Abstract
An analysis is made of the effects on the diffusion of Brownian particles whose Knudsen number is large compared to unity, of nonuniformities in the host gas. As examples, in one type of nonuniformity of the host gas, the Chapman-Enskog velocity distribution function for the gas molecules is used; in the other, the host gas is a free-molecule Couette flow. In both cases, a new force on the Brownian particles appears. Two techniques are used (extending Kramers' method and utilizing the Chapman-Enskog method) to transform the new Fokker-Planck equation into generalized Smoluchowski and convective diffusion equations. In these equations, the diffusion coefficient appears as a second-order tensor. Thus, it is demonstrated that Brownian diffusion in a nonuniform gas is anisotropic.
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References
A. Einstein,Brownian Diffusion, R. Furth, ed. (Dover Publications, New York, 1956).
M. V. Smoluchowski,Ann. Physik 21:756 (1960).
P. Langevin,Compt. Rend. 146:530 (1908).
G. E. Uhlenbeck and L. S. Ornstein,Phys. Rev. 36:823 (1930); reprinted inSelected Papers on Noise and Stochastic Processes, N. Wax, ed. (Dover Publications, New York, 1954).
M. C. Wang and G. E. Uhlenbeck,Rev. Mod. Phys. 17:323 (1945); reprinted in N. Wax,loc. cit.
S. Chandrasekhar,Rev. Mod. Phys. 15:1 (1943), reprinted in N. Wax,loc. cit.
S. F. Shen and W. G. N. Slinn,Bull. Am. Phys. Soc. 13:1593 (1968).
W. G. N. Slinn, Ph.D. dissertation, Cornell University, Ithaca, New York, and Battelle-North-west, BNWL-987, Richland, Washington, 1969.
W. G. N. Slinn and S. F. Shen,J. Geophys. Res. 75:2267 (1970).
L. Waldmann and K. H. Schmitt, inAerosol Science, C. N. Davies, ed. (Academic Press, New York, 1966).
S. Bell and S. A. Schaaf,J. Am. Rocket Soc. 23:314 (1953).
R. M. Mazo,J. Stat. Phys. 1:101 (1969).
D. N. Zubarev and A. G. Bashkirov,Physica 39:334 (1968).
A. A. Markoff (Markov),Wahrscheinlichkeitrechnung (Leipzig, 1912).
A. I. Khinchin,Mathematical Foundations of Statistical Mechanics (Dover Publications, New York, 1949).
W. G. N. Slinn, S. F. Shen, and R. M. Mazo,J. Stat. Phys. 2:251 (1970).
P. S. Epstein,Phys. Rev. 23:710 (1924).
M. S. Green,J. Chem. Phys. 19:1036 (1951).
W. G. Vincenti and C. H. Kruger,Physical Gas Dynamics (John Wiley and Sons, New York, 1965).
S. Chandrasekhar, private communication.
H. A. Kramers,Physica 7:284 (1940).
R. M. Mazo, private communication.
J. L. Lebowitz, H. L. Frisch, and E. Helfand,Phys. Fluids 3:325 (1960).
P. M. V. Resibois,Electrolyte Theory (Harper and Row, New York, 1968).
W. P. Allis,Motion of Ions and Electrons, T. R. 299, Massachusetts Institute of Technology, Cambridge, Massachusetts, 1956.
N. A. Fuchs,The Mechanics of Aerosols (Pergamon Press, New York, 1964).
I. Prigogine,Introduction to Thermodynamics of Irreversible Processes (Charles C. Thomas, Springfield, Illinois, 1955).
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The work of Slinn was financially supported in part by Battelle Memorial Institute and in part by U.S. Atomic Energy Commission Contract AT(45-1)-1830. The work of Shen was supported in part by U.S. Air Force Office of Scientific Research Contract 49(638)-1346.
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Slinn, W.G.N., Shen, S.F. Brownian diffusion in a nonuniform gas. J Stat Phys 3, 291–305 (1971). https://doi.org/10.1007/BF01011385
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DOI: https://doi.org/10.1007/BF01011385