Reduced mass flow rates of a rarefied Couette and Poiseuille flow in a long rectangular channel are calculated in the whole range of the gas rarefaction and a wide range of the width to height ratio. Furthermore, walls may be made of different materials so that different tangential momentum accommodation coefficients (TMACs) may be applied. Analytical solutions are given for the slip regime, where all four surrounding walls may have a different TMAC. Due to a simplified modeling assumption, these solutions can be used to correct the well-known flow rates of a fully diffuse channel for different TMACs in the whole range of the gas rarefaction. If the slip solution and the diffuse solution are known, the procedure can principally be adapted for any channel shape. The results of the analytical model expressions are validated with simulation data of the plane Couette and Poiseuille flow and the Poiseuille flow through a pipe, which are found in the literature. In addition, the analytical solution is compared to results of the Direct Simulation Monte Carlo (DSMC) method of a Couette and a Poiseuille flow in a rectangular channel, which are provided as tabulated data for a variation of the gas rarefaction parameter at different aspect ratios and different combinations of TMACs. The procedure to calculate the mass flow rate of the certain flow as well as the application limits are discussed.
References
1.
F.
Sharipov
, P.
Fahrenbach
, and A.
Zipp
, “Numerical modeling of the Holweck pump
,” J. Vac. Sci. Technol. A
23
, 1331
(2005
).2.
H.
Pleskun
, T.
Jünemann
, T.
Bode
, and A.
Brümmer
, “Modelling of inhomogeneous chamber states in rotary positive displacement vacuum pumps
,” IOP Conf. Ser.: Mater. Sci. Eng.
1180
, 012009
(2021
).3.
C.
Huck
, H.
Pleskun
, and A.
Brümmer
, “Measurement and simulation of rarefied Couette Poiseuille flow with variable cross section
,” J. Vac. Sci. Technol. A
36
, 031606
(2018
).4.
T.
Jünemann
and A.
Brümmer
, “One-dimensional simulation of Couette–Poiseuille flow with variable cross section for the full range of gas rarefaction
,” J. Vac. Sci. Technol. B
38
, 044201
(2020
).5.
Q.
Yang
, Y.
Liu
, H.
Zhang
, and C.
Zuo
, “Derivation of generalized Maxwell velocity slip model
,” AIP Conf. Proc.
1628
, 382
(2014
).6.
T.
Missoni
, H.
Yamaguchi
, I.
Graur
, and S.
Lorenzani
, “Extraction of tangential momentum and normal energy accommodation coefficients by comparing variational solutions of the Boltzmann equation with experiments on thermal creep gas flow in microchannels
,” Fluids
6
, 445
(2021
).7.
G.
Narendran
, D. A.
Perumal
, and N.
Gnanasekaran
, “A review of lattice Boltzmann method computational domains for micro-and nanoregime application
,” Nanosci. Technol.
11
, 343
(2020
).8.
F.
Sharipov
, “Rarefied gas flow through a long rectangular channel
,” J. Vac. Sci. Technol. A
17
, 3062
(1999
).9.
F.
Sharipov
, “Non-isothermal gas flow through rectangular microchannels
,” J. Micromech. Microeng.
9
, 394
(1999
).10.
H.
Pleskun
, T.
Bode
, and A.
Brümmer
, “Couette flow in a rectangular channel in the whole range of the gas rarefaction
,” Phys. Fluids
34
, 032004
(2022
).11.
S. K.
Loyalka
, “Kinetic theory of thermal transpiration and mechanocaloric effect
,” J. Chem. Phys.
63
, 4054
(1975
).12.
E. B.
Arkilic
, K. S.
Breuer
, and M. A.
Schmidt
, “Mass flow and tangential momentum accommodation in silicon micromachined channels
,” J. Fluid Mech.
437
, 29
–43
(2001
).13.
T.
Acharya
, J.
Falgoust
, I.
Schoegl
, and M. J.
Martin
, “Measurement of variation of momentum accommodation coefficients with molecular mass and structure
,” J. Thermophy. Heat Transfer
33
, 733
(2019
).14.
F.
Sharipov
, Rarefied Gas Dynamics—Fundamentals for Research and Practice
(Wiley-VCH
, 2016
).15.
J. C.
Maxwell
, “On stresses in rarified gases arising from inequalities of temperature
,” Philos. Trans. R. Soc.
170
, 231
(1879
).16.
C.
Cercignani
and M.
Lampis
, “Kinetic models for gas-surface interactions
,” Transp. Theory Stat. Phys.
1
, 101
(1971
).17.
R. G.
Lord
, in 17th Symposium on Rarefied Gas Dynamics
, edited by A. E.
Beylich
(VCH
, Weinheim
, 1991
), p. 1427
.18.
M. S.
Woronowicz
and D. F. G.
Rault
, “Cercignani-Lampis-Lord gas-surface interaction model: Comparisons between theory and simulation
,” J. Spacecr.
31
, 532
(1994
).19.
H.
Struchtrup
, “Maxwell boundary condition and velocity dependent accommodation coefficient
,” Phys. Fluids
25
, 112001
(2013
).20.
S. K.
Loyalka
and K. A.
Hickey
, “Kinetic theory of thermal transpiration and the mechanocaloric effect: Planar flow of a rigid sphere gas with arbitrary accommodation at the surface
,” J. Vac. Sci. Technol. A
9
, 158
(1991
).21.
F.
Sharipov
and V.
Seleznev
, “Data on internal rarefied gas flows
,” J. Phys. Chem. Ref. Data
27
, 657
(1998
).22.
N. T. P.
Le
, C.
White
, J. M.
Reese
, and R. S.
Myong
, “Langmuir-Maxwell and Langmuir-Smoluchowski boundary conditions for thermal gas flow simulations in hypersonic aerodynamics
,” Int. J. Heat Mass Transfer
55
, 5032
(2012
).23.
T.
Jünemann
, H.
Pleskun
, and A.
Brümmer
, “Maxwell velocity slip and Smoluchowski temperature jump boundary condition for ANSYS CFX
,” IOP Conf. Ser.: Mater. Sci. Eng.
1180
, 012037
(2021
).24.
S. K.
Loyalka
, “Momentum and temperature-slip coefficients with arbitrary accommodation at the surface
,” J. Chem. Phys.
48
, 5432
(1968
).25.
S. K.
Loyalka
, “Approximate method in the kinetic theory
,” Phys. Fluids (1958–1988)
14
, 2291
(1971
).26.
C. E.
Siewert
and F.
Sharipov
, “Model equations in rarefied gas dynamics: Viscous-slip and thermal-slip coefficients
,” Phys. Fluids
14
, 4123
(2002
).27.
F.
Sharipov
, “Application of the Cercignani–Lampis scattering kernel to calculations of rarefied gas flows. II. Slip and jump coefficients
,” Eur. J. Mech. B: Fluids
22
, 133
(2003
).28.
I. A.
Graur
, P.
Perrier
, W.
Ghozlani
, and J. G.
Méolans
, “Measurements of tangential momentum accommodation coefficient for various gases in plane microchannel
,” Phys. Fluids
21
, 102004
(2009
).29.
W.
Su
, P.
Wang
, H.
Liu
, and L.
Wu
, “Accurate and efficient computation of the Boltzmann equation for Couette flow: Influence of intermolecular potentials on Knudsen layer function and viscous slip coefficient
,” J. Comp. Phys.
378
, 573
(2019
).30.
S. M.
Nejand
, E.
Iype
, S.
Nedea
, A.
Frijns
, and D.
Smeulders
, “Modeling rarefied gas-solid surface interactions for Couette flow with different wall temperatures using an unsupervised machine learning technique
,” Phys. Rev. E
104
, 015309
(2021
).31.
J. G.
Méolans
, M. H.
Nacer
, M.
Rojas
, P.
Perrier
, and I.
Graur
, “Effects of two transversal finite dimensions in long microchannel: Analytical approach in slip regime
,” Phys. Fluids
24
, 112005
(2012
).32.
I.
Graur
and M. T.
Ho
, “Rarefied gas flow through a long rectangular channel of variable cross section
,” Vacuum
101
, 328
(2014
).33.
V. A.
Titarev
and E. M.
Shakhov
, “Kinetic analysis of the isothermal flow in a long rectangular microchannel
,” Comput. Math. Math. Phys.
50
, 1221
(2010
).34.
J.
Jang
and T.
Wereley
, “Effective heights and tangential momentum accommodation coefficients of gaseous slip flows in deep reactive ion etching rectangular microchannels
,” J. Micromech. Microeng.
16
, 493
(2006
).35.
F.
Sharipov
, “Application of the Cercignani–Lampis scattering kernel to calculations of rarefied gas flows. III. Poiseuille flow and thermal creep through a long tube
,” Eur. J. Mech. B Fluids
22
, 145
(2003
).36.
O. A.
Aksenova
and I. A.
Khalidov
, “Fractal and statistical models of rough surface interacting with rarefied gas flow
,” AIP Conf. Proc.
762
, 993
(2005
).37.
O. A.
Aksenova
and I. A.
Khalidov
, “Analytic model of the effect of poly-gaussian roughness on rarefied gas flow near the surface
,” AIP Conf. Proc.
1786
, 100007
(2016
).38.
O. A.
Aksenova
and I. A.
Khalidov
, “Inverse problem of finding surface roughness parameters in rarefied gas flow
,” AIP Conf. Proc.
2132
, 170005
(2019
).39.
T. C.
Lilly
, J. A.
Duncan
, S. L.
Nothnagel
, S. F.
Gimelshein
, and N. E.
Gimelshein
, “Numerical and experimental investigation of microchannel flows with rough surfaces
,” Phys. Fluids
19
, 106101
(2007
).40.
C.
Zhang
, Y.
Chen
, Z.
Deng
, and M.
Shi
, “Role of rough surface topography on gas slip flow in microchannels
,” Phys. Rev. E
86
, 016319
(2012
).41.
H.
Yan
, W.-M.
Zhang
, and Z.-K.
Peng
, “Effect of random surface topography on the gaseous flow in microtubes with an extended slip model
,” Microfluid. Nanofluid.
18
, 897
–910
(2015
).42.
W.
Su
, H.
Liu
, Y.
Zhang
, and L.
Wu
, “Rarefaction cloaking: Influence of the fractal rough surface in gas slider bearings
,” Phys. Fluids
29
, 102003
(2017
).43.
O.
Sazhin
, “Regarding a benchmark problem: Rarefied gas flow through a rough-surfaced channel
,” Comp. Math. Math. Phys.
58
, 1640
–1646
(2018
).44.
O.
Sazhin
, “The effect of surface roughness on internal free molecular gas flow
,” Vacuum
159
, 287
–292
(2019
).45.
O.
Sazhin
, “Rarefied gas flow through a rough channel into a vacuum
,” Microfluid. Nanofluid.
24
, 27
(2020
).46.
A.
Taassob
, A.
Bordbar
, S.
Kheirandish
, A.
Zanaghsh
, R.
Kamall
, and A. S.
Rana
, “A review of rarefied gas flow in irregular micro/nanochannels
,” J. Micromech. Microeng.
31
, 113002
(2021
).47.
C.
Cercignani
, M.
Lampis
, and S.
Lorenzani
, “Variational approach to gas flows in microchannels
,” Phys. Fluids
16
, 3426
(2004
).48.
C.
Cercignani
, M.
Lampis
, and S.
Lorenzani
, “Plane Poiseuille-Couette problem in micro-electro-mechanical systems applications with gas-rarefaction effects
,” Phys. Fluids
18
, 087102
(2006
).49.
E. B.
Arkilic
, M. A.
Schmidt
, and K. S.
Breuer
, “Gaseous slip flow in long microchannels
,” J. Microelectromech. Syst.
6
, 167
(1997
).50.
B.
John
, X.-J.
Gu
, and D. R.
Emerson
, “Nonequilibrium gaseous heat transfer in pressure-driven plane Poiseuille flow
,” Phys. Rev. E
88
, 013018
(2013
).51.
V.
Varade
, V. S.
Duryodhan
, A.
Agrawal
, A. M.
Pradeep
, A.
Ebrahimi
, and E.
Roohi
, “Low Mach number slip flow through diverging microchannel
,” Comput. Fluids
111
, 46
–61
(2015
).52.
A.
Ebrahimi
, V.
Shahabi
, and E.
Roohi
, “Pressure-driven nitrogen flow in divergent microchannels with isothermal walls
,” Appl. Sci.
11
, 3602
(2021
).53.
54.
55.
E.
Truckenbrodt
, Fluidmechanik—Band 1: Gundlagen Und Elementare Strömungsvorgänge Dichtebeständiger Fluide
(Springer-Verlag
, Berlin/Heidelberg
, 2008
), Vol. 4
.56.
I. A.
Graur
, J. G.
Méolans
, and D. E.
Zeitoun
, “Analytical and numerical description for isothermal gas flows in microchannels
,” Microfluid. Nanofluid.
2
, 64
–77
(2006
).57.
A.
Ebrahimi
and E.
Roohi
, “Flow and thermal fields investigation in divergent micro/nano channels
,” J. Therm. Eng.
2
, 709
–714
(2016
).58.
A.
Ebrahimi
and E.
Roohi
, “DSMC investigation of rarefied gas flow through diverging micro- and nanochannels
,” Microfluid. Nanofluid.
21
, 18
(2017
).59.
E.
Buckingham
, “On physically similar systems; Illustrations of the use of dimensional equations
,” Phys. Rev.
4
, 345
(1914
).60.
W. H.
Press
, S. A.
Teukolsky
, W. T.
Vetterling
, and B. P.
Flannery
, Numerical Recipes: The Art of Scientific Computing
(Cambridge University Press
, Cambridge
, 2007
).61.
J. H.
Spurk
and N.
Aksel
, Strömungslehre—Einführung in Die Theorie Der Strömungen
(Springer-Verlag
, Berlin/Heidelberg
, 2010
), Vol. 8
.62.
C.
Huck
and A.
Brümmer
, “Limits of one dimensional modeling of rarefied Couette Poiseuille clearance flow in vacuum pumps
,” IOP Conf. Ser.: Mater. Sci. Eng.
425
, 012027
(2018
).63.
A. H.
Shapiro
, The Dynamics and Thermodynamics of Compressible Fluid Flow
(Ronald
, New York
, 1953
), Vol. 1
.64.
R.
Müller
, “Spaltströmung mit wärmeübertragung in vakuumpumpen
,” Dissertation (Universität Kaiserslautern
, 2013
).65.
G. A.
Bird
, Molecular Gas Dynamics and the Direct Simulation of Gas Flows
(Clarendon Press
, Oxford
, 1994
).66.
C.
White
, M. K.
Borg
, T. J.
Scanlon
, S. M.
Longshaw
, B.
John
, D. R.
Emerson
, and J. M.
Reese
, “dsmcfoam+: An openfoam based direct simulation Monte Carlo solver
,” Comp. Phys. Commun.
224
, 22
(2018
).67.
G. A.
Bird
, “Sophisticated DSMC
,” in Notes for a Short Course at the DSMC07 Meeting
(2007
).68.
L.
Fahrmeir
, C.
Heumann
, R.
Künstler
, I.
Pigeot
, and G.
Tutz
, Statistik – Der Weg Zur Datenanalyse
(Springer
, Spektrum
, 2016
), Vol. 8
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