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Macroscopic description of arbitrary Knudsen number flow using Boltzmann–BGK kinetic theory. Part 2

Published online by Cambridge University Press:  16 June 2010

HUDONG CHEN ,
STEVEN A. ORSZAG  and
ILYA STAROSELSKY
HUDONG CHEN
Affiliation:
Exa Corporation, 55 Network Drive, Burlington, MA 01803, USA
STEVEN A. ORSZAG*
Affiliation:
Exa Corporation, 55 Network Drive, Burlington, MA 01803, USA Department of Mathematics, P.O. Box 208283, Yale University, New Haven, CT 06520-8283, USA
ILYA STAROSELSKY
Affiliation:
Exa Corporation, 55 Network Drive, Burlington, MA 01803, USA
*
†Email address for correspondence: [email protected]
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Abstract

We extend our previous analysis of closed-form equations for finite Knudsen number flow and scalar transport that result from the Boltzmann–Bhatnagar–Gross–Krook (BGK) kinetic theory with constant relaxation time. Without approximation, we obtain closed-form equations for arbitrary spatial dimension and flow directionality which are local differential equations in space and integral equations in time. These equations are further simplified for incompressible flow and scalars. The particular case of no-flow scalar transport admits analytical solutions that exhibit ballistic behaviour at short times while behaving diffusively at long times. It is noteworthy that, even with constant relaxation time BGK microphysics, quite complex macroscopic descriptions result that would be difficult to obtain using classical constitutive models or continuum averaging.

JFM classification

Mathematical Foundations: General fluid mechanics Micro-/Nano-fluid dynamics: Non-continuum effects Non-Newtonian Flows: Non-Newtonian Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 658 , 10 September 2010 , pp. 294 - 309
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Bender, C. M. & Orszag, S. A. 1999 Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory. SpringerGoogle Scholar
Bhatnagar, P. L., Gross, E. & Krook, M. 1954 A model for collision processes in gases. Phys. Rev. 94, 511.Google Scholar
Cercignani, C. 1969 Mathematical Methods in Kinetic Theory. Elsevier.CrossRefGoogle Scholar
Cercignani, C. 1975 Theory and Application of the Boltzmann Equation. Elsevier.Google Scholar
Chen, H., Kandasamy, S., Orszag, S., Shock, R., Succi, S. & Yakhot, V. 2003 Extended Boltzmann kinetic equation for turbulent flows. Science 301, 633.CrossRefGoogle ScholarPubMed
Chen, H., Orszag, S. & Staroselsky, I. 2007 Macroscopic description of arbitrary Knudsen number flow using Boltzmann–BGK kinetic theory. J. Fluid Mech. 574, 495.Google Scholar
Chen, H., Orszag, S., Staroselsky, I. & Succi, S. 2004 Expanded analogy between Boltzmann kinetic theory of fluids and turbulence. J. Fluid Mech. 519, 301.CrossRefGoogle Scholar
Colosqui, C., Chen, H., Shan, X., Staroselsky, I. & Yakhot, V. 2009 Propagating high-frequency shear waves in simple fluids. Phys. Fluids 21, 013105.Google Scholar
Karabacak, D., Yakhot, V. & Ekinci, K. 2007 High-frequency nanofluidics: an experimental study using nanomechanical resonators. Phys. Rev. Lett. 98, 254505.Google Scholar
Reif, F. 1985 Fundamentals of Statistical and Thermal Physics. McGraw-Hill.Google Scholar
Shen, S. F. 1963 A general transfer-equation approach for the transition regime of rarefied-gas flows and some of its applications. In Rarefied Gas Dynamics (ed. Laurmann, J. A.), vol. II, p. 112. Academic.Google Scholar
Struchtrup, H. & Torillon, M. 2007 H Theorem, regularization and boundary conditions for linearized 13 moment equations. Phys. Rev. Lett. 99, 014502.CrossRefGoogle ScholarPubMed
Toschi, F. & Succi, S. 2005 Lattice Boltzmann method at finite Knudsen numbers. Europhys. Lett. 69, 549.Google Scholar
Xu, K., Martinelli, L. & Jameson, A. 1994 Gas-kinetic finite volume methods, flux-vector splitting, and artificial diffusion. J. Comput. Phys. 120, 48.CrossRefGoogle Scholar
Yakhot, V. & Colosqui, C. 2007 Stokes’ second flow problem in a high frequency limit: application to nanomechanical resonators. J. Fluid Mech. 586, 249.Google Scholar
Zhou, Y., Zhang, R., Staroselsky, I., Chen, H., Kim, W. T. & Jhon, M. S. 2006 Simulation of micro- and nano-scale flows via the lattice Boltzmann method. Physica A 362, 68.Google Scholar