Large eddy simulation of coherent structures in rectangular methane non-premixed flame

Large eddy simulation of a three-dimensional spatially developing transitional free methane non-premixed flame is performed. The solver of the governing equations is based upon a projection method. The Smagorinsky model is utilized for the turbulent subgrid scale terms. A global reaction mechanism is applied for the simulation of methane/air combustion. Simulation results clearly illustrate the coherent structure of the rectangular non-premixed flame, consisting of three distinct zones in the near field. Periodic characteristics of the coherent structures in the rectangular non-premixed flame are discussed. The predicted structure of the flame is in good agreement with the experimental results. Distributions of species concentrations across the flame surfaces are illustrated and typical flame structures in the far field are analyzed. Local mass fraction analysis and flow visualization indicate that the black spots of the flames are due to strong entrainment of oxygen into the central jet by streamwise vortices, and breaking up of the flame is caused by an enormous amount of entrainment of streamwise vortices as well as stretching of spanwise vortices at the bottom of the flame.     

Solution of the linear and nonlinear advection–diffusion problems on a sphere

Various linear advection–diffusion problems and nonlinear diffusion problems on a sphere are considered and solved using the direct, implicit and unconditionally stable finite-volume method of second-order approximation in space and time. In the absence of external forcing and dissipation, the method preserves the total mass of the substance and the norm of the solution. The component wise operator splitting allows us to develop the direct (noniterative) and fast numerical algorithm. The split problems in the longitudinal direction are solved using the Sherman-Morrison formula and Thomas algorithm. The direct solution of the split problems in the latitudinal direction requires the use of the bordering method for a block matrix, and the preliminary determination of the solution at the poles. The resulting systems with tridiagonal matrices are solved by the Thomas algorithm. The numerical experiments demonstrate that the method correctly describes the local advection–diffusion processes on the sphere, in particular, through the poles, and accurately simulate blow-up regimes (unlimited growing solutions) of nonlinear combustion, the propagation of nonlinear temperature and spiral waves, and solutions to Gray-Scott reaction–diffusion model.     

Second-order thermal boundary-layer on a blunted wedge

Second-order thermal boundary-layer solutions are obtained for flow past a blunted wedge with constant wall temperature. Contributions due to longitudinal curvature and displacement effect are obtained by employing the G?rtler power series method. The first five terms of the series for each of the effects are computed. Since the region of validity of the results thus obtained is restricted in the streamwise direction, Eulerization technique is used to extend the radius of convergence to infinity.     

Combining linear and fast diffusion in a nonlinear elliptic equation

In this paper we analyse an elliptic equation that combines linear and nonlinear fast diffusion with a logistic type reaction function. We prove existence and non-existence results of positive solutions using bifurcation theory and sub-supersolution method. Moreover, we apply variational methods to obtain a pair of ordered positive solutions.     

Fast-diffusion limit with large noise for systems of stochastic reaction-diffusion equations

We consider a class of a stochastic reaction-diffusion equations with additive noise. In the limit of fast diffusion, one can approximate solutions of the stochastic reaction–diffusion equations by the solution of a suitable system of ordinary differential equation only describing the reactions, but due to nonlinear interaction of large diffusion and fluctuations in the limit new effective reaction terms appear. We focus on systems with polynomial nonlinearities and illustrate the result by applying it to a predator-prey system and a cubic auto-catalytic reaction between two chemicals.     

Global existence and blowup of a nonlocal problem in space with free boundary

This paper concerns a double fronts free boundary problem for the reaction–diffusion equation with a nonlocal nonlinear reaction term in space. For such a problem, we mainly study the blowup property and global existence of the solutions. Our results show that if the initial value is sufficiently large, then the blowup occurs, while the global fast solution exists for a sufficiently small initial data, and the intermediate case with a suitably large initial data gives the existence of the global slow solution.     

Stabilization to a positive equilibrium for some reaction–diffusion systems

The aim of this paper is to study the asymptotic behavior of solutions for some reaction–diffusion systems in biology. First, we establish a Liouville type theorem for entire solutions of these reaction–diffusion systems. Based on this theorem, we derive the stabilization of the solutions of the reaction–diffusion system to the unique positive constant state, under the condition that this positive constant state is globally stable in the corresponding kinetic systems. Two specific examples about spreading phenomena from ecology and epidemiology are given to illustrate the application of this theory.     

The Behavior of Solutions to a Special Abel Equation of the Second Kind near a Nodal Singular Point

Computational Mathematics and Mathematical Physics - The propagation of a diffusion–reaction plane traveling wave (for example, a flame front), the charge distribution inside a heavy atom in...     

一类弱耦合动力系统的参数识别问题

研究一类弱耦合反应－扩散动力系统的参数识别问题。通过构造上下解，证明了反应－扩散方程组解的存在惟一性；给出了求解参数识别问题的最优化系，从而可以选取适当的梯度法或者共轭梯度法，实现对系统参数的识别。     

Cross-Diffusion Limit for a Reaction-Diffusion System with Fast Reversible Reaction

We consider a reaction-diffusion system which models a fast reversible reaction of type C ₁ + C ₂?C ₃ between mobile reactants inside an isolated vessel. Assuming mass action kinetics, we study the limit when the reaction speed tends to infinity in case of unequal diffusion coefficients and prove convergence of a subsequence of solutions to a weak solution of an appropriate limiting pde-system, where the limiting problem turns out to be of cross-diffusion type. The proof combines the L ²-approach to reaction-diffusion systems having at most quadratic reaction terms with a thorough exploitation of the entropy functional for mass action systems. The limiting cross-diffusion system has unique local strong solutions for sufficiently regular initial data, while uniqueness of weak solutions is in general open but is shown to be valid under restrictions on the diffusivities.     

Analytical examples of diffusive waves generated by a traveling wave

We construct analytical solutions for a system composed of a reaction–diffusion equation coupled with a purely diffusive equation. The question is to know if the traveling wave solutions of the reaction–diffusion equation can generate a traveling wave for the diffusion equation. Our motivation comes from the calcic wave, generated after fertilization within the egg cell endoplasmic reticulum, and propagating within the egg cell. We consider both the monostable (Fisher–KPP type) and bistable cases. We use a piecewise linear reaction term so as to build explicit solutions, which leads us to compute exponential tails whose exponents are roots of second-, third-, or fourth-order polynomials. These raise conditions on the coefficients for existence of a traveling wave of the diffusion equation. The question of positivity and monotonicity is only partially answered.     

Propagation in Fisher–KPP type equations with fractional diffusion in periodic media

We are interested in the time asymptotic location of the level sets of solutions to Fisher–KPP reaction–diffusion equations with fractional diffusion in periodic media. We show that the speed of propagation is exponential in time, with a precise exponent depending on a periodic principal eigenvalue, and that it does not depend on the space direction. This is in contrast with the Freidlin–Gärtner formula for the standard Laplacian.     

Cauchy problem for fast diffusion equation with localized reaction

This paper studies the Cauchy problem for the fast diffusion equation with a localized reaction. We establish the Fujita type theorem to the problem, and then obtain the diffusion-independent blow-up rate for the non-global solutions. Moreover, we prove that the blow-up set for the problem consists of a single point under large initial data. These conclusions are quite different from those for the slow diffusion case.     

Global attractor of a coupled finite difference reaction diffusion system with delays

In the study of asymptotic behavior of solutions for reaction diffusion systems, an important concern is to determine whether and when the system has a global attractor which attracts all positive time-dependent solutions. The aim of this paper is to investigate the global attraction problem for a finite difference system which is a discrete approximation of a coupled system of two reaction diffusion equations with time delays. Sufficient conditions are obtained to ensure the existence and global attraction of a positive solution of the corresponding steady-state system. Applications are given to three types of Lotka-Volterra reaction diffusion models, where time-delays may appear in the opposing species.     

Global existence for degenerate parabolic equations with a non-local forcing

We establish local existence and comparison for a model problem which incorporates the effects of non-linear diffusion, convection and reaction. The reaction term to be considered contains a non-local dependence, and we show that local solutions can be obtained via monotone limits of solutions to appropriately regularized problems. Utilizing this construction, it is further shown that, under conditions of either ‘weak reaction’ or ‘sufficiently small’ initial mass, solutions exist for all time. Finally, we provide an alternative analysis of global existence and investigate blow up in finite time for the case of power law diffusion and convection. These results show the extent to which the assumption of weak reaction may be relaxed and still obtain global existence. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Traveling wave solutions in delayed diffusion systems via a cross iteration scheme

This paper is concerned with the existence of traveling wave solutions for delayed reaction diffusion systems which contain the competition diffusion systems with time lags. By using a cross iteration scheme, we reduce the existence of traveling wave solutions to the existence of a pair of admissible upper and lower solutions, which also provides a constructive process of the traveling wave solutions. To illustrate our conclusion, we consider a delayed diffusion system with the Gilpin–Ayala type nonlinearity and establish the existence of its traveling wave solutions, which cannot be answered by the existing results.     

Bogoliubov averaging principle of stochastic reaction–diffusion equation

The purpose of this paper is to establish Bogoliubov averaging principle of stochastic reaction–diffusion equation with a stochastic process and a small parameter. The solutions to stochastic reaction–diffusion equation can be approximated by solutions to averaged stochastic reaction–diffusion equation in the sense of convergence in probability and in distribution. Namely, we establish a weak law of large numbers for the solution of stochastic reaction–diffusion equation.     

An energy balance climate model with hysteresis

Energy balance climate models of Budyko type lead to reaction–diffusion equations with slow diffusion and memory on the 2-sphere. The reaction part exhibits a jump discontinuity (at the snow line). Here we introduce a Babuška–Duhem hysteresis in order to account for a frequent repetition of sudden and fast warming followed by much slower cooling as observed from paleoclimate proxy data. Existence of global solutions and of a trajectory attractor will be established for the resulting system of a parabolic differential inclusion and an ode.     

Fast Reaction Limit of the Discrete Diffusive Coagulation–Fragmentation Equation

Abstract

The local mass of weak solutions to the discrete diffusive coagulation–fragmentation equation is proved to converge, in the fast reaction limit, to the solution of a nonlinear diffusion equation, the coagulation and fragmentation rates enjoying a detailed balance condition.     

A fast precipitation and dissolution reaction for a reaction–diffusion system arising in a porous medium

This paper is devoted to the study of a fast reaction–diffusion system arising in reactive transport. It extends the articles [R. Eymard, T. Gallouët, R. Herbin, D. Hilhorst, M. Mainguy, Instantaneous and noninstantaneous dissolution: Approximation by the finite volume method, ESAIM Proc. (1998); J. Pousin, Infinitely fast kinetics for dissolution and diffusion in open reactive systems, Nonlinear Anal. 39 (2000) 261–279] since a precipitation and dissolution reaction is considered so that the reaction term is not sign-definite and is moreover discontinuous. Energy type methods allow us to prove uniform estimates and then to study the limiting behavior of the solution as the kinetic rate tends to infinity in the special situation of one aqueous species and one solid species.