Mathematical Sciences: Mathematical Topics in Combustion

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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. ***