ANALYSIS AND DATA PROCESSING SYSTEMS

A reliable prediction of characteristics of  liquid and gas flows is important both for theory and practice. There have been no methods of modeling the transfer coefficients of rarefied gases until recently. We have developed a stochastic algorithm for modeling the dynamics of a rarefied gas with the help of which the corresponding coefficients of self-diffusion and viscosity were calculated. In this paper, this algorithm is generalized for modeling binary mixtures of rarefied gases. We consider systems of molecules interacting with one another through the potential of hard spheres. At the initial moment of time, all molecules in some arbitrary order are listed. Then, successively for each molecule, the collision process is realized. The pairs of colliding molecules are selected randomly. The algorithm was tested using the example of calculating the diffusion coefficient. Because the calculation of the Green-Kubo diffusion coefficient requires information only about the molecular velocities, the calculation time can be substantially reduced by eliminating the phase of processing molecular displacements. The diffusion of binary mixtures Kr-Ar, Xe-Kr, Xe-Ar under normal conditions is considered. The diameters of the molecules were determined on the basis of the kinetic theory of rarefied gases. Comparison of modeling data of the diffusion coefficient with the experimental data shows that the proposed algorithm makes it possible to obtain a fairly good accuracy of about 3% with 3200 molecules. However, this accuracy can be easily increased by increasing the number of molecules in the system. The accuracy also increases with the increase in the number of phase trajectories of the system based on which the calculated transport coefficients are averaged.

1. Chapman S., Cowling T.G. The mathematical theory of non-uniform gases. An account of the kinetic theory of viscosity, thermal conduction, and diffusion in gases. Cambridge, Cambridge University Press, 1952. 506 p. (Russ. ed.: Chepmen S., Kauling T.D. Matematicheskaya teoriya neodnorodnykh gazov. Translated from English E.V. Malinovskaya. Moscow, Inostrannaya literatura Publ., 1960. 510 p.).

2. Hirschfelder J.O., Curtiss Ch.F., Bird R.B. Molecular theory of gases and liquids. New York, John Wiley and Sons, London, Chapman and Hall, 1954. 892 p.

3. Ferziger J.H., Kaper H.G. Mathematical theory of transport processes in gases. Amsterdam, North-Holland Publ., 1972. 428 p.

4. Burnett D. The distribution of velocities in a slightly non-uniform gas. Proceedings of the London Mathematical Society, 1935, vol. 39, iss. 1, pp. 385–430.

5. Resibois P., Leener M. de. Classical kinetic theory of fluids. New York, Wiley, 1977 (Russ. ed.: Rezibua P., Lenner M. de. Klassicheskaya kineticheskaya teoriya zhidkostei i gazov. Moscow, Mir Publ., 1980. 424 p.).

6. Zhdanov V.M., Alievskii M.Ya. Protsessy perenosa i relaksatsii v molekulyarnykh gazakh [Transport and relaxation processes in molecular gases]. Moscow, Nauka Publ., 1989. 336 p.

7. Rudyak V.Ya., Lezhnev E.V. Stokhasticheskoe modelirovanie koeffitsientov perenosa plotnykh gazov [Stochastic simulation of transport coefficients of dense gases]. Doklady Akademii nauk vysshei shkoly Rossiiskoi Federatsii – Proceedings of the Russian higher school academy of sciences, 2016, no. 4 (33), pp. 22–32.

8. Rudyak V.Ya., Lezhnev E.V. Stochastic method for modeling of the rarefied gas transport coefficients. Journal of Physics: Conference Series, 2016, vol. 738, p. 012086.

9. Rudyak V.Ya., Lezhnev E.V. Stokhasticheskii metod modelirovaniya koeffitsientov perenosa razrezhennogo gaza [Stochastic simulation of rarefied gas transport coefficients]. Matematicheskoe modelirovanie – Mathematical Models and Computer Simulations, 2017, vol. 29, no. 3, pp. 113–122. (In Russian).

10. Kubo R., Yokota M., Nakajima S. Statistical-mechanical theory of irreversible processes. II. Reaction on thermal disturbances. Journal of the Physical Society of Japan, 1957, vol. 12, no. 11, pp. 1203–1226.

11. Green H.S. Theories of transport in fluids. Journal of Mathematical Physics, 1961, vol. 2, no. 3, pp. 344–348.

12. Zubarev D.N. Neravnovesnaya statisticheskaya termodinamika [Nonequilibrium statistical thermodynamics]. Moscow, Nauka Publ., 1974. 398 p.

13. Rudyak V.Ya. Statisticheskaya aerogidromekhanika gomogennykh i geterogennykh sred. T. 2. Gidromekhanika [Statistical aerohydrodynamics of homogeneous and heterogeneous media. Vol. 2. Hydromechanics]. Novosibirsk, NGASU Publ., 2005. 468 p.

14. Krishna R., Wesselingh J.A. The Maxwell-Stefan approach to mass transfer. Chemical Engineering Science, 1997, vol. 52, no. 6, pp. 861–911.

15. Rudyak V.Ya., Belkin A.A., Ivanov D.A., Egorov V.V. Modelirovanie protsessov perenosa na osnove metoda molekulyarnoi dinamiki. Koeffitsient samodiffuzii [The simulation of transport processes using the method of molecular dynamics. Self-diffusion coefficient]. Teplofizika vysokikh temperature – High Temperature, 2008, vol. 46, no. 1, pp. 30–39. (In Russian).

16. Heijningen R.J.J. van, Harpe J.P., Beenakker J.J.M. Determination of the diffusion coefficients of binary mixtures of the noble gases as a function of temperature and concentration. Physica, 1968, vol. 38, pp. 1–34.

17. Grigor'ev I.S., Meilikhova E.Z. Fizicheskie velichiny: spravochnik [Physical quantities: handbook]. Moscow, Energoatomizdat Publ., 1991. 1234 p.