Hostname: page-component-7479d7b7d-767nl Total loading time: 0 Render date: 2024-07-11T05:22:43.207Z Has data issue: false hasContentIssue false
  • English
  • Français

Semiclassical Lattice Boltzmann Simulations of Rarefied Circular Pipe Flows

Published online by Cambridge University Press:  20 August 2015

Jaw-Yen Yang ,
Li-Hsin Hung  and
Yao-Tien Kuo
Jaw-Yen Yang*
Affiliation:
Center for Quantum Science and Engineering, National Taiwan University, Taipei 106, Taiwan Institute of Applied Mechanics, National Taiwan University, Taipei 106, Taiwan
Li-Hsin Hung*
Affiliation:
Institute of Applied Mechanics, National Taiwan University, Taipei 106, Taiwan
Yao-Tien Kuo*
Affiliation:
Institute of Applied Mechanics, National Taiwan University, Taipei 106, Taiwan
*
Corresponding author.Email:yangjy@spring.iam.ntu.edu.tw
Email:D95543014@ntu.edu.tw
Email:t24621@yahoo.com.tw
Get access

Abstract

Computations of microscopic circular pipe flow in a rarefied quantum gas are presented using a semiclassical axisymmetric lattice Boltzmann method. The method is first derived by directly projecting the Uehling-Uhlenbeck Boltzmann-BGK equations in two-dimensional rectangular coordinates onto the tensor Hermite polynomials using moment expansion method and then the forcing strategy of Halliday et al. [Phys. Rev. E., 64 (2001), 011208] is adopted by adding forcing terms into the resulting microdynamic evolution equation. The determination of the forcing terms is dictated by yielding the emergent macroscopic equations toward a particular target form. The correct macroscopic equations of the incompressible axisymmetric viscous flows are recovered through the Chapman-Enskog expansion. The velocity profiles and the mass flow rates of pipe flows with several Knudsen numbers covering different flow regimes are presented. It is found the Knudsen minimum can be captured in all three statistics studied. The results also indicate distinct characteristics of the effects of quantum statistics.

Keywords

76P0582B4082D0582C4047.11.-j51.10.+y47.45.Ab67.10.JnSemiclassical lattice Boltzmann methodaxisymmetric flowsrarefied pipe flowKnudsen minimum
Type
Research Article
Information
Communications in Computational Physics , Volume 10 , Issue 2 , August 2011 , pp. 405 - 421
Copyright
Copyright © Global Science Press Limited 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] McNamara, G. and Zanetti, G., Use of the Boltzmann equation to simulate lattice-gas automata, Phys. Rev. Lett., 61 (1988), 2332–2335.CrossRefGoogle ScholarPubMed
[2] Qian, Y. H, D’Humieres, D., and Lallemand, P., Lattice BGK models for Navier-Stokes equation, Europhys. Lett., 17 (1992), 479–484.CrossRefGoogle Scholar
[3] Chen, H., Chen, S., and Matthaeus, W. H., Recovery of the Navier-Stokes equations using a lattice-gas Boltzmann method, Phys. Rev. A., 45 (1992), R5339–R5342.CrossRefGoogle ScholarPubMed
[4] Qian, Y. H., and Orszag, S. A., Lattice BGK models for the Navier-Stokes equation: nonlinear deviation in compressible regimes, Europhys. Lett., 21 (1993), 255–259.CrossRefGoogle Scholar
[5] Bhatnagar, P. L., Gross, E. P., and Krook, M., A model for collision processes in gases I: small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94 (1954), 511–525.CrossRefGoogle Scholar
[6] Rothman, D. H., and Zaleski, S., Lattice-gas models of phase separation: interfaces, phase transitions, and multiphase flow, Rev. Mod. Phys., 66 (1994), 1417–1479.CrossRefGoogle Scholar
[7] He, X., and Luo, L.-S., A priori derivation of the lattice Boltzmann equation, Phys. Rev. E., 55 (1997), R6333.CrossRefGoogle Scholar
[8] Luo, L.-S., Comment on discrete Boltzmann equation for microfluidics, Phys. Rev. Lett., 92 (2004), 139401.CrossRefGoogle ScholarPubMed
[9] Shan, X., and He, X., Discretization of the velocity space in the solution of the Boltzmann equation, Phys. Rev. Lett., 80 (1998), 65.CrossRefGoogle Scholar
[10] Shan, X., Yuan, X.-F., and Chen, H., Kinetic theory representation of hydrodynamics: a way beyond the Navier-Stokes equation, J. Fluid. Mech., 550 (2006), 413.CrossRefGoogle Scholar
[11] Grad, H., On the kinetic theory of rarefied gases, Commun. Pure. Appl. Math., 2 (1949), 331–407.CrossRefGoogle Scholar
[12] Chen, G., Nanosclae Energy Transport and Conversion, Oxford University Press, 2005.CrossRefGoogle Scholar
[13] Uehling, E. A., and Uhlenbeck, G. E., Transport phenomena in Einstein-Bose and Fermi-Dirac gases I, Phys. Rev., 43 (1933), 552–561.CrossRefGoogle Scholar
[14] Lundstrom, M., Fundamentals of Carrier Transport, Cambridge University Press, 2nd ed., 2000.CrossRefGoogle Scholar
[15] Yang, J. Y., and Hung, L. H., Lattice Uehling-Uhlenbeck Boltzmann-BGK hydrodynamics of quantum gases, Phys. Rev. E., 79 (2009), 056708.CrossRefGoogle Scholar
[16] Halliday, I., Hammond, L. A., Care, C. M., Good, K., and Stevens, A., Lattice Boltzmann equation hydrodynamics, Phys. Rev. E., 64 (2001), 011208.CrossRefGoogle ScholarPubMed
[17] Peng, Y. C., Shu, C., Chew, Y. T., and Qiu, J., Numerical investigation of flows in Czochralski crystal growth by an axisymmetric lattice Boltzmann method, J. Comput. Phys., 186 (2003), 295–307.CrossRefGoogle Scholar
[18] Niu, X. D., Shu, C., and Chew, Y. T., An axisymmetric lattice Boltzmann model for simulation of Taylor-Couette flows between two concentric cylinders, Int. J. Mod. Phys. C., 6 (2003), 785–796.Google Scholar
[19] Premnath, K. N., and Abraham, J., Lattice Boltzmann model for axisymmetric multiphase flows, Phys. Rev. E., 71 (2005), 056706.CrossRefGoogle ScholarPubMed
[20] Lee, T. S., Huang, H., and Shu, C., An axisymmetric incompressible lattice Boltzmann model for pipe flow, Int. J. Mod. Phys. C., 17 (2006), 645–661.CrossRefGoogle Scholar
[21] Reis, T., and Philips, T. N., Modified lattice Boltzmann model for axisymmetric flows, Phys. Rev. E., 75 (2007), 056703.CrossRefGoogle ScholarPubMed
[22] Mukherjee, S., and Abraham, J., Lattice-Boltzmann simulations of two-phase flow with high density ratio in axially symmetric geometry, Phys. Rev. E., 75 (2007), 026701.CrossRefGoogle ScholarPubMed
[23] Reis, T., and Philips, T. N., Erratum: modified lattice Boltzmann model for axisymmetric flows [Phys. Rev. E., 75 (2007), 056703], Phys. Rev. E., 76 (2007), 059902.CrossRefGoogle Scholar
[24] Reis, T., and Philips, T. N., Numerical validation of a consistent axisymmetric lattice Boltz-mann model, Phys. Rev. E., 77 (2008), 026703.CrossRefGoogle ScholarPubMed
[25] Zhou, J. G., Axisymmetric lattice Boltzmann method, Phys. Rev. E., 78 (2008), 036701.CrossRefGoogle ScholarPubMed
[26] Guo, Z. L., Han, H., Shi, B., and Zheng, C., Theory of the lattice Boltzmann equation: lattice Boltzmann model for axisymmetric flows, Phys. Rev. E., 79 (2009), 046708.CrossRefGoogle ScholarPubMed
[27] Knudsen, M., The law of the molecular flow and viscosity of gases moving through tubes, Ann. Phys., 28 (1909), 75–130.Google Scholar
[28] Knudsen, M., The Kinetic Theory of Gases, Methuen & Co. London, 1934.Google Scholar
[29] Shi, Y. H., and Yang, J. Y., J. Comput. Phys., 343 (2008), 552.Google Scholar
[30] Chapman, S., and Cowling, T. G., The Mathematical Theory of Non-Uniform Gases, 3rd ed., Cambridge University Press, 1970.Google Scholar
[31] Rice, M. J., Size-dependent transport-coefficient effects in fermi liquids, Phys. Rev., 165 (1968), 288–292.CrossRefGoogle Scholar
[32] Eu, B. C., Normal-stress effects in tube flow of a non-Newtonian fluid, Phys. Rev. A., 40 (1989), 1497–1506.CrossRefGoogle ScholarPubMed
[33] Parpia, J. M., and Rhodes, T. L., First observation of the Knudsen minimum in normal liquid 3He , Phys. Rev. Lett., 5 (1983), 805–808.Google Scholar
[34] Sawkey, D., and Harrison, J. P., Volume flow in liquid 3He in the Knudsen and Poiseuille regions, Phys. B., 112 (2003), 329–333.Google Scholar
[35] Kardar, M., Statistical Physics of Particles, Cambridge University Press, 2007.CrossRefGoogle Scholar