A new model of a laminar combustion process is constructed based on its thermodynamic analysis. Under controlled growth of temperature at the inlet to the combustion chamber, depending on the structure of the standard chemical potential, high-frequency oscillations of the thermal explosion resonance occur in the model. Resonance modes in the case of heat pumping are modeled, the nature of their origin is established depending on the structure of the standard chemical potential, and numerical experiments exhibiting these modes are presented.

Based on thermodynamic analysis of combustion processes, we build a new model of laminar combustion process of vibration combustion. If the passive component velocity for the two-component mixture is controlled at the inlet (the interior energy is pumped at the inlet of the combustion chamber), high-frequency resonance oscillations of vibration combustion are shown to develop depending on the structure of the standard chemical potential. The resonance is driven by hydrodynamic instability. Numerical experiment models the regimes of resonance with pumped internal energy, the nature of their nucleation depending on the structure of the standard chemical potential is revealed. For low temperatures, the appearance of detonation combustion with substantial pressure increase is established near resonance.

Асташова И.В., Барабанов Е.А., Боровских А.В., Бутузов В.Ф., Быков В.В., Ветохин А.Н., Глызин С.Д., Горицкий А.Ю., Денисова Н.В., Изобов Н.А., Ильин А.В., Ильяшенко Ю.С., Капустина Т.О., Кигурадзе И.Т., Козлов В.В., Колесов А.Ю., Коньков А.А., Ломов И.С., Моисеев Е.И., Палин В.В. и др.//Дифференциальные уравнения, 2021

The author’s method of thermodynamic analysis is used to single out two equations of state for the laminar combustion process: the classical Hugoniot adiabat, which determines the pressure, and the equation of state, which determines the entropy. This allows constructing a new mathematical model of the laminar process of vibrational combustion of a two-component mixture by closing the classical models of continuum mechanics. The model is phenomenological, which requires its verification. For numerical verification, the well-known experimental fact is chosen, the appearance of high-frequency acoustic vibrations described by B.V. Raushenbakh. The conditions for the origin of high-frequency oscillations are obtained in terms of the standard chemical potential. They can substantially disturb the combustion process and may cause a catastrophic break-up of the furnace of the engine structure. A numerical experiment established critical values of the standard chemical potential when high-frequency vibrations lead to destruction.

On the basis of thermodynamic analysis, new mathematical models of the combus- tion process (thermal theory) and vibrational combustion are constructed. A global inhomo- geneity of the system can be described as an inhomogeneous distribution of the enthalpy over a two-component mixture. In this case, for the combustion process in the phase space of the variables (%, P, T, n, S, E), an increase in the enthalpy is not a total differential. An increase in the enthalpy is a total differential on the local equilibrium manifold (a laminar combustion process). These two assertions, which allow one to single out in the phase space the corre- sponding adiabatic of the combustion process (the Hugoniot adiabatic) and the equation for the entropy, close the classical mathematical model of the combustion process. The above numerical experiments show that two regimes of the combustion process (deflagration and detonation) depend on the structure of the standard chemical potential Moreover, a control of the passive component velocity at the inlet results in (depending on the structure of the standard chemical potential) high-frequency oscillations, which are responsible for a blow-up.

This article is an attempt to study the process of Rayleigh–Benard convective instability by the methods used for mathematical modeling of critical phenomena as nonequilibrium phase transitions in their initial stages of spinodal decomposition. We show that it is possible to extend the formalism adopted in the Cahn–Hillard theory of nonequilibrium phase transitions and perfected on problems of highgradient crystallization to other types of problems, in particular, those pertaining to the Rayleigh–Benard convective instability. For the initial stage of instability, a model is constructed that represents it as a nonequilibrium phase transition due to diffusive stratification. It is shown that the Gibbs free energy of deviation from the homogeneous state (with respect to the instability under consideration) is an analogue of the Ginsburg–Landau potential. Numerical experiments, by means of boundary temperature control, have been conducted with regard to self-excitation of the homogeneous state. Numerical analysis shows that convective flows may appear and proceed from regular forms (the so-called regular structures) to nonregular flows through a chaotization of the process. External factors, such as temperature growth, may lead to chaos via period doubling bifurcations.

For the laminar–turbulent transition, a model of reconstructing the initial stage of an instability treated as a nonequilibrium phase transition is developed. Its mechanism is based on diffusion stratification. It is shown that the Gibbs free energy of the deviation from the homogeneous state (with respect to the instability under consideration) is an analogue of the Ginzburg–Landau potentials. Numerical experiments concerning the self-excitation of a homogeneous state by applying a boundary control condition in the form of an increasing velocity were performed. Under an external influence (an increase in the velocity as input), the system exhibits a transition to chaos through period-doubling bifurcations similar to the Feigenbaum period-doubling cascade.

A model was constructed for reconstructing the initial stage of solidification of binary alloys treated as a nonequilibrium phase transition with a diffusion stratification mechanism. Numerical experiments concerning the self-excitation of a homogeneous state by applying a melt cooling boundary control condition were performed.

A model was constructed for the reconstruction of the initial stage of crystallization of binary alloys as a nonequilibrium phase transition, the mechanism of which is diffusion stratification. Numerical experiments were performed. Self-excitation of a homogeneous state by the edge control melt cooling condition.

Построена модель реконструкции начальной стадии кристаллизации бинарных сплавов как неравновесного фазового перехода, механизмом которого является диффузионное расслоение. Проведены численные эксперименты самовозбуждения однородного состояния управлением краевым условием охлаждения расплава.

For the laminar-turbulent transition, we construct a model of reconstruction of the initial stage of instability qua a nonequilibrium transition with diffusion separation mechanism. It is shown that the free Gibbs energy of departure from the homogeneous state (with respect to the instability under consideration) is an analogue of the Ginzburg-Landau potential. Numerical experiments for self-excitation of the homogeneous state with control of the boundary condition of velocity increase were carried out, which showed the appearance of the laminar-turbulent transition and its development from regular forms (the so-called dissipative structures) with subsequent transition to irregular flows via chaotization of the process. An external action (an increase in velocity) results in a transition to chaos in terms of period-doubling bifurcations similarly to the Feigenbaum cascade of period-doubling bifurcations. The chaotization of the process transforms regular forms (dissipative structures) into the two-velocity regime (the regime of two shock waves), which was called the Riemann-Hugoniot catastrophe by Prigogine and Nicolis. This transformation depends substantially on gravitation. The perturbation is shown to be nonlocal, which indications that the classical perturbation theory is inapplicable in this case.

The initial stage of instability in the failure of a structural material treated as a nonequilibrium phase transition is reconstructed. Its mechanism is based on spinodal decomposition (diffusion separation).

Sufficient conditions are found for the existence of stabilizing solutions of the Riccati differential equation y′ = (y − y1(x)) (y − y2(x)) with given y1(x) and y2(x). For various types of stabilizing solutions, the number of points of extremum is examined.

The initial phase of instability in the structure of a solid in the form of a fracture process of the structure material as a nonequilibrium phase transition which depends on the spinodal decomposition (diffusion separation into two pases with different degradation factor) is reconstructed.

We consider one-dimensional Carleman and Godunov–Sultangazin equations and obtain local equilibrium conditions for solutions of the Cauchy problem with finite energy and periodic initial data. Moreover, we prove the exponential stabilization to the equilibrium state.

The initial phase of instability in the structure of a solid in the form of a fracture process of the structure material as a nonequilibrium phase transition which depends on the spinodal decomposition (diffusion separation into two pases with different degradation factor) is reconstructed.

The occurrence of convective currents and their development from regular forms with the subsequent transition to irregular turbulent currents draw attention to the fact that they are responsible for the efficiency of many technological processes of heat and mass transfer. Such technological processes are basic in the chemical, petrochemical, power, metallurgical and other industries. Convective flows arise in liquids and gases in the gravitational field in the presence of spatial inhomogeneity of the density created by the inhomogeneity of the temperature and the concentration of components arising during, for example, chemical reactions or other causes. With increasing temperature difference, the resting liquid loses its stability, which then leads to the appearance of a convective flow (Rayleigh-Benard instability). A further increase in the temperature difference leads to an instability of the primary convective flow, and the hydrodynamic crisis leads to a heat transfer crisis. The paper reconstructs the early stage of the Rayleigh-Benard convective instability considered as a nonequilibrium phase transition with the spinodal decomposition (diffusion separation) mechanism.