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.     

On one method of finding the equilibrium state

An algorithm for solving a linear “production-exchange” model is described. The model reduces to a parametric model with strictly concave utility functions. Some properties of the parametric model are studied. Since the excessive demand map satisfies the discovered preference condition, it is possible to apply the second form of Chebyshev centers. For a sufficiently small parameter, the algorithm converges to the equilibrium state of the initial linear model. Bibliography:11 titles. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 80, 1996, pp. 59–67.     

On instability of the equilibrium state of nonholonomic systems

We establish a criterion of instability for the equilibrium state of nonholonomic systems, in which gyroscopic forces may dominate over potential forces. We show that, similarly to the case of holonomic systems, the evident domination of gyroscopic forces over potential ones is not sufficient to ensure the equilibrium stability of nonholonomic systems. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 3, pp. 389–397, March, 1999.     

On the behavior of a slightly rarefied gas mixture over plane boundaries

Summary The behavior of a slightly rarefied gas mixture bounded by plane boundaries is investigated on the basis of the linearized Boltzmann equation ofB-G-K type for gas mixtures under the diffusive boundary condition. A useful result of the present analysis is that the macroscopic equations and the appropriate boundary conditions in terms of slip and jump are obtained together with the Knudsen-layer corrections near the boundaries. This system of equations makes possible the treatment at fluid dynamic level for various problems of gas mixtures with plane geometry which require kinetic theory consideration. As an application of this system, some basic flow problems of a slightly rarefied gas mixture, namely, Couette flow, thermal slip flow and diffusion slip flow between two plates are taken up. The total velocity distributions of these concrete problems are explicitly obtained for the first time, and their dependence on the properties and concentration of the component gases in the mixture are clarified in some detail.

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Zusammenfassung Das Verhalten einer verdünnten Gasmischung, bei kleiner aber nicht vernachlässigbarer Knudsen Zahl, zwischen zwei parallelen Platten wird analytisch untersucht. Die linearisierten Boltzmann Gleichungen desB-G-K Typs für Gasmischungen mit diffusiven Randbedingungen werden angewendet. Aus der vorliegenden Untersuchung resultieren brauchbare makroskopische Gleichungen mit den zugehörigen Randbedingungen, — mit Gleitgeschwindigkeit und Temperatursprung formuliert, — sowie die Korrekturen für die wandnahe Knudsen-Schicht. Verschiedene Strömungen von Gasmischungen, bei denen gas-kinetische Effekte eine Rolle spielen, entlang ebener Begrenzungen können mit den Gleichungssystemen auf strömungsmechanischem Niveau behandelt werden. Das System wird auf die Couette Strömung und die thermal-slip und diffusion-slip Strömungen angewendet. Zum ersten Mal werden die Geschwindigkeitsprofile dieser elementaren Strömungen explizit berechnet. Der Einfluß der Gaseigenschaften und Konzentrationen auf diese Profile werden weitgehend erklärt.

     

On the stressed state of concrete at low temperatures

We give the results of experimental study of the influence of internal stresses that arise in concrete under cyclic freezing on its frost-resistance. On this basis we exhibit the causes of destruction of concrete tested for frost-resistance in a working temperature range.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 27, 1988, pp. 81–83.     

Stability of the equilibrium state of the equation system of a viscous barotropic gas in the model of atmosphere

Abstract We consider the system of equations of viscous gas motion whose pressure is related to the density by the law with 1<γ <2, in a domain defined by two levels of geopotential. Under the force due to geopotential and the Coriolis force, we prove the stability of the equilibrium state in a suitable Sobolev space. Keywords: Viscous barotropic gas, Equilibrium state, Coriolis force Mathematics Subject Classification (2000): 35Q35, 76N15     

Isoperimetric inequality and equilibrium states of a two-phase medium

A two-sided estimate for the volume of each phase of a two-phase equilibrium state is derived provided that the equilibrium displacement field of the one-phase problem is smooth. The obtained estimate is uniform with respect to the temperature and surface tension coefficient. Bibliography: 8 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 36, 2007, pp. 81–88.     

A thermodynamic framework for a mixture of two liquids

In this study, we extend a thermodynamic framework that has been used with some success for describing the response of a variety of single constituent continua. Using the thermodynamic framework, we obtain a model for the mixture of two compressible fluids that has a much simpler structure than the model obtained earlier within the context of mixture theory. We also investigate the response of a mixture of two fluids that is constrained to have a constant volume, using the same thermodynamic framework.     

一个核为零齐次的Hilbert型不等式

通过引入权系数,应用实分析的方法,建立一个具有最佳常数因子的零齐次核的Hilbert型不等式,同时还考虑了其等价形式及逆向形式.     

On a generalization of the entropy inequality

In the present paper, the classical entropy inequality is generalized. To this end, a sharp integral inequality is proved.     

On a generalization of the bessel inequality

Translated from Matematicheskie Zametki, Vol. 51, No. 5, pp. 149–150, May, 1992.     

On the thermal-creep flow of a binary gas mixture over a plane wall with imperfect accommodation

The thermal-creep flow of a binary gas mixture over a plane wall is investigated analytically on the basis of the linearized Boltzmann equation of BGK type under the boundary condition of Maxwell's type or diffuse-specular reflection type. By an accurate analysis of the Knudsen layer formed near the wall, the Knudsen-layer structure of the velocity field has been clarified and, hence, the velocity distribution over the whole flow region is given explicitly together with the macroscopic slip coefficient. For future comparison with experimental data which may become available, the values of the slip coefficient of thermal-creep flow for several pairs of gases, Ne-Ar, He-Ne, He-Ar, N₂-Ar and N₂-O₂ are also given and listed in Table together with the values calculated based on the result given by other authors.     

Poiseuille flow of a rarefied gas mixture

The flow of a rarefied binary gas mixture is governed by transport equations which correspond to a coupled set of Fredholm integral equations of the second kind.By means of the Fourier transform technique and by expanding the transformed kernels in terms or orthogonal functions the problem is reduced to the solution of a system of linear algebraic equations.As a significant application, the case of a plane Poiseuille flow of a binary mixture following Sirovich-Hamel's model is considered.The convergence of the method is excellent.

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Riassunto Il flusso di una miscela binaria di gas rarefatti è retto da equazioni del trasporto che corrispondono ad un sistema di equazioni integrali di Fredholm di seconda specie. Mediante la tecnica della trasformata di Fourier e sviluppando i kernel trasformati in termini di funzioni ortogonali, il problema si riduce alla soluzione di un sistema di equazioni algebriche lineari. Quale applicazione significative si considera il flusso piano di Poiseuille di una miscela binaria che segue il modello di Sirovich-Hamel. La convergenza del metodo e'eccellente.

     

Boundary slip phenomena in a binary gas mixture

The slip phenomena in gas mixtures are of fundamental significance in the specification of boundary conditions for flows in the slip regime. In a recent paper, new explicit results for the slip coefficients appropriate to binary gas mixtures were reported. The present work being reported extends the previous work to a higher level of accuracy by involving a higher order Chapman-Enskog expansion. In particular, new expressions for the slip coefficients are presented which are applicable for arbitrary models of the intermolecular interaction. Limiting expressions for the slip coefficients are given (for a simple gas) and the accuracy of the theory is discussed. Numerical calculations of the slip coefficients for different binary gas mixtures using the first and second order Chapman-Enskog approximations and the rigid sphere and Lennard-Jones (12-6) potential models have been carried out. The thermal creep and diffusion slip coefficients are found to be sensitive to the order of the approximation and to the potential model used. A comparison of the new higher order results with some of our previously obtained experimental data for the thermal transpiration effect has also been carried out and shows excellent agreement between the theory and the experiments which confirms the accuracy of the theory.     

On an approach to the description of the local thermodynamic state of solid bodies

We introduce extensive thermodynamic state parameters corresponding to the process of deformation of a solid body. We write the basic thermodynamic relations for deformable systems in terms of these parameters. By passing to the specific extensive quantities and their densities we obtain the corresponding local thermodynamic relations.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 57–62.