Abstract
A general procedure to obtain analytic solutions of the linearized lattice Boltzmann equation for simple flows is developed. As examples, the solutions for the Poiseuille and the plane Couette flows in two-dimensional space are obtained and studied in detail. The solutions not only have a component which is the solution of the Navier–Stokes equation, they also include a kinetic component which cannot be obtained by the Navier–Stokes equation. The kinetic component of the solutions is due to the finite-mean-free-path effect. Comparison between the analytic results and the numerical results of lattice-gas simulations is made, and they are found to be in accurate agreement.
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REFERENCES
L.-S. Luo, H. Chen, S. Chen, G. D. Doolen, and Y. C. Lee, Phys. Rev. A 43:7097 (1991).
L.-S. Luo, Ph.D. thesis, Georgia Institute of Technology, 1993.
S. P. Das, H. J. Bussemaker and M. H. Ernst, Phys. Rev. E 48:245 (1993).
P. Grosfils, J. P. Boon, R. Brito and M. H. Ernst, Phys. Rev. E 48:2655 (1993).
O. Behrend, R. Harris and P. B. Warren, Phys. Rev. E 50:4586 (1994).
U. Frisch, B. Hasslacher, and Y. Pomeau, Phys. Rev. Lett. 56:1505 (1986).
S. Wolfram, J. Stat. Phys. 45:471 (1986).
U. Frísch, D. d'Humières, B. Hasslacher, P. Lallemand, Y. Pomeau, and J.-P. Rivet, Complex Systems 1:649 (1987).
G. D. Doolen, editor, Lattice Gas Methods for Partial Differential Equations (Addison-Wesley, Redwood City, CA, 1990).
G. R. McNamara and G. Zanetti, Phys. Rev. Lett. 61:2332 (1988).
P. J. Davis, Circulant Matrices (Wiley, New York, 1979).
L. P. Kadanoff, G. R. McNamara, and G. Zanetti, Complex Systems 1:791 (1987).
W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran (Cambridge, New York, 1992), 2nd Edition, Chapter 12.
K. Huang, Statistical Mechanics (Wiley, New York, 1963).
M. Ernst and T. Naitoh, J. Phys. A 24:2555 (1991).
J. C. Maxwell, Philos. Trans. Roy. Soc. London 170:231 (1879).
D. K. Bhattacharya and G. C. Lie, Phys. Rev. Lett. 62:897 (1989).
G. Mo and F. Rosenberger, Phys. Rev. A 42:4688 (1990).
I. Ginzbourg and P. M. Adler, J. Phys. II France 4:191 (1994).
B. J. Alder and W. E. Alley, Phys. Rev. A 1:18 (1970).
B. J. Alder and W. E. Alley, Phys. Today 37(1):56 (1984).
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Luo, LS. Analytic Solutions of Linearized Lattice Boltzmann Equation for Simple Flows. Journal of Statistical Physics 88, 913–926 (1997). https://doi.org/10.1023/B:JOSS.0000015178.19008.78
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DOI: https://doi.org/10.1023/B:JOSS.0000015178.19008.78