Modelling and solutions to the linear stability of a detonation wave in the kinetic frame

Artigo publicado num número especial da revista.The analysis of linear stability of a steady detonation wave is formulated for the first time at the kinetic level in the frame of the Boltzmann equation extended to reacting gases. Within this context and for a reversible reaction, the stability problem is carried out, in agreement with most classical papers on gas detonation, through a normal mode approach for the one-dimensional disturbances of the steady wave solution, and an acoustic radiation condition at the final equilibrium as closure condition. The proposed modelling leads to an initial value problem, constituted by the linearized reactive Euler equations in the perturbed shock frame with related Rankine-Hugoniot conditions, which can be solved by means of a proper numerical technique. An application is provided for an elementary bimolecular reaction.Centro de Matemática da Universidade do MinhoFundação para a Ciência e a Tecnologia (FCT)Italian INDAM-GNF

Kinetic approach to transport properties of a reacting gas

A multicomponent reacting gas with reversible reactions is studied at a kinetic level with the main objective of deriving the reactive Navier-Stokes equations in dependence on the macroscopic variables, and characterizing the dissipative terms related to shear viscosity, heat conduction and thermal diffusion. A step-by-step procedure, which can be applied to a quite large variety of reactive flows, is proposed in order to identify the transport coefficients, basically resorting to a first-order density approximation of Chapman-Enskog type.Fundação para a Ciência e Tecnologia (FCT) - Programa Operacional "Ciência, Tecnologia, Inovação" (POCTI).National Research Project PRIN 2003

A reactive BGK-type model: influence of elastic collisions and chemical interactions

A BGK‐type model for a reactive multicomponent gas undergoing chemical bimolecular reactions is here presented. The mathematical and physical consistency of the model is stated in detail. The relaxation process towards local Maxwellians, depending on mass and numerical densities of each species, as well as on common mean velocity and temperature, is investigated with respect to chemical equilibrium. Such a trend is numerically tested within the hydrogen‐air reaction mechanism

A nonlinear transport problem of monochromatic photons in resonance with a gas

A transport problem arising from the dynamics of a gas in a radiation field, recently modelled in kinetic theory, is formulated and the trend to equilibrium of the gas-photon system is studied. A computational technique matching relevant mathematical aspects of differential quadrature and spectral methods is applied. The numerical results are then compared with those of other models known in literatur

On the kinetic modelling of reactive waves in the hydrodynamic limit

The present study deals with the extension of the exact and approximate models of the Boltzmann equation to a gas mixture of four constituents undergoing a reversible bimolecular reaction, and its application to wave propagation problems. The paper intends to highlight how Boltzmann type models, on the basis of a shared kinetic framework, can be adopted as the starting point for a consistent derivation of the reactive hydrodynamic equations, at both the Euler and Navier Stokes limit. At this scope, a proper mathematical procedure is applied to obtain an approximate solution to the model equations, which is necessary in order to derive the closed system of the governing equations in the above said hydrodynamic limits. Resorting to this unified kinetic approach which is presented in detail, one can recognize how the dynamics of rather different wave problems well known in literature, as the steady detonation wave and its linear stability, the sound wave propagation and the light scattering phenomena, match a satisfactory description with care for the chemical mechanism at the microscopic scale. The knowledge of the chemical process at this level permits to evaluate the influence of the chemical reaction on the fundamental aspects of the reacting gas system, reinforcing the proposed kinetic approach. Accordingly, some propagation wave problems, recently studied by the authors in this context, are in turn here reviewed at the end of focusing how their formulation and solution depend on the proposed hydrodynamic closure procedure.Fundação para a Ciência e a Tecnologia (FCT

Spectral distribution of scattered light from a chemical relaxation system

The aim of this work is to describe the light scattering spectra of a quaternary reacting gas mixture from the macroscopic field equations derived from the kinetic BGK-type model proposed by the authors in a previous paper. The study is developed in a hydrodynamic regime for which the system of the field equations of constituent number densities, momentum and temperature of the mixture is closed by the constitutive equations for rate of reaction, diffusion velocities, pressure tensor and heat flux vector. The spontaneous Rayleigh-Brillouin scattering is calculated from the constituent density perturbations of the linearized field equations, and its line shape is drawn for two different mixtures of the Hydrogen-Chlorine system showing the induced chemical reaction effect.Brazilian Research Council (CNPq)Universidade do Minho. Centro de Matemática(CMat)Fundação para a Ciência e a Tecnologia (FCT) - FCT-PTDC/MAT/68615/2006Italian National Project GNFM 2009/1

A kinetic approach to propagation and stability of detonation waves

The problem of the steady propagation and linear stability of a detonation wave is formulated in the kinetic frame for a quaternary gas mixture in which a reversible bimolecular reaction takes place. The reactive Euler equations and related Rankine‐Hugoniot conditions are deduced from the mesoscopic description of the process. The steady propagation problem is solved for a Zeldovich, von Neuman and Doering (ZND) wave, providing the detonation profiles and the wave thickness for different overdrive degrees. The one‐dimensional stability of such detonation wave is then studied in terms of an initial value problem coupled with an acoustic radiation condition at the equilibrium final state. The stability equations and their initial data are deduced from the linearized reactive Euler equations and related Rankine‐Hugoniot conditions through a normal mode analysis referred to the complex disturbances of the steady state variables. Some numerical simulations for an elementary reaction of the hydrogen‐oxygen chain are proposed in order to describe the time and space evolution of the instabilities induced by the shock front perturbation

Equilibrium and stability properties of detonation waves in the hydrodynamic limit of a kinetic model

A shock wave structure problem, like the one which can be formulated for the planar detonation wave, is analyzed here for a binary mixture of ideal gases undergoing the symmetric reaction A1+A1=A2+A2 . The problem is studied at the hydrodynamic Euler limit of a kinetic model of the reactive Boltzmann equation. The chemical rate law is deduced in this frame with a second-order reaction rate, in a hemical regime such that the gas flow is not far away from the chemical equilibrium. The caloric and the thermal equations of state for the specific internal energy and temperature are employed to close the system of balance laws. With respect to other approaches known in the kinetic literature for detonation problems with a reversible reaction, this paper aims to improve some aspects of the wave solution. Within the mathematical analysis of the detonation model, the equation of the equilibrium Hugoniot curve of the final states is explicitly derived for the first time and used to define the correct location of the equilibrium Chapman–Jouguet point in the Hugoniot diagram. The parametric space is widened to investigate the response of the detonation solution to the activation energy of the chemical reaction. Finally, the mathematical formulation of the linear stability problem is given for the wave detonation structure via a normal-mode approach, when bidimensional disturbances perturb the steady solution. The stability equations with their boundary conditions and the radiation condition of the considered model are explicitly derived for small transversal deviations of the shock wave location. The paper shows how a second-order chemical kinetics description, derived at the microscopic level, and an analytic deduction of the equilibrium Hugoniot curve, lead to an accurate picture of the steady detonation with reversible reaction, as well as to a proper bidimensional linear stability analysis.Brazilian Research Council (CNPq), by Italian Research Council GNFM-INdAM, and by the Research Centre of Mathematics of the University of Minho with the Portuguese Funds of FCT, project PEstOE/MAT/UI0013/2014

Hydrodynamic bidimensional stability of detonation wave solutions for reactive mixtures

The structure of a planar detonation wave is analyzed for an Eulerian mixture of ideal gases undergoing the symmetric reversible explosive reaction A1 + A1 = A2 + A2. The chemical rate law is derived from the reactive Boltzmann equation, showing a detailed chemical kinetics in terms of a second-order reaction rate. The hydrodynamic bidimensional stability of the detonation wave is also investigated using a normal mode approach, when small time-space transverse disturbances affect the shock wave location. A suitable numerical technique is here proposed in order to solve the stability problem and numerical results are provided illustrating the detonation wave structure and its instability spectrum.The paper is partially supported by Brazilian Research Council (CNPq), by Italian Research Council GNFM-INdAM, and by Portuguese Funds of FCT, CMAT project UID/MAT/00013/2013

Search for the standard model Higgs boson in the H to ZZ to 2l 2nu channel in pp collisions at sqrt(s) = 7 TeV

A search for the standard model Higgs boson in the H to ZZ to 2l 2nu decay channel, where l = e or mu, in pp collisions at a center-of-mass energy of 7 TeV is presented. The data were collected at the LHC, with the CMS detector, and correspond to an integrated luminosity of 4.6 inverse femtobarns. No significant excess is observed above the background expectation, and upper limits are set on the Higgs boson production cross section. The presence of the standard model Higgs boson with a mass in the 270-440 GeV range is excluded at 95% confidence level.Comment: Submitted to JHE