A rarefied gas in a channel of two parallel plane walls with a spatially periodic steady temperature distribution, on which a series of ditches is dug periodically, is considered, and the behavior of a flow induced by the temperature variation on the walls is investigated numerically for various Knudsen numbers and ditch sizes by the standard direct simulation Monte Carlo method (Bird’s method). It is found that a steady one‐way flow is induced through the channel without any pressure gradient being applied externally. The pumping effect owing to the one‐way flow is discussed.
REFERENCES
1.
J. C.
Maxwell
, “On stress in rarefied gases arising from inequalities of temperature
,” Philos. Trans. R. Soc. Part I
2
, 255
(1879
).2.
E. H. Kennard, Kinetic Theory of Gases (McGraw-Hill, New York, 1938), p. 327.
3.
L. Loeb, Kinetic Theory of Gases (Dover, New York, 1961), Secs. 83, 84.
4.
Y.
Sone
, “Thermal creep in rarefied gas
,” J. Phys. Soc. Jpn.
21
, 1836
(1966
).5.
T.
Ohwada
, Y.
Sone
, and K.
Aoki
, “Numerical analysis of the shear and thermal creep flows of a rarefied gas over a plane wall on the basis of the linearized Boltzmann equation for hard-sphere molecules
,” Phys. Fluids A
1
, 1588
(1989
).6.
Y.
Sone
, “A simple demonstration of a rarefied gas flow induced over a plane wall with a temperature gradient
,” Phys. Fluids A
3
, 997
(1991
).7.
Y.
Sone
, K.
Sawada
, and H.
Hirano
, “A simple experiment on the strength of thermal creep flow of a rarefied gas over a flat wall
,” Eur. J. Mech. B/ Fluids
13
, 299
(1994
).8.
Y. Sone and K. Aoki, Molecular Gas Dynamics (Asakura, Tokyo, 1994), Chaps. 1 and 3, Appendix 2 (in Japanese).
9.
Y.
Sone
, “Flow induced by thermal stress in rarefied gas
,” Phys. Fluids
15
, 1418
(1972
).10.
T.
Ohwada
and Y.
Sone
, “Analysis of thermal stress slip flow and negative thermophoresis using the Boltzmann equation for hard-sphere molecules
,” Eur. J. Mech. B/Fluids
11
, 389
(1992
).11.
M. N.
Kogan
, V. S.
Galkin
, and O. G.
Fridlender
, “Stresses produced in gasses by temperature and concentration inhomogeneities. New types of free convection
,” Sov. Phys. Usp.
19
, 420
(1976
).12.
K.
Aoki
, Y.
Sone
, and T.
Yano
, “Numerical analysis of flow induced in a rarefied gas between noncoaxial cylinder with different temperatures for entire range of the Knudsen number
,” Phys. Fluids A
1
, 363
(1989
).13.
M. N.
Kogan
, “Kinetic theory in aerothermodynamics
,” Progr. Aerosp. Sci.
29
, 271
(1992
).14.
K. Aoki, Y. Sone, and N. Masukawa, “A rarefied gas flow induced by a temperature field,” in Rarefied Gas Dynamics, edited by J. Harvey and G. Lord (Oxford University Press, Oxford, 1995), Vol. I, p. 35.
15.
Y. Sone and K. Aoki, “Forces on a spherical particle in a slightly rarefied gas,” in Rarefied Gas Dynamics, edited by J. L. Potter (AIAA, New York, 1977), Part I, p. 417.
16.
S. K.
Loyalka
, “Mechanics of aerosols in nuclear reactor safety: A review
,” Prog. Nucl. Energy
12
, 1
(1983
).17.
L. Talbot, “Thermophoresis—A review,” in Rarefied Gas Dynamics, edited by S. S. Fisher (AIAA, New York, 1981), Part I, p. 467.
18.
S.
Beresnev
and V.
Chernyak
, “Thermophoresis of a spherical particle in a rarefied gas: Numerical analysis based on the model kinetic equations
,” Phys. Fluids
7
, 1743
(1995
).19.
S.
Takata
and Y.
Sone
, “Flow induced around a sphere with non-uniform surface temperature in a rarefied gas, with application to the drag and thermal force problems of a spherical particle with arbitrary thermal conductivity
,” Eur. J. Mech. B/Fluids
14
, 487
(1995
).20.
D. C. Wadsworth, E. P. Muntz, G. Pham-Van-Diep, and P. Keeley, “Crookes’ radiometer and micromechanical actuators,” in Ref. 14, p. 708.
21.
M. Ota and N. Kawata, “Direct simulation of gas flows around rarefied gas dynamics engines for micro-machine,” in Ref. 14, p. 722.
22.
M.
Knudsen
, “Eine Revision der Gleichgewichtsbedingung der Gase. Thermische Molekularströmung
,” Ann. Phys.
31
, 205
(1910
).23.
M.
Knudsen
, “Thermischer Molekulardruck der Gase in Röhren
,” Ann. Phys.
33
, 1435
(1910
).24.
C. A.
Huber
, “Nanowire array composite
,” Science
263
, 800
(1994
).25.
M.
Esashi
, S.
Shoji
, and A.
Nakano
, “Normally closed microvalve and micropump fabricated on silicon wafer
,” Sensors Actuators
20
, 163
(1989
).26.
G. Pham-Van-Diep, P. Keeley, E. P. Muntz, and D. P. Weaver, “A micromechanical Knudsen compressor,” in Ref. 14, p. 715.
27.
Y.
Sone
and K.
Yamamoto
, “Flow of rarefied gas through a circular pipe
,” Phys. Fluids
11
, 1672
(1968
);28.
H.
Niimi
, “Thermal creep flow of rarefied gas between two parallel plates
,” J. Phys. Soc. Jpn.
30
, 572
(1971
).29.
T.
Kanki
and S.
Iuchi
, “Poiseuille flow and thermal creep of a rarefied gas between parallel plates
,” Phys. Fluids
16
, 594
(1973
).30.
S. K.
Loyalka
, “Comments on Poiseuille flow and thermal creep of a rarefied gas between parallel plates
,” Phys. Fluids
17
, 1053
(1974
).31.
Y.
Sone
and E.
Itakura
, “Analysis of Poiseuille and thermal transpiration flows for arbitrary Knudsen numbers by a modified Knudsen number expansion method and their data base
,” J. Vacuum Soc. Jpn.
38
, 92
(1990
) (in Japanese).32.
T.
Ohwada
, Y.
Sone
, and K.
Aoki
, “Numerical analysis of the Poiseuille and thermal transpiration flows between two parallel plates on the basis of the Boltzmann equation for hard-sphere molecules
,” Phys. Fluids A
1
, 2042
(1989
);33.
Y.
Sone
, “Highly rarefied gas around a group of bodies with various temperature distributions. II. Arbitrary temperature variation
,” J. Méc. Théor. Appl.
4
, 1
(1985
).34.
Recently S. E. Vargo and E. P. Muntz reported an experimental work on the Knudsen compressor (APS/DFD Meeting, Irvine, California, 19–21 November 1995). It is an experiment of a single step through a porous membrane, and the cascade system is not used.
35.
As is obvious from the preceding discussion, the temperature gradient on the average is defined by where is the difference of the wall temperature over the distance s of the channel.
36.
Strictly, the error from the end region is included.
37.
C. Cercignani, The Boltzmann Equation and Its Applications (Springer-Verlag, Berlin, 1988), Chaps. II, III.
38.
G. A. Bird, Molecular Gas Dynamics (Oxford University Press, Oxford, 1976).
39.
G. A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows (Oxford University Press, Oxford, 1994).
40.
W.
Wagner
, “A convergence proof for Bird’s direct simulation Monte Carlo method for the Boltzmann equation
,” J. Stat. Phys.
66
, 1011
(1992
).41.
M. N.
Kogan
, “On the equation of motion of a rarefied gas
,” Appl. Math. Mech.
22
, 597
(1958
).42.
The results are the average of data at every two time steps for and 0.25, every three steps for and the same steps as in Problem I for other
43.
C.
Cercignani
and F.
Sernagiotto
, “Cylindrical Poiseuille flow of a rarefied gas
,” Phys. Fluids
9
, 40
(1966
).
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© 1996 American Institute of Physics.
1996
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