Non-equilibrium phenomena in gas mixture flow through a plane channel in near-continuum regime. (English) Zbl 07464404
Summary: Non-equilibrium phenomena in the flow of a binary gas mixture between two parallel plates, driven by gradients of the pressure, temperature and concentration, are studied. The phenomenological relationships that describe the energy and mass transfer in the flow of a gas mixture through a channel are analyzed by using methods of the non-equilibrium thermodynamics of discontinuous systems. The solution of the linearized moment equations in the 13-moments approximation of the Grad’s method and the relation for the slip velocity previously found in Zhdanov (Phys. Rev. E95 (2017), 033106) are applied to deduce the expression for the mass-average velocity of the mixture flowing in a plane channel in the near-continuum regime. Expressions for the diffusion and heat fluxes are determined from the initial third-order moment equations for the gas mixture, averaged over the channel cross-section, with the subsequent evaluation of several moments of the distribution function at the channel wall by using the modified Maxwell method. It is shown that the employed approach automatically ensures the validity of the reciprocal relations for the cross terms in the Onsager matrix. Analytical formulas for kinetic coefficients of the Onsager matrix, expressed in terms of known transport coefficients of the gas mixture and slip coefficients at the channel wall, are obtained. The results of calculations of the matrix coefficients for several binary mixtures of noble gases (Ne-Ar and He-Xe) and their comparison with the results of the numerical calculations on the basis of the linearized Boltzmann equation with the McCormack model are presented.
MSC:
| 82-XX | Statistical mechanics, structure of matter |
Keywords:
gas mixture; channel flow; non-equilibrium phenomena; near-continuum flow regime; moment equations; transport and slip coefficientsReferences:
| [1] | Kogan, M. N., Rarefied Gas Dynamics (1969), Plenum: Plenum New York |
| [2] | Cercignani, C., The Boltzmann Equation and Its Application (1988), Springer: Springer New York · Zbl 0646.76001 |
| [3] | Sharipov, F.; Seleznev, V., Data on internal rarefied gas, J. Phys. Chem. Ref. Data, 27, 3, 657-706 (1998) |
| [4] | (Jousten, K., Handbook of Vacuum Technology (2008), Wiley-VCN: Wiley-VCN Weinheim) |
| [5] | Ho, C. M.; Tai, Y. C., Micro-electro-mechanical systems (MEMS) and fluid flows, Annu. Rev. Fluid Mech., 30, 579-612 (1998) |
| [6] | Kandlikar, S. G.; Garimella, S., Heat Transfer and Fluid Flow in Minichannels and Microchannels (2006), Elsevier: Elsevier Oxford |
| [7] | de Groot, S. R.; Mazur, P., Non-Equilibrium Thermodynamics (1962), North-Holland Publ. Comp.: North-Holland Publ. Comp. Amsterdam · Zbl 1375.82003 |
| [8] | Ferziger, J. H.; Kaper, H. G., Mathematical Theory of Transport Processes in Gases (1972), North-Holland: North-Holland Amsterdam |
| [9] | McCormack, F. J., Construction of linearized kinetic models for gaseous mixture and molecular gases, Phys. Fluids, 16, 2095-2105 (1973) · Zbl 0274.76054 |
| [10] | Hamel, B. B., Kinetic model for binary gas mixtures, Phys. Fluids, 8, 418-425 (1965) |
| [11] | Chapman, S.; Cowling, T. G., The Mathematical Theory of Non-Uniform Gases (1970), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0098.39702 |
| [12] | Grad, H., On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2, 325-407 (1949) · Zbl 0037.13104 |
| [13] | Grad, H., Principles of the kinetic theory of gases, (Flugge, S., Handbuch der Physik, Vol. 12 (1958), Springer: Springer Berlin) · Zbl 0101.43704 |
| [14] | Zhdanov, V.; Kagan, Yu.; Sazykin, A., Effect of viscous transfer of momentum on diffusion in a gas mixture, Sov. Phys.—JETP, 15, 596-604 (1962) |
| [15] | Zhdanov, V. M., Transport Processes in Multicomponent Plasma (2002), Tailor & Francis: Tailor & Francis London-New York |
| [16] | Chernyak, V. G.; Kalinin, V. V.; Suetin, P. E., The kinetic phenomena in non-isothermal motion of a binary gas mixture through a plane channel, Int. J. Heat Mass Transfer, 27, 8, 1189-1196 (1984) · Zbl 0546.76122 |
| [17] | Chernyak, V. G., Zh. Prikl. Mekhaniki i Tekhnicheskoi Fiz, 5, 50-56 (1982), (in Russian) |
| [18] | Naris, S.; Valougeorgis, D.; Kalempa, D.; Sharipov, F., Gaseous mixture flow between two parallel plates in the whole range of the gas rarefaction, Physica A, 336, 294-321 (2004) |
| [19] | Sharipov, F.; Kalempa, D., Gaseous mixture flow through a long tube at arbitrary Knudsen number, J. Vac. Sci. Technol. A, 20, 3, 814-822 (2002) |
| [20] | Siewert, C. E.; Valougeorgis, D., The McCormack model: channel flow of a binary gas mixture driven by temperature, pressure and concentration gradients, Eur. J. Mech. B Fluids, 23, 645-664 (2004) · Zbl 1060.76631 |
| [21] | Sharipov, F., Data on the velocity slip and temperature jump on a gas-solid interface, J. Phys. Chem. Ref. Data, 40, 3, Article 023101 pp. (2011) |
| [22] | Zhdanov, V. M., Slip and barodiffusion phenomena in slow flows of a gas mixture, Phys. Rev. E, 95, Article 033106 pp. (2017) |
| [23] | Loyalka, S. K., Velocity slip coefficient and the diffusion slip velocity for a multicomponent gas mixture, Phys. Fluids, 14, 2599-2604 (1971) · Zbl 0247.76070 |
| [24] | Loyalka, S. K., Approximate method in the kinetic theory, Phys. Fluids, 14, 2291-2294 (1971) · Zbl 0247.76069 |
| [25] | Ivchenko, I. N.; Loyalka, S. K.; Tompson, R. V., Slip coefficients for binary gas mixture, J. Vac. Sci. Technol. A, 15, 4, 2375-2381 (1997) |
| [26] | Ivchenko, I. N.; Loyalka, S. K.; Tompson, R. V., Boundary slip phenomena in a binary gas mixture, Z. Angew. Math. Phys., 53, 58-72 (2002) · Zbl 1041.76063 |
| [27] | Lang, H., Second order slip effect in poiseuille flow, Phys. Fluids, 19, 366-3710 (1976) · Zbl 0339.76047 |
| [28] | Loyalka, S. K.; Hickey, K. A., Plane Poiseuille flow near continuum results for a rigid sphere gas, Physica A, 160, 395-408 (1989) |
| [29] | Zhdanov, V. M.; Roldughin, V. I., Moment method and rarefied gas flow in channels. General relations, Physica A, 184, 169-186 (1992) |
| [30] | Zhdanov, V. M.; Roldughin, V. I., Moment method and rarefied gas flow in channels. II. Diffusion in a multicomponent gas mixture, Physica A, 199, 291-298 (1993) |
| [31] | Zhdanov, V. M.; Roldughin, V. I., On the theory of nonequilibrium phenomena in the flow of a gas mixture in a capillary, Sov. Phys.—JETP, 78, 170-178 (1994) |
| [32] | Stepanenko, A. A.; Zaznoba, V. A.; Zhdanov, V. M., Boundary slip phenomena in multicomponent gas mixtures, Phys. Fluids, 31, Article 062105 pp. (2019) |
| [33] | Kolodner, I. I., Moment Description of Gas Mixture-IReport (1957), New York University |
| [34] | Burgers, J. M., Flow Equations for Composite Gases (1969), Academic: Academic New York · Zbl 0214.25207 |
| [35] | Seleznev, V. D.; Tokmantsev, V. I., Nonequilibrium Statistical Thermodynamics of Rarefied Gases (2005), UB RAS: UB RAS Ekaterinburg, (in Russian) |
| [36] | Bakanov, S. P.; Roldughin, V. I., Boundary value problems of the kinetic theory of gases and irreversible thermodynamics, Appl. Math. Mech., 41, 665-673 (1977) · Zbl 0431.76061 |
| [37] | Roldughin, V. I.; Zhdanov, V. M., Nonequilibrium thermodynamics and kinetic theory of gas mixtures in the presence of interfaces, Adv. Colloid Interface Sci., 98, 121-215 (2002) |
| [38] | Müller, I.; Ruggeri, T., Rational Extended Thermodynamics (1998), Springer · Zbl 0895.00005 |
| [39] | Loyalka, S. K., Kinetic theory of thermal transpiration and mechanocaloric effect. I, J. Chem. Phys., 55, 4497-4503 (1971) |
| [40] | Ten Bosch, B. L.M.; Beenakker, J. J.M.; Kuscer, I., Onsager symmetries in field-depended flows of rarefied molecular gases, Physica A, 123, 443-462 (1984) · Zbl 0598.76093 |
| [41] | Sharipov, F., Onsager-Casimir reciprocity relations for open gaseous systems at arbitrary rarefaction: II. Application of the theory for single gas, Physica A, 203, 457-485 (1994) |
| [42] | Sharipov, F., Onsager-Casimir reciprocity relations for open gaseous systems at arbitrary rarefaction: III. Theory and its application for gaseous mixtures, Physica A, 209, 457-476 (1994) |
| [43] | Zhdanov, V. M.; Roldughin, V. I., Non-equilibrium thermodynamics and kinetic theory of rarefied gases, Phys.-Usp., 41, 4, 349-378 (1998) |
| [44] | Ruggeri, T.; Sugiyama, M., Rational Extended Thermodynamics beyond the Monatomic Gas (2015), Springer · Zbl 1330.76003 |
| [45] | Goebel, C. J.; Harris, S. M.; Johnson, E. A., Two-temperature disparate-mass gas mixture. A thirteen moment description, Phys. Fluids, 19, 627-635 (1976) · Zbl 0353.76051 |
| [46] | Kestin, J.; Knierim, K.; Mason, E. A.; Naja, B.; Ro, S. T.; Waldman, M., Equilibrium and transport properties of the noble gases and their mixture at low densities, J. Phys. Chem. Ref. Data, 13, 1, 229-303 (1984) |
| [47] | Landau, L. D.; Lifshitz, E. M., Fluid Mechanics (Course of Theoretical Physics) (1980), Pergamon Press: Pergamon Press Oxford |
| [48] | Zhdanov, V. M.; Zaznoba, V. A., Nonisothermical flow of gas mixture in a channel at intermediate Knudsen numbers, J. Appl. Math. Mech., 45, 801-808 (1981) · Zbl 0543.76132 |
| [49] | Zhdanov, V. M., Baro- and thermal diffusion of gas mixture in a capillary, J. Appl. Mech. Tech. Phys., 23, 201-204 (1982) |
| [50] | Zhdanov, V. M.; Zaznoba, V. A., Gas mixture flow in a cylindrical channel at intermediate Knudsen numbers, J. Eng. Phys. Thermophys., 45, 998-1002 (1983) |
| [51] | Loyalka, S. K., The slip problems for a simple gas, Z. Naturforschg., 26a, 964-972 (1971) |
| [52] | Loyalka, S. K., Slip in the thermal creep flow, Phys. Fluids, 14, 21-24 (1971) |
| [53] | Waldmann, L., Z. Naturforschg., 22a, 1269-1280 (1967) |
| [54] | Zhdanov, V. M., Barodiffusion in slow flows of a gas mixture, Tech. Phys., 64, 5, 596-605 (2019) |
| [55] | Breton, J. P., The diffusion equation in discontinuous system, Physica A, 50, 365-379 (1970) |
| [56] | Sharipov, F.; Kalempa, D., Velocity slip and temperature jump coefficients for gaseous mixtures. I. Viscous slip coefficient, Phys. Fluids, 15, 1800-1806 (2003) · Zbl 1186.76469 |
| [57] | Sharipov, F.; Kalempa, D., Velocity slip and temperature jump coefficients for gaseous mixtures. II. Thermal slip coefficient, Phys. Fluids, 16, 759-764 (2004) · Zbl 1186.76470 |
| [58] | Sharipov, F.; Kalempa, D., Velocity slip and temperature jump coefficients for gaseous mixtures. III. Diffusion slip coefficient, Phys. Fluids, 16, 3779-3785 (2004) · Zbl 1187.76478 |
| [59] | Szalmas, L., DSMC simulation of binary gas flows between parallel plates and comparision to the other methods, AIP Conf. Proc., 1333, 348-353 (2011) |
| [60] | Gupta, V. K.; Struchtrup, H.; Torrilhon, M., Regularized moment equations for binary gas mixtures: derivation and linear analysis, Phys. Fluids, 28, Article 042003 pp. (2016) |
| [61] | Gupta, V. K.; Torrilhon, M., Higher order moment equations for rarefied gas mixtures, Proc. R. Soc. A, 471, Article 20140754 pp. (2015) · Zbl 1371.82088 |
| [62] | Gu, X.-J.; Emerson, D. R.; Tang, G.-H., Analysis of the slip coefficient and defect velocity in the knudsen layer of a rarefied gas using the linearized moment equations, Phys. Rev. E, 81, Article 016313 pp. (2010) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
