The computational predictions of channel and pipe flows with classical models and no-slip condition at the wall reach excellent results for lower Knudsen numbers (Kn) only. Linear slip models reach a very good approximation of measurement results over the region of 10−3 < Kn < 10−1. The numerical results of higher-order slip models match experimental data up to Kn ≈ 1. The present work derives an analytical model for the transition from the slip regime to the free-molecular flows by the superposition of diffuse molecular boundary reflection and the molecular diffusion inside the bulk flow. The methodology of the present publication models the mass flow resulting from the molecular diffusion for the approximation of the mass flow in microchannels and micropipes for the regime of molecular mass flows (1 < Kn < 100) in an excellent way. The present model shows good agreement with the former models, measurement data, and direct simulation Monte Carlo results for the complete region from the transitional regime up to free-molecular flow (10−2 < Kn < 102).
REFERENCES
1.
J. C.
Maxwell
, “On stresses in rarified gases arising from inequalities of temperature
,” Philos. Trans. R. Soc. London
170
, 231
(1879
).2.
3.
C.
Cercignani
and A.
Daneri
, “Flow of a rarefied gas between two parallel plates
,” J. Appl. Phys.
34
, 3509
(1963
).4.
R. G.
Deissler
, “An analysis of second-order slip flow and temperature-jump boundary conditions for rarefied gases
,” Int. J. Heat Mass Transfer
7
(6
), 681
(1964
).5.
C.
Cercignani
and S.
Lorenzani
, “Variational derivation of second-order slip coefficients on the basis of the Boltzmann equation for hard-sphere molecules
,” Phys. Fluids
22
, 062004
(2010
).6.
H.
Struchtrup
, “Maxwell boundary condition and velocity dependent accommodation coefficient
,” Phys. Fluids
25
, 112001
(2013
).7.
A.
Montessori
, P.
Prestininzi
, M.
LaRocca
, and S.
Succi
, “Lattice Boltzmann approach for complex nonequilibrium flows
,” Phys. Rev. E
92
, 043308
(2015
).8.
M.
Sadr
and M. H.
Gorji
, “A continuous stochastic model for non-equilibrium dense gases
,” Phys. Fluids
29
, 122007
(2017
).9.
P.
Jenny
, S.
Küchlin
, and H.
Gorji
, “Controlling the bias error of Fokker–Planck methods for rarefied gas dynamics simulations
,” Phys. Fluids
31
, 062005
(2019
).10.
S. A.
Rooholghdos
and E.
Roohi
, “Extension of a second order velocity slip/temperature jump boundary condition to simulate high speed micro/nanoflows
,” Comput. Math. Appl.
67
, 2029
–2040
(2014
).11.
N. T. P.
Le
and E.
Roohi
, “A new form of the second-order temperature jump boundary condition for the low-speed nanoscale and hypersonic rarefied gas flow simulations
,” Int. J. Therm. Sci.
98
, 51
–59
(2015
).12.
T.
Veltzke
, On Gaseous Microflows under Isothermal Conditions
(Universität Bremen
, 2013
), https://media.suub.uni-bremen.de/handle/elib/470.13.
S.
Tison
, “Experimental data and theoretical modeling of gas flows through metal capillary leaks
,” Vacuum
44
, 1171
(1993
).14.
G. A.
Bird
, Molecular Gas Dynamics and the Direct Simulation of Gas Flows
(Clarendon
, Oxford
, 1994
).15.
S.
Chapman
and T. G.
Cowling
, The Mathematical Theory of Non-Uniform Gases
(Cambridge University Press
, Cambridge
, 1970
).16.
Y.
Sone
, Kinetic Theory and Fluid Mechanics
(Birkhäuser
, Boston
, 2002
).17.
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
).18.
T.
Ewart
, P.
Perrier
, I. A.
Graur
, and J. G.
Méolans
, “Mass flow rate measurements in a microchannel, from hydrodynamic to near free molecular regimes
,” J. Fluid Mech.
584
, 337
(2007
).19.
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
).20.
G. E.
Karniadakis
, A.
Beskok
, and N.
Aluru
, Microflows and Nanoflows: Fundamentals and Simulations
(Springer-Verlag
, New York
, 2005
).21.
S. S.
Lo
and S. K.
Loyalka
, “An efficient computation of near-continuum rarefied gas flows
,” Z. Angew. Math. Phys.
33
, 419
(1982
).22.
J.
Maurer
, P.
Tabeling
, P.
Joseph
, and H.
Willaime
, “Second-order slip laws in microchannels for helium and nitrogen
,” Phys. Fluids
15
, 2613
(2003
).23.
S.
Colin
, P.
Lalonde
, and R.
Caen
, “Validation of a second-order slip flow model in rectangular microchannels
,” Heat Transfer Eng.
25
, 23
(2004
).24.
S.
Albertoni
, C.
Cercignani
, and L.
Gotusso
, “Numerical evaluation of the slip coefficient
,” Phys. Fluids
6
, 993
(1963
).25.
J.
Gomez
and R.
Groll
, “Pressure drop and thrust predictions for transonic micronozzle flows
,” Phys. Fluids
28
, 022008
(2016
).26.
A.
Beskok
and G. E.
Karniadakis
, “Report: A model for flows in channels, pipes and ducts at micro and nano scales
,” Microscale Thermophys. Eng.
3
, 43
(1999
).27.
S.
Varoutis
, S.
Naris
, V.
Hauer
, C.
Day
, and D.
Valougeorgis
, “Computational and experimental study of gas flows through long channels of various cross sections in the whole range of the Knudsen number
,” J. Vac. Sci. Technol., A
27
, 89
(2009
).28.
M. A.
Gallis
and J. R.
Torczynski
, “Direct simulation Monte Carlo-based expressions for the gas mass flow rate and pressure profile in a microscale tube
,” Phys. Fluids
24
, 012005
(2012
).29.
S.
Gruener
and P.
Huber
, “Knudsen diffusion in silicon nanochannels
,” Phys. Rev. Lett.
100
, 064502
(2008
).30.
F.
Sharipov
and V.
Seleznev
, “Data on internal rarefied gas flows
,” J. Phys. Chem. Ref. Data
27
, 657
(1998
).31.
P.
Perrier
, I. A.
Graur
, T.
Ewart
, and J. G.
Méolans
, “Mass flow rate measurements in microtubes: From hydrodynamic to near free molecular regime
,” Phys. Fluids
23
, 042004
(2011
).© 2020 Author(s).
2020
Author(s)
You do not currently have access to this content.
