a priori physical assumptions concerning the structure of the combustion wave. These works have led to numerous approximate formulae for the propagation velocity. However, very few works have been concerned with a careful assessment of their accuracy or applicability. As a result, different formulae for the propagation velocity of the same combustion wave derived in different papers do not necessarily agree with one another. New approaches to the determination of the combustion velocity, based on minimax principles have been developed in our previous work. These approaches allow us not only to derive an approximate formula for the propagation velocity in a number of combustion problems but also to determine the accuracy of this formula. A number of problems of gaseous combustion, with both simple and complex chemistry, are proposed to study using these novel methods. The second topic is a study of combustion synthesis waves with wide reaction zones, which are often observed in experiments. All previous theories of the process, which is used in materials production, are based on the assumption of a narrow reaction zone. This is due to mathematical difficulties in the study of synthesis waves with wide reaction zones, and the absence of well developed approaches to the analysis. Development of the theory of wide reaction zone combustion waves is the primary goal of the proposed project. Preliminary results show that though waves with wide reaction zones do have very complex structures, new asymptotic methods allow one to successfully treat the problem. Specific topic s discussed in the proposal include nonadiabatic combustion with wide reaction zones, synthesis waves with complex chemistry and phase transitions as well as stability of the propagating wave. In each of these problems the structure of the combustion wave, its propagation velocity, burning temperature, degree of conversion of reactants, etc., will be studied in order to understand the fundamental mechanisms of the synthesis process with wide reaction zones. %%% Combustion synthesis of materials is a process that uses combustion waves for materials production. In the simplest manifestation of this process, a sample consisting of a powder mixture of reactants is ignited at one end. A high-temperature thermal wave, having a frontal structure, then propagates through the sample, converting reactants to desired products. The process enjoys certain advantages over conventional technology, in which the mixture is placed in a furnace. These include (i) shorter synthesis times, (ii) less expense, since the internal chemical energy of the reactants is used rather than the external energy of the oven, (iii) the use of simpler equipment, and (iv) purer products, since the high-temperature wave burns off volatile impurities. A research program designed to further our knowledge of the mechanisms involved in combustion synthesis is proposed. The main object of the proposed research is combustion synthesis waves involving wide reaction zones, which, though they often occur due to the heterogeneous nature of the chemical conversion, have not been theoretically studied because of the mathematical complexity of the problem. The goal of this project is to understand the fundamental mechanisms of the synthesis process with wide reaction zones, and thus to be able to make recommendations for effective control of the composition and quality of the products. ***