Abstract
Molecular hydrogen is modeled by numerically solving the Wang Chang–Uhlenbeck equation. The differential scattering cross sections of molecules are calculated using the quantum mechanical scattering theory of rigid rotors. The collision integral is computed by applying a fully conservative projection method. Numerical results for relaxation, heat conduction, and a one-dimensional shock wave are presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
C. S. Wang Chang and G. E. Uhlenbeck, “Transport phenomena in polyatomic gases,” Research Report No. CM-681 (Univ. of Michigan, 1951).
J. H. Ferziger and H. G. Kaper, Mathematical Theory of Transport Processes in Gases (North-Holland, Amsterdam, 1972; Mir, Moscow, 1976).
R. F. Snider, “Quantum-mechanical modified Boltzmann equation for degenerate internal states,” J. Chem. Phys. 32 (4), 1051–1060 (1960).
M. W. Thomas and R. F. Snider, “Boltzmann equation and angular momentum conservation,” J. Stat. Phys. 2 (1), 61–81 (1970).
K. Koura, “Monte Carlo direct simulation of rotational relaxation of diatomic molecules using classical trajectory calculations: Nitrogen shock wave,” Phys. Fluids 9 (11), 3543–3549 (1997).
F. G. Tcheremissine, “Method for solving the Boltzmann kinetic equation for polyatomic gases,” Comput. Math. Math. Phys. 52 (2), 252–268 (2012).
Yu. A. Anikin and O. I. Dodulad, “Solution of a kinetic equation for diatomic gas with the use of differential scattering cross sections computed by the method of classical trajectories,” Comput. Math. Math. Phys. 53 (7), 1026–1043 (2013).
K. Takayanagi, “The production of rotational and vibrational transitions in encounters between molecules,” Adv. At. Mol. Phys. 1, 149–194 (1965).
D. K. Veirs and G. M. Rosenblatt, “Raman line positions in molecular hydrogen: H2, HD, HT, D2, DT, and T2,” J. Mol. Spectrosc. 121 (2), 401–419 (1987).
S. Green, “Rotational excitation in H2–H2 collisions: Close-coupling calculations,” J. Chem. Phys. 62 (6), 2271–2277 (1975).
P. Diep and J. K. Johnson, “An accurate H2–H2 interaction potential from first principles, J. Chem. Phys. 112 (10), 4465–4473 (2000); Erratum: J. Chem. Phys. 113 (8), 3480–3481 (2000).
B. Maté, F. Thibault, G. Tejeda, J. M. Fernández, and S. Montero, “Inelastic collisions in para-H2: Translation- rotation state-to-state rate coefficients and cross sections at low temperature and energy,” J. Chem. Phys. 122 (6), 064313 (2005).
B. R. Johnson, “The multichannel log-derivative method for scattering calculations,” J. Comput. Phys. 13 (3), 445–449 (1973).
Yu. Ya. Milenko, R. M. Sibileva, and M. A. Strzhemechny, “Natural ortho-para conversion rate in liquid and gaseous hydrogen,” J. Low Temp. Phys. 107 (1), 77–92 (1997).
J. M. Blatt and L. C. Biedenharn, “The angular distribution of scattering and reaction cross sections,” Rev. Mod. Phys. 24 (4), 258–272 (1952).
J. Schaefer, “Transport coefficients of dilute hydrogen gas, calculations and comparisons with experiments,” Chem. Phys. 368 (1–2), 38–48 (2010).
J. M. Hutson and S. Green, MOLSCAT version 14, 1994. Collaborative Comput. Project no. 6 of the UK Sci. Eng. Research Council. http://www.giss.nasa.gov/tools/molscat/
Yu. A. Anikin, “On the accuracy of the projection computation of the collision integral,” Comput. Math. Math. Phys. 52 (4), 615–636 (2012).
M. J. Assael, S. Mixafendi, and W. A. Wakeham, “The viscosity and thermal conductivity of normal hydrogen in the limit of zero density,” J. Phys. Chem. Ref. Data 15 (4), 1315–1322 (1986).
J. W. Leachman, R. T. Jacobsen, S. G. Penoncello, and M. L. Huber, “Current status of transport properties of hydrogen,” Int. J. Thermophys. 28 (3), 773–795 (2007).
R. M. Jonkman, G. J. Prangsma, I. Ertas, H. F. P. Knaap, and J. J. M. Beenakker, “Rotational relaxation in mixtures of hydrogen isotopes and noble gases,” Physica A 38 (3), 451–455 (1968).
C. G. Sluijter, H. F. P. Knaap, and J. J. M. Beenakker, “Determination of rotational relaxation times of hydrogen isotopes by sound absorption measurement at low temperatures,” Physica A 30 (4), 745–762 (1964).
P. W. Huber and A. Kantrowitz, “Heat-capacity lag measurements in various gases,” J. Chem. Phys. 15 (5), 275–284 (1957).
R. J. Gallagher and J. B. Fenn, “Rotational relaxation of molecular hydrogen,” J. Chem. Phys. 60 (9), 3492–3498 (1974).
T. G. Winter and G. L. Hill, “High-temperature ultrasonic measurements rotational relaxation in hydrogen, deuterium, nitrogen and oxygen,” J. Acoust. Soc. Am. 42 (4), 848–858 (1967).
Yu. A. Anikin, “Numerical study of radiometric forces via the direct solution of the Boltzmann kinetic equation,” Comput. Math. Math. Phys. 51 (7), 1251–1266 (2011).
M. J. Assael, J.-A. M. Assael, M. L. Huber, R. A. Perkins, and Y. Takata, “Correlation of the thermal conductivity of normal and parahydrogen from the triple point to 1000 K and up to 100 MPa,” J. Phys. Chem. Ref. Data 40 (3), 033101 (2011).
E. F. Greene and D. F. Hornig, “The shape and thickness of shock fronts in argon, hydrogen, nitrogen, and oxygen,” J. Chem. Phys. 21 (4), 617–624 (1953).
L. D. Landau and E. M. Lifshitz, Statistical Physics (Nauka, Moscow, 1995; Butterworth-Heinemann, Oxford, 1980), Part 1.
S. Takata and H. Umetsu, “Numerical study on effective configurations of the Knudsen pump for separation and compression,” AIP Conf. Proc. 1333 (1), 998–1003 (2011).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © Yu.A. Anikin, 2017, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2017, Vol. 57, No. 6, pp. 1061–1079.
Rights and permissions
About this article
Cite this article
Anikin, Y.A. Solution of the Wang Chang–Uhlenbeck equation for molecular hydrogen. Comput. Math. and Math. Phys. 57, 1048–1065 (2017). https://doi.org/10.1134/S0965542517060033
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0965542517060033