Gibbs equation in the nonlinear nonequilibrium thermodynamics of dilute nonviscous gases

This paper deals with the derivation of the Gibbs equation for a nonviscous gas in the presence of heat flux. The analysis aims to shed some light on the physical interpretation of thermodynamic potentials far from equilibrium. Two different definitions for the chemical potential and thermodynamic pressure far from equilibrium are introduced: nonequilibrium chemical potential and nonequilibrium thermodynamic pressure at constant heat flux q and nonequilibrium chemical potential and nonequilibrium thermodynamic pressure at constant J = Vq, where V is the specific volume.     

On the Properties of a Quasihydrodynamic System of Equations for a Homogeneous Gas Mixture with a Common Regularizing Velocity

Differential Equations - We study a quasihydrodynamic system of equations for a homogeneous (with common velocity and temperature) multicomponent gas mixture in the absence of chemical reactions...     

Effective method for numerical integration of the equations describing the flow of high-temperature multicomponent gas mixtures in thermochemical equilibrium

A new stable iterative method is described for computing the flow of a high-temperature multicomponent gas mixture in thermochemical equilibrium. The method is illustrated by computing the thermochemical destruction of a carbon material in a high-temperature airflow.     

A consistent numerical method for the solution of the Boltzmann equation for inelastic and reactive scattering

A spatially homogeneous gas mixture is considered in which inelastic collisions and chemical reactions may occur. The corresponding Boltzmann equation is transformed to a system of scalar kinetic equations. A method is presented for the numerical solution of this set of integro-differential equations. It is shown that the method is consistent with the Boltzmann equation in the sense that it is conserving and preserves theH-theorem, that the equilibrium solution is a discretized Maxwellian, and that the equilibrium densities satisfy the generalized law of mass action.

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Zusammenfassung Es wird ein räumlich homogenes Gasgemisch betrachtet, dessen Moleküle durch elastische und inelastische Stöße, sowie durch chemische Umwandlungsprozesse miteinander wechselwirken. Die entsprechende Boltzmann-Gleichung wird in ein System skalarer kinetischer Gleichungen umgeformt. Eine Methode zur numerischen Lösung dieses Systems von Integrodifierentialgleichungen wird präsentiert. Wie sich zeigen läßt, ist das numerische Verfahren konsistent, d.h. es gelten die Erhaltungssätze, einH-Theorem und ein verallgemeinertes Massenwirkungsgesetz und als Gleichgewichtslösung ergibt sich eine diskretisierte Maxwell-Verteilung.

     

The hydrodynamic equations for chemically equilibrium flows of a multielement plasma with exact transport coefficients

Explicit expressions for all of the effective transport coefficients are derived for thermochemically equilibrium flows using the exact mass and heat transfer equations, which are resolved with respect to the “forces” (the gradients of the hydrodynamic variables) via the flukes. It is shown that, in a mixture where the components have different diffusion properties, separation (diffusion) of the chemical elements occurs which leads to a state of affairs where the equilibrium concentrations, and together with them, the effective transport coefficients will be functions not only of the pressure and temperature but will also depend on the concentrations of the elements, determined when solving the problem (self-consistent concentrations of the elements). It is shown that the existence of an electric current and lack of quasineutrality (flow around electrically conducting walls—electrodes) does not change the structure of the expressions for the effective transport coefficients and does not add anything new. The approximate and incomplete treatment of thermochemically equilibrium flows of multicomponent gas mixtures and a plasma in previously published papers are especially noted. Numerical estimates of the effective transport coefficients are presented for an air plasma and the domains in the pressure-temperature plane with the required number of approximations in order to obtain results with an error of no worse than 5% are indicated.     

Numerical Investigation of Cavitation Interacting with Pressure Wave

A computational fluid dynamics solver based on homogeneous cavitation model is employed to compute the two-phase cavitating flow. The model treats the two-phase regime as the homogeneous mixture of liquid and vapour which are locally assumed to be under both kinetic and thermodynamic equilibrium. As our focus is on pressure wave formation, propagation and its impact on cavitation bubble, the compressibility effects of liquid water have to be accounted for and hence the flow is considered to be compressible. The cavitating flow disturbed by the introduced pressure wave is simulated to investigate the unsteady features of cavitation due to the external perturbations. It is observed that the cavity becomes unstable, locally experiencing deformation or collapse, which depends on the shock wave intensity and freestream flow speed.     

Primal-Dual Active-Set Algorithm for Chemical Equilibrium Problems Related to the Modeling of Atmospheric Inorganic Aerosols

A general equilibrium model for multiphase multicomponent inorganic atmospheric aerosols is proposed. The thermodynamic equilibrium is given by the minimum of the Gibbs free energy for a system involving an aqueous phase, a gas phase, and solid salts. A primal-dual algorithm solving the Karush-Kuhn-Tucker conditions is detailed. An active set/Newton method permits to compute the minimum of the energy and tracks the presence or not of solid salts at the equilibrium. Numerical results show the efficiency of our algorithm for the prediction of multiphase multireaction chemical equilibria.Communicated by R. GlowinskiThis work has been partially supported by the United States Environmental Protection Agency through Cooperative Agreement X-83234201 to the University of Houston. The second author was supported by the Swiss National Science Foundation, Grant PBEL2-103152.     

Entropic structure of multicomponent reactive flows with partial equilibrium reduced chemistry

In this paper, we investigate the system of equations modelling multicomponent reactive flows with detailed transport and complex chemistry in the limit of partial equilibrium. The reduced system is obtained using a projection step compatible with the chemical entropy production. The reduced multicomponent transport and convection fluxes are shown to be compatible with the mathematical entropy thus providing a symmetric form as well as normal forms for the reduced system. This yields global existence and asymptotic stability around constant equilibrium states for the Cauchy problem on the partial equilibrium manifold in all space dimensions. Copyright © 2004 John Wiley & Sons, Ltd.     

Diffusion limits of the linear boltzmann equation in extended kinetic theory: Weak and strong inelastic collisions

We consider the asymptotic analysis for the linear Boltzmann equation with elastic and inelastic scattering. The physical model describes the motion of test particles propagating by elastic and inelastic collisions through a host medium in the Lorentz gas limit. The background is in thermodynamical equilibrium with only two internal energy levels. We apply the compressed Chapman-Enskog procedure to derive the diffusive-type approximations in the cases of dominant elastic and dominant inelastic collisions. Then we present numerical examples showing the time evolution of the distribution function in some physically relevant cases. In the appendix the successive overrelaxation method is briefly cutlined. Conferenza tenuta il giorno 27 Settembre 1999 da G. Frosali     

Multicomponent steady flows in porous media with phase transitions

Steady multiphase flow of a multicomponent mixture in a porous medium with phase transitions is considered. It is shown that, in a wide class of cases, the thermodynamic problem separates from the filtration problem and the latter is integrated in quadratures. The class of exact solutions which has been found is used to interpret indicator curves. Solutions are presented in an analytic form for systems of the “gas-condensate” and “oil-gas” type.     

Seepage of a gas condensate mixture when using the geoloosening method

The problem of the seepage of a two-phase multicomponent hydrocarbon mixture for evaluating the efficiency of the use of the geoloosening method in gas condensate deposits is considered. The geoloosening method is a technology for increasing the productivity of wells, developed at the Institute of Problems in Mechanics of the Russian Academy of Sciences, and it ensures an increase in the permeability of the critical zone of a well because of directed relief of the stratum. The initiation of the geoloosening process requires the creation of deep depressions at the well bottom and, as a result, there is an accumulation of retrograde condensate in the neighbourhood of the well, which leads to a decrease in the phase permeability with respect to the gas. It is necessary to take account of the existence of these two processes, which are mutually counter directed from the point of view of the change in permeability, when this method is used for gas condensate deposits. Due to the change in the chemical composition of the mixture in the condensation process and the action of capillary forces, the gas content at each point and each instant is not the equilibrium content and, consequently, cannot be directly determined from the phase diagram of the substance. A differential scheme is used to describe the seepage of the mixture, according to which, unlike an integral scheme, the relation for the transition into the liquid phase is specified for increments and not for the pressure and volume values themselves. Numerical calculations of the steady seepage of a hydrocarbon mixture are carried out for the necessary depression levels for the conditions in the Astrakhan gas condensate deposit and the effectiveness of the use of the geoloosening method there is demonstrated.     

Spinodal decomposition for the cahn-hilliard equation

The Cahn-Hilliard equation is a fourth-order parabolic partial differential equation that is one of the leading models for the study of phase separation in isothermal, isotropic, binary mixtures, such as molten alloys. When a spatially homogeneous alloy is rapidly quenched in a physical experiment, a fine-grained decomposition into two distinct phases is frequently observed; this phenomenon is known as spinodal decomposition. A simple linear analysis about an unstable homogeneous equilibrium of the one-dimensional Cahn-Hilliard equation gives heuristic evidence that most solutions that start with initial data near such an equilibrium exhibit a behavior corresponding to spinodal decomposition. In this paper we formulate this conjecture in a mathematically precise way, using geometric and measure-theoretic techniques, and prove its validity. We believe that this is the first rigorous treatment of this phenomenon.     

Gas-dynamic equations for spatially inhomogeneous gas mixtures with internal degrees of freedom. II. General representation for one-temperature reaction rates

In our previous paper (Kolesnichenko and Gorbachev, 2010) [1] the general approach for solving kinetic equations for gas mixtures with internal degrees of freedom and for getting corresponding gas-dynamic equations was developed. In the present article we continue our studies and focus on formulating expressions for reaction rates arising in zero-order (Euler) gas-dynamic equations for a one-temperature case. Derived expressions take into account all non-equilibrium effects, that we understand as deviation of the distribution function from its quasi-equilibrium value. As was shown in Kolesnichenko and Gorbachev (2010, 2011)  and , for zero-order approximation these effects can be subdivided into two groups. The first group contain the effects caused by the perturbation of quasi-equilibrium distribution function by the physical–chemical processes. We call them “scalar” non-equilibrium effects. Only these effects remain in the spatially homogeneous case. The second group consists of the terms proportional to the velocity divergence and therefor is indicated as spatially inhomogeneous. Both the above-mentioned effects are described via additive corrections to the quasi-equilibrium distribution function. Thus they can be treated separately and give rise to the corresponding terms in the expressions for reaction rates. Those non-equilibrium reaction rates are functions of concentrations of all species presented in the mixture and of the whole set of equilibrium rate constants. This leads to the necessity of developing new approaches to obtaining the reaction rates from experimental data. Traditionally defined an equilibrium constant, that is the function only of the thermodynamic state of the system, can be introduced only in a spatially homogeneous case. It can partially simplify the problem of getting information on reaction rates. In general no such value can be introduced. Final expressions show strong correlations between parallel reactions. One such reaction can affect another one until it vanishes. This can be caused not only by the “scalar” non-equilibrium effects, as it was shown in Kolesnichenko and Gorbachev (2011) [2], but also by spatially inhomogeneous. This means that the derived non-equilibrium terms are not small corrections, but can dramatically change chemical kinetics of the reacting mixture.     

Similarity solution for a cylindrical shock wave in a rotational axisymmetric dusty gas with heat conduction and radiation heat flux

The propagation of shock waves in a rotational axisymmetric dusty gas with heat conduction and radiation heat flux, which has a variable azimuthally fluid velocity together with a variable axial fluid velocity, is investigated. The dusty gas is assumed to be a mixture of non-ideal (or perfect) gas and small solid particles, in which solid particles are continuously distributed. It is assumed that the equilibrium flow-condition is maintained and variable energy input is continuously supplied by the piston (or inner expanding surface). The fluid velocities in the ambient medium are assume to be vary and obey power laws. The density of the ambient medium is assumed to be constant, the heat conduction is express in terms of Fourier’s law and the radiation is considered to be of the diffusion type for an optically thick grey gas model. The thermal conductivity K and the absorption coefficient α_R are assumed to vary with temperature and density. In order to obtain the similarity solutions the angular velocity of the ambient medium is assume to be decreasing as the distance from the axis increases. The effects of the variation of the heat transfer parameter and non-idealness of the gas in the mixture are investigated. The effects of an increase in (i) the mass concentration of solid particles in the mixture and (ii) the ratio of the density of solid particles to the initial density of the gas on the flow variables are also investigated.     

Multicomponent flow modeling

We present multicomponent flow models derived from the kinetic theory of gases and investigate the symmetric hyperbolic-parabolic structure of the resulting system of partial differential equations.We address the Cauchy problem for smooth solutions as well as the existence of deflagration waves,also termed anchored waves.We further discuss related models which have a similar hyperbolic-parabolic structure,notably the SaintVenant system with a temperature equation as well as the equations governing chemical equilibrium flows.We next investigate multicomponent ionized and magnetized flow models with anisotropic transport fluxes which have a different mathematical structure.We finally discuss numerical algorithms specifically devoted to complex chemistry flows,in particular the evaluation of multicomponent transport properties,as well as the impact of multicomponent transport.     

Analysis of the diffraction from periodic chiral structures

Consider a time-harmonic electromagnetic plane wave incident on a biperiodic structure in R³. The periodic structure separates two homogeneous regions. The medium inside the structure is chiral and heterogeneous. In general, wave propagation in the chiral medium is governed by Maxwell's equations together with the Drude-Born-Fedorov (constitutive) equations. In this Note, it is shown that for all but possibly a discrete set of parameters, there is a unique quasiperiodic weak solution to the diffraction problem. Our proof is based on a Hodge decomposition, a compact imbedding result, as well as the Lax-Milgram Lemma.     

A two-person game of timing with random termination

We consider a marksmanship contest in which the first contestant to hit his target wins and the contest is to be terminated at a random timeT with cdfH(t). The model is evidently an extension of the classical discrete fire duel to the timing problem under an uncertain environment. It is shown that the uncertainty on the termination of the contest has influence on the equilibrium strategies and the equilibrium values.     

The thermodynamic functions of an ideal gas mixture in an equilibrium state

We find expressions for the entropy and thermodynamic potentials (chemical, internal energy, and the like) for a one-component ideal gas, and also for a mixture of ideal gases in equilibrium.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 3, 1997, pp. 134–140.     

Approximation and optimization of Darboux type differential inclusions with set-valued boundary conditions

The present paper is devoted to an optimal control problem given by hyperbolic discrete (P _D ) and differential inclusions (P _C ) of generalized Darboux type and ordinary discrete inclusions. The results are extended to non-convex problems. An approach concerning necessary and sufficient conditions for optimality is proposed. In order to formulate sufficient conditions of optimality for problem (P _C ) the approximation method is used. Formulation of these conditions is based on locally adjoint mappings. Moreover for construction of adjoint partial differential inclusions the equivalence theorems of locally adjoint mappings are proved. One example with homogeneous boundary conditions is considered.     

Punctured plane partitions and the q-deformed Knizhnik-Zamolodchikov and Hirota equations

We consider partial sum rules for the homogeneous limit of the solution of the q-deformed Knizhnik-Zamolodchikov equation with reflecting boundaries in the Dyck path representation of the Temperley-Lieb algebra. We show that these partial sums arise in a solution of the discrete Hirota equation, and prove that they are the generating functions of τ²-weighted punctured cyclically symmetric transpose complement plane partitions where τ=−(q+q−1). In the cases of no or minimal punctures, we prove that these generating functions coincide with τ²-enumerations of vertically symmetric alternating sign matrices and modifications thereof.