Plane thermal transpiration of a rarefied gas between two walls of Maxwell-type boundaries with different accommodation coefficients is studied based on the linearized Boltzmann equation for a hard-sphere molecular gas. The Boltzmann equation is solved numerically using a finite difference method, in which the collision integral is evaluated by the numerical kernel method. The detailed numerical data, including the mass and heat flow rates of the gas, are provided over a wide range of the Knudsen number and the entire range of the accommodation coefficients. Unlike in the plane Poiseuille flow, the dependence of the mass flow rate on the accommodation coefficients shows different characteristics depending on the Knudsen number. When the Knudsen number is relatively large, the mass flow rate of the gas increases monotonically with the decrease in either of the accommodation coefficients like in Poiseuille flow. When the Knudsen number is small, in contrast, the mass flow rate does not vary monotonically but exhibits a minimum with the decrease in either of the accommodation coefficients. The mechanism of this phenomenon is discussed based on the flow field of the gas.

References

References
1.
Kennard
,
E. H.
,
1938
,
Kinetic Theory of Gases
,
McGraw-Hill
,
New York
.
2.
Sone
,
Y.
,
1966
, “
Thermal Creep in Rarefied Gas
,”
J. Phys. Soc. Jpn.
,
21
, pp.
1836
1837
.10.1143/JPSJ.21.1836
3.
Niimi
,
H.
,
1971
, “
Thermal Creep Flow of Rarefied Gas Between Two Parallel Plates
,”
J. Phys. Soc. Jpn.
,
30
, pp.
572
574
.10.1143/JPSJ.30.572
4.
Loyalka
,
S. K.
,
1971
, “
Kinetic Theory of Thermal Transpiration and Mechanocaloric Effect. I
,”
J. Chem. Phys.
,
55
, pp.
4497
4503
.10.1063/1.1676780
5.
Kanki
,
T.
, and
Iuchi
,
S.
,
1973
, “
Poiseuille Flow and Thermal Creep of a Rarefied Gas Between Parallel Plates
,”
Phys. Fluids
,
16
, pp.
594
599
.10.1063/1.1694393
6.
Loyalka
,
S. K.
,
1974
, “
Comments on Poiseuille Flow and Thermal Creep of a Rarefied Gas Between Parallel Plates
,”
Phys. Fluids
,
17
, pp.
1053
1055
.10.1063/1.1694820
7.
Ohwada
,
T.
,
Sone
,
Y.
, and
Aoki
,
K.
,
1989
, “
Numerical Analysis of the Poiseuille and Thermal Transpiration Flows Between Two Parallel Plates on the Basis of the Boltzmann Equation for a Hard-Sphere Molecules
,”
Phys. Fluids A
,
1
, pp.
2042
2049
.10.1063/1.857478
8.
Sone
,
Y.
, and
Itakura
,
E.
,
1990
, “
Analysis of Poiseuille and Thermal Transpiration Flows for Arbitrary Knudsen Numbers by a Modified Knudsen Number Expansion Method and Their Database
,”
J. Vac. Soc. Jpn.
,
33
, pp.
92
94
(in Japanese).10.3131/jvsj.33.92
9.
Sharipov
,
F.
, and
Seleznev
,
V.
,
1998
, “
Data on Internal Rarefied Gas Flows
,”
J. Phys. Chem. Ref. Data
,
27
, pp.
657
706
.10.1063/1.556019
10.
Sone
,
Y.
,
2002
,
Kinetic Theory and Fluid Dynamics
,
Birkhäuser
,
New York
.
11.
Karniadakis
,
G.
,
Beskok.
A.
, and
Aluru
,
N.
,
2005
,
Microflows and Nanoflows: Fundamentals and Simulation
,
Springer-Verlag
,
New York
.
12.
Sone
,
Y.
,
2007
,
Molecular Gas Dynamics
,
Birkhäuser
,
New York
.
13.
Chen
,
C. C.
,
Chen
,
I. K.
,
Liu
,
T.-P.
, and
Sone
,
Y.
,
2007
, “
Thermal Transpiration for the Linearized Boltzmann Equation
,”
Commun. Pure. Appl. Math.
,
60
, pp.
147
163
.10.1002/cpa.20167
14.
Sharipov
,
F.
, and
Bertoldo
,
G.
,
2009
, “
Poiseuille Flow and Thermal Creep Based on the Boltzmann Equation With the Lennard–Jones Potential Over a Wide Range of the Knudsen Number
,”
Phys. Fluids
,
21
, p.
067101
.10.1063/1.3156011
15.
Takata
,
S.
, and
Hunagane
,
H.
,
2011
, “
Poiseuille and Thermal Transpiration Flows of a Highly Rarefied Gas: Over-Concentration in the Velocity Distribution Function
,”
J. Fluid Mech.
,
669
, pp.
242
259
.10.1017/S0022112010005021
16.
Doi
,
T.
,
2012
, “
Plane Thermal Transpiration of a Rarefied Gas in the Presence of Gravitation
,”
Vacuum
,
86
, pp.
1541
1546
.10.1016/j.vacuum.2012.02.034
17.
Doi
,
T.
,
2012
, “
Effect of Weak Gravitation on the Plane Thermal Transpiration of a Slightly Rarefied Gas
,”
Fluid Dyn. Res.
,
44
, p.
065503
.10.1088/0169-5983/44/6/065503
18.
Fukui
,
S.
, and
Kaneko
,
R.
,
1987
, “
Analysis of Ultra-Thin Gas Film Lubrication Based on the Linearized Boltzmann Equation (Influence of Accommodation Coefficient)
,”
JSME Int. J.
,
30
, pp.
1660
1666
.10.1299/jsme1987.30.1660
19.
Sharipov
,
F.
,
2002
, “
Application of the Cercignani–Lampis Scattering Kernel to Calculations of Rarefied Gas Flows. I. Plane Flow Between Parallel Plates
,”
Euro. J. Mech. B/Fluids
,
21
, pp.
113
123
.10.1016/S0997-7546(01)01160-8
20.
Veijola
,
T.
,
Kuisma
,
H.
, and
Lahdenperä
,
J.
,
1998
, “
The Influence of Gas-Surface Interaction on Gas-Film Damping in a Silicon Accelerometer
,”
Sens. Actuators A
,
66
, pp.
83
92
.10.1016/S0924-4247(97)01732-9
21.
Fukui
,
S.
, and
Kaneko
,
R.
,
1988
, “
Analysis of Ultra-Thin Gas Film Lubrication Based on Linearized Boltzmann Equation: First Report—Derivation of a Generalized Lubrication Equation Including Thermal Creep Flow
,”
ASME J. Tribol.
,
110
, pp.
253
262
.10.1115/1.3261594
22.
Bhatnagar
,
P. L.
,
Gross
,
E. P.
, and
Krook
,
M.
,
1954
, “
A Model for Collision Processes in Gases: I. Small Amplitude Processes in Charged and Neutral One-Component Systems
,”
Phys. Rev.
,
94
, pp.
511
525
.10.1103/PhysRev.94.511
23.
Welander
,
P.
,
1954
, “
On the Temperature Jump in a Rarefied Gas
,”
Ark. Fys.
,
7
, pp.
507
553
.
24.
Cercignani
,
C.
,
Lampis
,
M.
, and
Lorenzani
,
S.
,
2004
, “
Variational Approach to Gas Flows in Microchannels
,”
Phys. Fluids
,
16
, pp.
3426
3437
.10.1063/1.1764700
25.
Cercignani
,
C.
,
Lampis
,
M.
, and
Lorenzani
,
S.
,
2004
, “
Plane Poiseuille Flow With Symmetric and Nonsymmetric Gas-Wall Interactions
,”
Transp. Theory Stat. Phys.
,
33
, pp.
545
561
.10.1081/TT-200053939
26.
Scherer
,
C. S.
,
Prolo Filho
,
J. F.
, and
Barichello
,
L. B.
,
2009
, “
An Analytical Approach to the Unified Solution of Kinetic Equations in Rarefied Gas Dynamics
,”
Z. Angew. Math. Phys.
,
60
, pp.
70
115
.10.1007/s00033-008-7084-4
27.
Gross
,
E. P.
, and
Jackson
,
E. A.
,
1959
, “
Kinetic Model and the Linearized Boltzmann Equation
,”
Phys. Fluids
,
2
, pp.
432
441
.10.1063/1.1724415
28.
Sone
,
Y.
,
1970
, “
A Note on Thermal Creep in Rarefied Gas
,”
J. Phys. Soc. Jpn.
,
29
, p.
1655
.10.1143/JPSJ.29.1655
29.
Wakabayashi
,
M.
,
Ohwada
,
T.
, and
Golse
,
F.
,
1996
, “
Numerical Analysis of the Shear and Thermal Creep Flows of a Rarefied Gas Over the Plane Wall of a Maxwell-Type Boundary on the Basis of the Linearized Boltzmann Equation for Hard-Sphere Molecules
,”
Euro. J. Mech. B/Fluids
,
15
, pp.
175
201
.
30.
Sone
,
Y.
,
1969
, “
Asymptotic Theory of Flow of Rarefied Gas Over a Smooth Boundary I
,”
Rarefied Gas Dynamics
, Vol.
1
,
L.
Trilling
and
H. Y.
Wachman
, eds.,
Academic Press
,
New York
, pp.
243
253
.
31.
Sone
,
Y.
,
1971
, “
Asymptotic Theory of Flow of Rarefied Gas Over a Smooth Boundary II
,”
Rarefied Gas Dynamics
, Vol.
2
,
D.
Dini
, ed.,
Editrice Tecnino Scientifica
,
Pisa, Italy
, pp.
737
749
.
32.
Aoki
,
K.
,
Inamuro
,
T.
, and
Onishi
,
Y.
,
1979
, “
Slightly Rarefied Gas Flow Over a Body With Small Accommodation Coefficient
,”
J. Phys. Soc. Jpn.
,
47
, pp.
663
671
.10.1143/JPSJ.47.663
33.
Sone
,
Y.
, and
Aoki
,
K.
,
1977
, “
Slightly Rarefied Gas Flow Over a Specularly Reflecting Body
,”
Phys. Fluids
,
20
, pp.
571
576
.10.1063/1.861914
34.
Sone
,
Y.
, and
Aoki
,
K.
,
1994
,
Molecular Gas Dynamics
,
Asakura
,
Tokyo
(in Japanese).
35.
Sone
,
Y.
,
Ohwada
,
T.
, and
Aoki
,
K.
,
1989
, “
Temperature Jump and Knudsen Layer in a Rarefied Gas Over a Plane Wall: Numerical Analysis of the Linearized Boltzmann Equation for a Hard-Sphere Molecules
,”
Phys. Fluids A
,
1
, pp.
363
370
.10.1063/1.857457
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