In this paper we consider a one-dimensional reaction-diffusion model with piecewise continuous reaction term that describes propagation of autoignition fronts in reactive co-flow jets in a certain parametric regime. The model is reduced to a free boundary problem with two interfaces. It is shown that this problem admits permanent traveling front solution which is unique up to translations. The result is obtained using dynamical system approach employing Stable Manifold Theorem and the Melnikov integral as the main tools.

We study a boundary value problem for the p-Laplacian in the perforated domain B(0,ρ)⊂ℝ^d, where ρ>1 and Γ is the union of many small compact cavities placed near the unit sphere. The cavities are separated at scale ε, asymptotically equidistributed on the sphere, and have cardinality of order ε^1-d. The cavities have diameters of order α(ε)ε, where α(ε)→0, and their relative p-capacity is comparable to the relative p-capacity of a ball of the same diameter. The solution is required to equal 1 on all cavities and 0 on ∂ B(0,ρ). We focus on the critical case p=d>1. We identify the critical scale through the parameter τ=lim_ε↓0[εlog(1/α(ε))]^-1∈[0,∞]. Thus, α(ε)=exp[-(1+o(1))/(τε)] when 0<τ<∞. Away from the unit sphere, the solutions converge to A_*U_ρ, where U_ρ(x)=min{1,1-log |x|/} is the radial d-harmonic potential of the unit ball in B(0,ρ). The constant A_* equals 0 when τ=0, equals 1 when τ=∞, and is explicit for 0<τ<∞. We construct an explicit ansatz that approximates the solution for sufficiently small ε in both L^∞ and in terms of d-capacity.

This paper is concerned with a study of a natural generalization of a classical Frank-Kamenetskii model of thermal explosion in the presence of a vortical flow in a two dimensional setting. This model describes possible stationary temperature distributions in a combustion vessel which boundary is maintained at a constant temperature. The model constitutes a Dirichlet boundary value problem for a certain semi-linear elliptic equation that depends on a parameter λ, called Frank-Kamenetskii parameter. A remarkable property of this problem is that it admits a classical minimal solution when the Frank-Kamenetskii parameter does not exceed some critical value λ^* and no classical solutions for λ>λ^*. The absence of a classical solution, in the framework of Frank-Kamenetskii theory, is associated with the thermal explosion event. Consequently, in the context of combustion, λ^*, commonly called an explosion threshold, is a maximal value of the Frank-Kamenetskii parameter which allows to attain a thermal equilibrium within a combustion vessel and thus provides a sharp characterization of the thermal explosion. A critical temperature distribution corresponding to λ^* is called an extremal solution. In this paper, we show that, under an assumption of sufficiently fast growth of the reaction term, there exists a regular vortical flow that allows to adjust an explosion threshold by reversing its direction, provided a combustion vessel is not a disk.We also give rather detailed description of extremal solutions. In particular, we show that extremal solutions are always classical.

The proposed study is motivated by experimental evidence, dating back to 1985, demonstrating the possibility of deflagration-to-detonation transition (DDT) in a fuel-air cloud. The detonation is initiated by a flame jet developed in a thin open-ended tube inserted into the cloud. Despite the experimental data, a first-principle understanding of the mechanism controlling the transition is still missing. The current research is aimed at resolution of this issue through a simple 2D formulation involving minimum physical ingredients.

In this paper we formulate and analyze an elementary model for the extinction of hydrothermal flames in laminar reactive jets experimentally observed in an aqueous environment at pressures exceeding the critical point of water. This work is motivated by experimental studies of the dynamics of hydrothermal flames performed at the high-pressure laboratory of the NASA Glenn Research Center. Guided by experimental observations, we use several simplifying assumptions that allow the derivation of a simple, yet experimentally feasible, mathematical model for the hydrothermal flame extinction. The analysis of the model shows that the principal parameters of the problem determining flame extinction are Damk & ouml;hler number and the injection velocity of the reactive components. In particular, we formulate a sharp condition for the extinction of hydrothermal flames in terms of Damk & ouml;hler number. The results of the present study may assist in the interpretation of existing experimental data and provide guidance for future experiments.

In this paper we consider a classical model of gasless combustion in a one dimensional formulation under the assumption of ignition temperature kinetics. We study the propagation of flame fronts in this model when the initial distribution of the solid fuel is a spatially periodic function that varies on a large scale. It is shown that in certain parametric regimes the model supports periodic traveling fronts. An accurate asymptotic formula for the velocity of the flame front is derived and studied. The stability of periodic fronts is also explored, and a critical condition in terms of parameters of the problem is derived. It is also shown that the instability of periodic fronts, in certain parametric regimes, results in a propagation-extinction-conduction-reignition pattern which is studied numerically. Novelty and significance statement: This work provides a closed form asymptotic description of periodic traveling fronts in a gasless combustion model with step-wise ignition temperature kinetics with a slowly varying concentration field. The stability analysis is performed, and the range of applicability of asymptotic formulas is given. A new propagation-extinction-conduction-reignition regime is identified. This regime emerges exclusively due to periodicity of the concentration field.

On a certain class of 16-dimensional manifolds a new class of Riemannian metrics, called octonionic Kähler, is introduced and studied. It is an octonionic analogue of Kähler metrics on complex manifolds and of HKT-metrics of hypercomplex manifolds. Then for this class of metrics an octonionic version of the Monge–Ampère equation is introduced and solved under appropriate assumptions. The latter result is an octonionic version of the Calabi–Yau theorem from Kähler geometry.

In this paper we formulate and analyze an elementary model for the propagation of advancing autoignition fronts in reactive co-flow fuel/oxidizer jets injected into an aqueous environment at high pressure. This work is motivated by the experimental studies of autoignition of hydrothermal flames performed at the high pressure laboratory of NASA Glenn Research Center. Guided by experimental observations, we use several simplifying assumptions that allow the derivation of a simple, still experimentally feasible, mathematical model for the propagation of advancing ignition fronts. The model consists of a single diffusion-absorption-advection equation posed in an infinite cylindrical domain with a non-linear condition on the boundary of the cylinder and describes the temperature distribution within the jet. This model manifests an interplay of thermal diffusion, advection and volumetric heat loss within a fuel jet which are balanced by the weak chemical reaction on the jet's boundary. We analyze the model by means of asymptotic and numerical techniques and discuss feasible regimes of propagation of advancing ignition fronts. In particular, we show that in the most interesting parametric regime when the advancing ignition front is on the verge of extinction this model reduces to a one dimensional reaction-diffusion equation with bistable non-linearity. We hope that the present study will be helpful for the interpretation of existing experimental data and guiding of future experiments.

We consider a boundary value problem for the p -Laplacian, posed in the exterior of small cavities that all have the same p -capacity and are anchored to the unit sphere in ℝ^d , where 10 . We show that the problem possesses a critical window characterized by τ :=lim _ε↓ 0α /α _c ∈ (0,∞ ) , where α _c=ε ^1/γ and γ = d-p/p-1. We prove that outside the unit sphere, as ε↓ 0 , the solution converges to A_*U for some constant A_* , where U(x)=min{1,|x|^-γ} is the radial p -harmonic function outside the unit ball. Here the constant A_* equals 0 if τ =0 , while A_*=1 if τ =∞ . In the critical window where τ is positive and finite, A_*∈ (0,1) is explicitly computed in terms of the parameters of the problem. We also evaluate the limiting p -capacity in all three cases mentioned above. Our key new tool is the construction of an explicit ansatz function u_A_*^ε that approximates the solution u^ε in L^∞(ℝ^d) and satisfies ‖∇ u^ε -∇ u_A_*^ε‖ _L^p(ℝ^d)→ 0 as ε↓ 0 .

In this paper we consider a classical model of gasless combustion in a one dimensional formulation under the assumption of ignition temperature kinetics. We study the propagation of flame fronts in this model when the initial distribution of the solid fuel is a spatially periodic function that varies on a large scale. It is shown that in certain parametric regimes the model supports periodic traveling fronts. An accurate asymptotic formula for the velocity of the flame front is derived and studied. The stability of periodic fronts is also explored, and a critical condition in terms of parameters of the problem is derived. It is also shown that the instability of periodic fronts, in a certain parametric regimes, results in a propagation-extinction-diffusion-reignition pattern which is studied numerically.

In this paper, we consider a reaction-diffusion system describing the propagation of flames under the assumption of ignition-temperature kinetics and fractional reaction order. It was shown in [3] that this system admits a traveling front solution. In the present work, we show that this traveling front is unique up to translations. We also study some qualitative properties of this solution using the combination of formal asymptotics and numerics. Our findings allow conjecture that the velocity of the propagation of the flame front is a decreasing function of all of the parameters of the problem: ignition temperature, reaction order and an inverse of the Lewis number.

The paper is concerned with identification of the key mechanisms controlling deflagration-to-detonation transition in stellar medium. The issue of thermal runaway triggered by positive feedback between the advancing flame and the flame-driven precompression is discussed in the framework of a one-dimensional flame-folding model. The paper is an extension of the authors' previous study dealing with the non-stoichiometric fusion, $fuel \to products$, kinetics (Phys.Rev.E, 103(2021)) over physically more relevant, $fuel+fuel \to products$, kinetics. Despite this change the runaway effect endures. The transition occurs prior to merging of the flame with the flame-supported precursor shock, i.e. the pretransition flame does not reach the threshold of Chapman-Jouguet deflagration.

A weakly nonlinear model for a self-accelerating outward propagating corrugated flame is formulated and explored. The self-acceleration is sustained by the intrinsic Darrieus-Landau and Rayleigh-Taylor instabilities until the Deshaies-Joulin deflagrability threshold is reached, followed by an abrupt transition to detonation. Emergence of the threshold is caused by positive feedback between the accelerating flame and the flame-driven pressure shock that results in the thermal runaway when the flame speed reaches a critical level. The model offers a simple mechanism that may be responsible for the transition to detonation in thermonuclear supernovae.

Nonlinear Rayleigh-Benard convection in an infinite-Prandtl-number fluid layer be-tween poorly conducting boundaries is considered as a model for convection in the earth's upper mantle. It is shown that accounting for the generally neglected impact of viscous dissipation may lead to the development of large-scale spatiotemporal chaotic dynamics governed by the familiar Kuramoto-Sivashinsky equation Phi(tau) + del(4)Phi + 2 del(2)Phi - (del Phi)(2) + alpha Phi = 0, known to occur in various physical systems.

The nature of thermonuclear explosions of white-dwarf stars is a fundamental astrophysical issue, the first principle interpretation of which is still commonly regarded as an unresolved problem. There is a general consensus that stellar explosions are a manifestation of the deflagration-to-detonation transition of an outward propagating self-accelerating thermonuclear flame subjected to instability-induced corrugations. A similar problem arises in unconfined terrestrial flames where a positive feedback mechanism leading to the pressure runaway has been identified. The present study is an application of this finding to the stellar environment. Notwithstanding a substantial modification of the equation of state the runaway effect endures. Approaching the runaway point the pretransition flame may stay perfectly subsonic, which challenges the view that to ensure the transition the flame should cross the threshold of Chapman-Jouguet deflagration.

In this paper we derive a simple, still experimentally feasible, model for propagation of traveling fronts of ignition which were recently observed in experimental studies of autoignition of co-flow reactive jets at high pressure. We obtain a closed form analytical solution for this model and present a detailed analysis of this solution. We also present a qualitative comparison of the model predictions with available experimental data.

We consider the model of auto-ignition (thermal explosion) of a free round reactive turbulent jet intro-duced in [11]. This model falls into the general class of Gelfand-type problems and constitutes a boundary value problem for a certain semi-linear elliptic equation that depends on two parameters: alpha characterizing the flow rate and lambda (Frank-Kamenetskii parameter) characterizing the strength of the reaction. Similarly to the classical Gelfand problem, this equation admits a solution when the Frank-Kamenetskii parameter lambda does not exceed some critical value lambda* (alpha) and admits no solutions for larger values of lambda. We obtain the sharp asymptotic behavior of the critical Frank-Kamenetskii parameter in the strong flow limit (alpha >> 1). We also provide a detailed description of the extremal solution (i.e., the solution corresponding to lambda*) in this regime. (C) 2020 Elsevier Inc. All rights reserved.

The paper is concerned with identification of the key mechanisms controlling deflagration-to-detonation transition in confined and unconfined gaseous systems. The issue of thermal runaway triggered by positive feedback between the advancing flame and the flame-driven precursor shock is revisited. A new mechanism for parametric transition based on the flame-speed sensitivity to pressure changes is discussed. Depending on the parameters of the system the transition may occur either within the subsonic, sonic or supersonic range of deflagrations. In the latter case the deflagration obeys the classical Chapman–Jouguet (CJ) condition. In a certain parameter range the adopted model reproduces the experimentally known situation where the transition occurs close to the sonic point.

The paper is concerned with the experimentally known phenomenon that the detonation velocity of a tubular charge may markedly exceed that of a homogeneous charge of the same explosive. It is shown that the effect may be successfully reproduced using a simple quasi-one-dimensional two-layer model assuming the gas-solid system to be isothermal and the volumetric fraction of the solid phase to be small. In view of a considerable drop of the burned gas pressure/density, compared to the Chapman-Jouguet case, fast detonation may be perceived as a variety of weak (undercompressed) detonation.

In this paper we consider a model of thermal explosion in porous media. The model consists of two reaction-diffusion equations in a bounded domain with Dirichlet boundary conditions and describes the initial stage of evolution of pressure and temperature fields. Under certain conditions, the classical solution of these equations exists only on finite time interval after which it forms a singularity and becomes unbounded (blows up). This behavior raises a natural question whether this solution can be extended, in a weak sense, after blow up time. We prove that the answer to this question is no, that is, the solution becomes unbounded in entire domain immediately after the singularity is formed. From a physical perspective our results imply that autoignition in porous materials occurs simultaneously in entire domain.