Travelling combustion fronts demonstrate the appearance of a variety of instabilities and structures and thus the revealing of the mechanisms of these phenomena is important both in theoretical and practical aspects. In this work, the mechanism of formation of spiral nonlinear wave structures in the process of propagation of a combustion wave in a rich hydrogen-air mixture at elevated pressure is studied analytically and numerically. A detailed model of the hydrogen oxidation reaction is reduced to a system of partial differential equations for the evolution of H, HO2, O2 and temperature profiles describing both low-temperature and high-temperature reactivity. It is shown to be able to successfully reproduce the characteristics of the diffusive-thermal pulsations emerging with the increase of pressure. A model for the dynamics of the low temperature flame region is separated from the reduced model by using a number of assumptions and appeared to be similar to the Sal'nikov model. It is shown that for the parameter values at which spiral waves are observed, the latter model turns to excitable regime and generates the spiral solutions. Thus it is concluded that the low temperature oxidation processes are responsible for emergence of the spiral structures at the combustion front. We think that the current work is an important step towards understanding the mechanisms of formation of spiral wave structures in expanding combustion fronts.

A model describing the dynamics of a one-dimensional chain of interacting nonlocally coupled neurons, based on properties of the fractional Laplacian with constant and variable order is proposed. As a result of numerical simulation of the action potential propagation for various types of nonlocal interactions (with a constant set of parameters corresponding to nonlinear functions of the Hindmarsh—Rose model, as well as diffusion coefficients), the appearance of various spatiotemporal modes is established. At given parameters, in the case of classical diffusion, the general trend to synchronous behavior is observed. In the superdiffusion case, the modes of cluster excitation of the action potential are formed. For systems with variable-order fractional Laplacian, spatial anisotropy of arising structures is characteristic. It is shown that the introduction of long-range couplings realized in the system by introducing the fractional Laplacian can provide additional possibilities of describing dynamic properties of interacting neurons.

In this paper, the complex dynamics of the pulsating regime of combustion waves propagation is numerically investigated within the Zel’dovich–Barenblatt–Dold model with two-step chain-branching reaction mechanism. It is shown that there exists a remarkable similarity in the dynamics of oscillations in this system and one-dimensional discrete-time maps such as the logistic map. In particular, both systems exhibit the period doubling route to chaos and the appearance of windows of stability as the bifurcation parameter is increased. For the first time for a model of combustion wave propagation the sequence of windows of stability is demonstrated and classified. Nevertheless, as the activation energy, which is taken as a bifurcation parameter, reaches a critical value corresponding to the dynamical quenching scenario the dynamics of the flame oscillations becomes multidimensional and therefore it cannot be described by a one-dimensional map anymore. The effect of the heat losses is also studied and it is demonstrated that the number of windows of stability decreases, the order of their appearance changes and the dynamical quenching appears for smaller values of the activation energy as the heat losses are increased.

In this work the performance of various detailed reaction mechanisms of combustion of methane–hydrogen fuel mixtures is studied. The investigation is focused on the burner stabilized flame configuration and is undertaken for the normal pressure conditions, which is a starting point for the analysis of the combustion chemistry at elevated pressures met in rocket engine chambers. The study presents the experimental setup, computational approach and results of comparison, which allow us to access the properties of complex combustion systems in variety of dynamical regimes. The comparison of the predicted and measured characteristics of these regimes is used to test the performance of the reaction models. In particular, the critical behavior, e.g. for blow-up, onset of pulsations and quasi-steady flame front propagation and bifurcation between these regimes may be used to characterize and to study properties of combustion systems with mixed fuels. A neutral stability boundary for onset of pulsations is suggested to be used in this study. A method for automatic identification of the boundary is proposed and implemented with several detailed mechanisms. Although for pure methane the results look acceptable, they show gradual divergence with the increase of percentage of hydrogen addition. Significant quantitative differences between experiments and modeling with respects to frequencies of oscillations are reported even for 2:1 ratio of hydrogen to methane in the unburned mixture. The results of the work indicate that current reaction mechanisms need to be improved in order to gain the quantitative predictability of methane–hydrogen–air combustion at normal pressure before proceeding to the conditions fuel-oxygen high pressure combustion more relevant for rocket chamber.

Precipitation patterns are commonly concentric rings forming in a Petri dish or parallel bands appearing in a test tube (Liesegang phenomenon). The rings frequently consist of a number of convex segments that are separated from each other by spaces devoid of precipitate resulting in small gaps (dislocations). Along these gaps, the so-called zig-zag structures can form, which connect one side of a gap with its opposite side. We observe that the occurrence of zig-zags requires a minimum thickness of the reactive layer (≥ 0.8 mm). This fact together with microscopic evidence indicates their three-dimensional character. One finds that at the very beginning of the precipitation reaction a curling process starts in the corresponding contour lines. These observations suggest structures of a helicoid with the axis perpendicular to the plane of the reaction-diffusion front to pass through the layer. Zig-zags are not parallel to the reaction plane, i.e., they are not formed periodically, but evolve continuously as a rotating spiral wave. Thus, their topology is closely related to helices in a test tube.

A possibility of generating a high degree of spin polarization of 13 C and 15 N nuclei in the cyanide ion, which forms the coordination bond with the metal ion, using parahydrogen is demonstrated for the first time for the new iridium carbene complex as an example. The spin–spin interaction constants in the synthesized complex and the structure of the hydride intermediate are determined by an analysis of the 13 С NMR spectra detected using broadband and selective heteronuclear decoupling. The cyanide ion is shown to coordinate to the metal ion by the carbon atom in one of two equatorial positions, and two pyridine molecules are arranged in the axial and equatorial positions. The signal amplification factors for 13 С and 15 N nuclei of the cyanide anion (5665 and –49 555, respectively) are estimated by NMR spectroscopy of the polarized substrate using the SABRE method from an ultralow magnetic field of 0.5 μT. This amplification corresponds to 15.5% polarization of nitrogen nuclei achieved within several seconds at room temperature.

In memory of Yurii Mikhailovich Romanovsky, Aksenteva M.S., Guria G.T., Ivanitskii G.R., Makarov V.A., Polezhaev A.A., Priezzhev A.V., Riznichenko G.Yu., Ritus V.I., Romanovsky M.Yu., Rudenko O.V., Sysoev N.N., Tuchin V.V.

The stability and dynamics of combustion waves in a system consisting of two exothermic solid fuel layers in thermal contact is numerically investigated within the framework of a one-dimensional model. The influence of the thermal interaction between the layers on the stability of combustion waves in the layers is investigated. The characteristics of instabilities and the dynamics of emerging pulsations of the reaction fronts are found to be different for weak and strong thermal coupling. It is shown that with a strong thermal contact between the layers, the stability boundary of a joint system can be significantly expanded.

In chemical systems, in particular, combustion systems, a variety of spatiotemporal modes is observed. For example, depending on the Lewis number defining the ratio of the heat transfer efficiency to the diffusion of chemical components, either autowave or cellular structures can arise, at the combustion front. Based on a simple mathematical model obtained by the complete model reduction describing the hydrogen combustion kinetics, formation features of such structures are studied. In this study, our interest is the description of cellular (Turing) structures at the combustion front within the model. A parametric analysis determining the Turing instability onset criteria is performed. It is found that a necessary condition is the Lewis number smaller than unity, which is consistent with experimental observations. In the found parameter range, numerical calculations are performed; as results, various versions of formed structures are shown.

In experimental studies of the propagation of combustion waves in gaseous media, it was found that, under certain conditions, autowave - spiral or target - patterns appear at the wave front. The purpose of the present study is to propose a mathematical model that can explain this phenomenon based on the known chemical kinetics of hydrogen combustion. Model. The original detailed model was first reduced to four equations that adequately describe the propagation of the combustion wave. To explain the structures at the combustion front, the model was further reduced to two equations. Results. An analytical study of the resulting model was carried out, which demonstrated that it can describe the occurrence of spiral waves, and the corresponding conditions for the parameters of the model were determined. These analytical results have been confirmed in numerical experiments. Conclusion. Thus, it has been demonstrated that the model constructed on the basis of the reduction of the known kinetic scheme of hydrogen combustion is capable of explaining the experimentally observed autowave patterns at the propagating combustion front.

In some chemical systems, the reaction proceeds in the form of a propagating wave. An example is the propagation of a combustion wave. At the front of such a wave, different oscillatory regimes and the appearance of spatiotemporal structures can be observed. We propose a qualitative mechanism for the formation of patterns at the front of the reaction. It is assumed that the reason is the interaction of two subsystems, one corresponding to the propagating front and the other describing the emerging patterns. The appropriate mathematical model contains two blocks: for the travelling front, we use a model of the Fisher-Kolmogorov-Petrovsky-Piskunov type, while patterns at the front are described by the FitzHugh-Nagumo type model. Earlier, we applied this approach to explain the occurrence of autowaves-target waves and spirals-at the front of the reaction. In the present paper, we demonstrate in numerical simulations that this approach also works effectively to explain stationary relative to the front patterns, the so-called Turing or cellular structures, that are observed experimentally, in particular, at the front of a combustion wave. We also investigate the dependence of these patterns on the thickness of the front and its speed, as well as on the degree of diffusion instability achieved within the front layer.

The classical concept for emergence of Turing patterns in reaction-diffusion systems requires that a system should be composed of complementary subsystems, one of which is unstable and diffuses sufficiently slowly while the other one is stable and diffuses sufficiently rapidly. In this work, the phenomena of emergence of Turing patterns are studied and do not fit into this concept, yielding the following results. (1) The criteria are derived, under which a reaction-diffusion system with immobile species should spontaneously produce Turing patterns under any diffusion coefficients of its mobile species. It is shown for such systems that under certain sets of types of interactions between their species, Turing patterns should be produced under any parameter values, at least provided that the corresponding spatially non-distributed system is stable. (2) It is demonstrated that in a reaction-diffusion system, which contains more than two species and is stable in absence of diffusion, the presence of a sufficiently slowly diffusing unstable subsystem is already sufficient for diffusion instability (i.e., Turing or wave instability), while its complementary subsystem can also be unstable. (3) It is shown that the presence of an immobile unstable subsystem, which leads to destabilization of waves within an infinite range of wavenumbers, in a spatially discrete case can result in the generation of large-scale stationary or oscillatory patterns. (4) It is demonstrated that under the presence of subcritical Turing and supercritical wave bifurcations, the interaction of two diffusion instabilities can result in the spontaneous formation of Turing structures outside the region of Turing instability.

We consider the model describing propagation of a combustion wave in a system of two layers of different exothermic reacting materials under conditions of thermal contact between them through a common surface. This system is directly related to synthesis of advanced materials via the Self-propagating High temperature Synthesis technology when one of the reactants serves as a heat source (donor layer) for the other reacting material (acceptor layer) and facilitates the chemical reaction in the latter. The reaction sheet approximation is used and the parametric study of the boundaries of existence and characteristics of combustion waves in the system of layers is undertaken. The parameters of the process are identified which allow to achieve significantly superadiabatic peak temperatures of combustion in the acceptor layer. (C) 2019 Elsevier Inc. All rights reserved.

A qualitative mechanism for autowave pattern formation at the reaction front, observed in certain chemical systems including combustion, is suggested. It is assumed that patterns are formed as a result of interaction of two subsystems, one of which is responsible for the reaction front propagation while the other determines the formation of waves at the front. A corresponding phenomenological model is constructed in which reaction front propagation is described by a submodel of the Fisher-Kolmogorov-Petrovskii-Piskunov type and waves on the front are described by a submodel of the FitzHugh-Nagumo type. In the three-dimensional numerical analysis, it is demonstrated that the model is able to qualitatively explain the emergence of wave patterns of both spiral and target types, which are experimentally observed at the reaction front. The dependence of these patterns on the velocity and thickness of the front is examined.

In this chapter, we provide mathematical data concerning the description of spirals. Before starting with mathematical equations, Albrecht Dürer'sDürer/Albrecht Dürer pioneering works are briefly introduced. Subsequently, we discuss some properties of different spirals in a plane which make them important in nature and for technical applications. Smooth spirals are usually described by equations which are formulated either in terms of the polar coordinates radius and angle, such spirals being called algebraic, or in terms of curvature and arc length; then they are referred to as pseudo-spirals. WeSpiralpseudo consider in detail a number of spirals of both classes emphasizing their most essential features. Besides 2D spirals we also discuss examples of 3D spirals, usually referred to as helices. To conclude the chapter we mention non-smooth spirals and fractalFractal spirals.

A qualitative mechanism of the formation of wave structures at the reaction front is proposed. It is assumed that the structures are formed as a result of the interaction of two subsystems, one of which is responsible for the front formation, and the other is responsible for the formation of structures themselves. Three models are considered; two-dimensional analogues of concentric and spiral waves are numerically demonstrated in each. Fitzhugh–Nagumo, Fisher–Kolmogorov–Petrovskii–Piscounov (Fisher–KPP), and Oregonator models were used as subsystems.

We investigate numerically the behavior of a two-component reaction-diffusion system of Fitzhugh-Nagumo type before the onset of subcritical Turing bifurcation in response to local rigid perturbation. In a large region of parameters, the initial perturbation evolves into a localized structure. In a part of that region, closer to the bifurcation line, this structure turns out to be unstable and covers all the available space over the course of time in a process of self-completion. Depending on the parameter values in two-dimensional (2D) space, this process happens either through generation and evolution of new peaks on oscillatory tails of the initial pattern, or through the elongation, deformation, and rupture of initial structure, leading to space-filling nonbranching snakelike patterns. Transient regimes are also possible. Comparison of these results with 1D simulations shows that the prebifurcation region of parameters where the self-completion process is observed is much larger in the 2D case.

In this paper, the properties and stability of combustion waves propagating in the composite solid energetic material of the shell-core type are numerically investigated within the one-dimensional diffusive-thermal model with heat losses to the surroundings. The flame speed is calculated as a function of the parameters of the model. The boundaries of stability are determined in the space of parameters by solving the linear stability problem and direct integration of the governing non-stationary equations. The results are compared with the characteristics of the combustion waves in pure solid fuel. It is demonstrated that a stable travelling combustion wave solution can exist for the parameters of the model for which the flame front propagation is unstable in pure solid fuel and it can propagate several times faster even in the presence of significant heat losses.

Памяти Александра Сергеевича Холодова(11

In this paper, we numerically investigate the stability of propagating combustion waves in the competitive exothermic–endothermic reaction model. The analysis is based on the Evans function method and direct numerical integration of the governing partial differential equations. The critical conditions for the onset of instability are found for a broad range of parameter values of the model. It is demonstrated that for the parameter values for which the combustion wave is unstable in the one-step reaction model, the inclusion of the endothermic step can lead to flame stabilization.