A Review On Slip Models For Gas Microflows: Wen-Ming Zhang, Guang Meng & Xueyong Wei
A Review On Slip Models For Gas Microflows: Wen-Ming Zhang, Guang Meng & Xueyong Wei
A Review On Slip Models For Gas Microflows: Wen-Ming Zhang, Guang Meng & Xueyong Wei
ISSN 1613-4982
Microfluid Nanofluid
DOI 10.1007/s10404-012-1012-9
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DOI 10.1007/s10404-012-1012-9
REVIEW PAPER
Abstract Accurate modeling of gas microflow is crucial reached on the correct and generalized form of higher-order
for the microfluidic devices in MEMS. Gas microflows slip expression. The influences of specific effects, such as
through these devices are often in the slip and transition effective mean free path of the gas molecules and viscosity,
flow regimes, characterized by the Knudsen number of the surface roughness, gas composition and tangential
order of 10-2*100. An increasing number of researchers momentum accommodation coefficient, on the hybrid slip
now dedicate great attention to the developments in the models for gas microflows are analyzed and discussed. It
modeling of non-equilibrium boundary conditions in the gas shows that although the various hybrid slip models are
microflows, concentrating on the slip model. In this review, proposed from different viewpoints, they can contribute to
we present various slip models obtained from different N–S equations for capturing the high Knudsen number
theoretical, computational and experimental studies for gas effects in the slip and transition flow regimes. Future studies
microflows. Correct descriptions of the Knudsen layer are also discussed for improving the understanding of gas
effect are of critical importance in modeling and designing microflows and enabling us to exactly predict and actively
of gas microflow systems and in predicting their perfor- control gas slip.
mances. Theoretical descriptions of the gas-surface inter-
action and gas-surface molecular interaction models are Keywords MEMS Microfluidic device
introduced to describe the boundary conditions. Various Gas microflow Slip coefficient Slip model
methods and techniques for determination of the slip coef-
ficients are reviewed. The review presents the considerable Abbreviations
success in the implementation of various slip boundary AB Augmented Burnett
conditions to extend the Navier–Stokes (N–S) equations MD Molecular dynamics
into the slip and transition flow regimes. Comparisons of BE Boltzmann equation
different values and formulations of the first- and second- MEMS Microelectromechanical systems
order slip coefficients and models reveal the discrepancies CL Cercignani–Lampis
arising from different definitions in the first-order slip MFP Mean free path
coefficient and various approaches to determine the second- HS Hard sphere
order slip coefficient. In addition, no consensus has been N–S Navier–Stokes
LBE Linearized Boltzmann equation
N–S–F Navier–Stokes–Fourier
W.-M. Zhang (&) G. Meng
LBM Lattice Boltzmann method
State Key Laboratory of Mechanical System and Vibration,
School of Mechanical Engineering, Shanghai Jiao Tong QGD Quasi-gas dynamic
University, 800 Dongchuan Road, Shanghai 200240, China BGK Bhatnagar Gross Krook
e-mail: wenmingz@sjtu.edu.cn TMAC Tangential momentum accommodation
coefficient
X. Wei
Department of Engineering, University of Cambridge, DSMC Direct Simulation Monte Carlo
Trumpington Street, Cambridge CB2 1PZ, UK VHS Variable hard sphere
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properties of the gas microflows. The KL effect on the gas thermodynamic equilibrium assumption in the N–S equa-
microflows and the capturing approaches are presented in tions can be related to the Knudsen number (Barber and
Sect. 3. Section 4 focuses on the theoretical descriptions of Emerson 2006). Gad-el-Hak (2003) also discussed that the
the gas-surface interaction and gas-surface molecular N–S equations remain to be valid when the three funda-
interaction models. Section 5 summarizes and reviews the mental assumptions (Newtonian framework, continuum
main theoretical, numerical, and experimental slip coeffi- approximation, and thermodynamic equilibrium) are sat-
cients and relative various slip models, including first- isfied. Typically, rarefaction effect can be characterized by
order, second-order, higher-order and hybrid types, which the Knudsen number, which is the key parameter to indi-
are widely used to extend the N–S equations in the slip cate the degree of rarefaction or state of non-equilibrium of
flow and transition regimes. The effects of effective MFP gas flows and defined as:
and viscosity, gaseous mixture, surface roughness and
k k dU0
TMAC on the slip coefficients and slip models are ana- Kn ¼ ð2Þ
L0 U dL
0 0
lyzed and discussed in detail. Finally, Sect. 6 provides final
remarks and conclusions. Throughout the review, we where L0 is the characteristic length of the microflow
emphasized the discrepancies among the various slip system and U0 is a quantity of interest, such as the gas
models in order to focus on future research efforts into density, pressure or temperature (Tang et al. 2008). Prac-
providing an understanding of the velocity slip boundary tically, a local Knudsen number can be used as a global
conditions for gas microflows through the microfluidic measure to avoid the ambiguity of selecting L0 in large or
devices in MEMS. complex systems (Oran et al. 1998). The ratio of L0/d
satisfies to be larger than 100 in order to obtain a statisti-
cally stable estimation of the macroscopic properties (Bird
2 General physics of gas microflows 1994).
The analysis and modeling of gas microflow depend on 2.2 Compressibility effect
some important characteristic length scales and parameters
(Barber and Emerson 2006; Colin 2005). Gad-el-Hak In general for gas microflows in MEMS, the effects of
(2001, 2006) reported the general flow physics in MEMS rarefaction and compressibility are coupled and tend to
and broadly reviewed available methodologies to model conflict with each other (Morini et al. 2004, 2005). The
the transport phenomena in microdevices. Colin (2005) and compressibility is significant when the Mach number
Barber and Emerson (2006) discussed and reviewed the approaches unity. From the classical kinetic theory, the
rarefaction and compressibility effects on gas microflows Knudsen number is related to the Reynolds number Re and
and provided several characteristic length scales. Mach number Ma by:
At the level of molecules, the relationship between the rffiffiffiffiffi
k pc Ma
mean molecular spacing d and mean molecular diameter d Kn ¼ ð3Þ
L0 2 Re
is an important parameter. For the dilute gases, gases sat-
isfy d/d [ 7 (Bird 1994; Gad-el-Hak 2003; Barber and where c is the ratio of specific heats of the gas. Li et al.
Emerson 2006) or d/d 1(Colin 2005). In these cases, (2000) demonstrated experimentally that the effect of
most of the intermolecular interactions are binary colli- compressibility can be neglected only for an average Mach
sions. Conversely, the gas can be regarded as a dense one. number lower than 0.3, while Colin (2005) recommended
The dilute gas approximations lead to the classical kinetic that the compressibility effect should be taken into account
theory of gases and the Boltzmann transport equation. when Ma [ 0.2.
For a gas of hard sphere (HS) molecules in thermody- For the rarefaction degree of the gas, it can calculate the
namic equilibrium, Bird (1994) gave the definition of the Reynolds number for which the Mach number is less than
MFP as: 0.2. For the Knudsen numbers in the slip flow and early
1 transition regimes, Fig. 1 illustrates the range of Reynolds
k ¼ pffiffiffi ð1Þ numbers for which the flow can be divided into two zones,
2png d2
i.e., incompressible and compressible flows. The dividing
where ng is the number density of the gas and ng = d-3. line is different for monatomic and diatomic gases. It is
evident that the compressibility effects can be neglected for
2.1 Rarefaction effect high Knudsen numbers only when the Reynolds number is
very low. However, the gas flow at lower Knudsen numbers
The rarefaction effect in microsystems is attributed to the can be considered incompressible within a larger range of
MFP of the gas microflow. In practice, the validity of the Reynolds numbers. Therefore, the coupled effects of
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Free
Continuum Slip Transition molecular
Collisionless
Molecular
Boltzmann equation Boltzmann
model equation
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Ο (λ )
u*s
us
uw
y
Fig. 4 Schematic of the Knudsen layer with gas microflow near a
solid wall. Actual velocity profile (continuous line) and velocity
profile (dash lines) predicted by N–S equations with a slip boundary
condition within the KL (Lockerby et al. 2005a; Watari 2010)
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7 y2 questionable (Cercignani 1988). Moreover, these higher-
uðyÞ ¼ ku y þ 1:1466k 1þ ð5Þ order models cannot even capture the KL for the simple
20 k
Kramer’s problem (Lockerby et al. 2005b). Lilley and
The wall function shows the velocity gradient equals to Sader (2008) discussed that the wall function, various
1.7 at the wall. However, this model does not consider the hydrodynamic models, and Fichman and Hetsroni models
accommodation coefficient and is limited at a relatively (Fichman and Hetsroni 2005) do not capture the asymptotic
low Knudsen number up to 0.1 for planar surfaces (Hare form of the velocity profile in the KL near the wall.
et al. 2007). Zheng et al. (2006) addressed a formulation of
the wall function incorporating the accommodation 3.1.3 Power-law model
coefficient to explain this problem.
The lattice Boltzmann method (LBM) can be used to
3.1.2 Higher-order continuum model resolve the KL; in this method the wall-function approach
can alter the dynamics near the wall by adjusting the
Higher-order continuum model can be regarded as an relaxation time or applying the mean-field theory (Zheng
appealing strategy to capture the KL effect (Guo et al. et al. 2006; Guo et al. 2006), and the higher-order con-
2007a, b). Some models beyond the N–S equations, such as tinuum models can be included in the LBM, such as the
the Burnett, BGK–Burnett, and super-Burnett models, had Burnett, super-Burnett, Grad 13-moment, or beyond
been proposed (Zhong et al. 1993; Jin and Slemrod 2001; (Aidun and Clausen 2010). However, the higher-order
Struchtrup and Torrilhon 2003; Balakrishnan 2004). These continuum model cannot provide a proper treatment of the
models are usually derived from LBE using the Chapman– boundary conditions at the wall (Gu and Emerson 2007;
Enskog expansion and truncating at certain orders (Cer- Struchtrup and Torrilhon 2008), and is formally valid
cignani 1988). Lockerby et al. (2005b) explained as to why outside the KL (Hadjiconstantinou 2006). The power-law
there are so many different higher-order models with three description is obtained from the LBE and DSMC solu-
basic reasons: (1) constitutive relations of higher-order tions of the BE (Lilley and Sader 2008), and it indicates
than the N–S equations have indicated potential in mod- that the velocity gradient singularity comes naturally from
eling rarefied flows; (2) all these models depend on the the BE.
numerical and physical instability; and (3) no single Lilley and Sader (2007) used the solutions from LBE
equation has the ability to predict the important non- and DSMC calculations to examine the KL, and discovered
equilibrium effects in the rarefied gas microflows. that the bulk gas velocity can be accurately described by
The velocity profile predicted by the super-Burnett the remarkably simple power-law behavior as:
equations can be given by (Lockerby et al. 2005b):
uðyÞ uð0Þ / yap ð8Þ
uðyÞ ¼ ks1 þ ku y þ ks2 cosðks3 yÞ þ ks4 sinðks3 yÞ ð6Þ where ap & 0.8 applies for HS molecules near a diffusely
where ks1*ks4 are constants. reflecting wall. The expression establishes the existence of
Lockerby et al. (2005b) considered the velocity gradient a velocity gradient singularity at the wall, which cannot be
from the wall and developed an expression to approximate captured by the higher-order continuum model (Lockerby
the KL as: et al. 2005b) and wall-function model (Lockerby et al.
" rffiffiffi # 2005a).
2 7 CL y=k
Lilley and Sader (2008) also presented the entire
uðyÞ ¼ ku y þ k þ k 1e ð7Þ
p 10CL velocity profile including the KL as:
apðr Þ
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3.2 Knudsen layer thickness mechanics of the slip boundary conditions based on the con-
cepts of Navier, Stokes, Reynolds, and Maxwell. The history
The KL thickness can be approximated by lc and given by and formulation of the slip boundary condition is listed in
(Gusarov and Smurov 2002): Table 2, in which the symbols refer to Badur et al. (2011).
kB T
lc ¼ ð10Þ 4.1 Gas-surface interaction: kinetic theory
pd2 p
where kB is the Boltzmann constant, T is the absolute In the kinetic theory, the gas-surface interaction forms a
temperature, and p is the gas pressure. boundary condition between the gas molecules and the
The thickness of KL is of the order of a few MFPs and solid wall. It is important to investigate the gas-surface
can be predicted with the kinetic theory and DSMC sim- interaction for understanding the gas microflows. The
ulation quantitatively. A comparison of the KL thickness microflow near a wall is strongly influenced by the gas-
obtained with various approaches is listed in Table 1. It can surface interaction, which can be governed by the typical
be seen that the solution obtained from the kinetic theory models, such as the Maxwell (elastic-diffuse) model or the
distinguishes from the DSMC data and MD simulation. Cercignani–Lampis (CL) model (McCormick 2005).
The thickness calculated by kinetic theory is about 1.4 k Although various gas-surface interaction models have been
and those obtained by various higher-order continuum proposed since Maxwell in 1879, the validity of these
models are in the range from 0.9 k to 4.9 k (Lockerby et al. models remains under question for rarefied flow conditions.
2005b), and that simulated by molecular dynamics (MD) The slip effect should be considered to make a correction
method is 2.5 k (Galvin et al. 2007). based on the degree of non-equilibrium near the wall.
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(a) No-slip (b) Partial slip second term is the order of OðKn2 Þ, which were often
modified as ou=oxt or neglected, or improved by adding a
correction term in theoretical analyses (Khadem et al.
2009; Hossainpour and Khadem 2010; Colin 2012).
To et al. (2010) derived a slip model for gas microflows
induced by external body forces based on Maxwell’s col-
lision theory between gas molecules and the wall. The slip
Gas boundary condition yields:
Solid Ls us
c oun
us uw ¼ rp k bT T þ ð17Þ
c on S
Fig. 5 Schematic diagram of slip at a gas–solid interface where cT is the molecular acceleration, c is the thermal
speed of the gas, bT denotes a constant, which denotes the
ous 3 l oT difference between the idealized condition used to derive
us uw ¼ rp k þ ð14Þ
on S 4 qT on S the slip model and the realistic one and is expected to be
close to unity.
where n is the coordinate normal to the wall, us is the slip
Second-order boundary conditions have been proposed
velocity, uw is the wall velocity, and rp ¼ as ð2 rv Þ=rv is
in the literature (Karniadakis et al. 2005; Barber and
the slip coefficient, in which as is a controversial coeffi-
Emerson 2006; Dongari et al. 2007; Weng and Chen 2008)
cient and will be discussed in Sect. 5. Maxwell (1879) first
and the general form can be expressed as:
estimated the coefficient and assumed that the incident
molecules have the same distributions as those in the midst
2
pffiffiffi ous o us
of the gas, and obtained as ¼ p=2. This slip boundary us uw ¼ C1 k þC2 k2 ð18Þ
on S on2 S
condition can provide useful prediction for certain gas
microflows (Hare et al. 2007). where C1 and C2 are the first- and second-order slip coef-
When the thermal creep effects due to the axial tem- ficients, respectively.
perature gradient are neglected, the first-order slip bound- From the tangential momentum flux analyses, Beskok
ary condition can be written as: and Karniadakis (1996) and Beskok and Karniadakis
(1999) derived a high-order slip boundary condition for an
ous
us uw ¼ rp k ð15Þ isothermal surface in the following form:
on S
1
In the case of wall with curvature, the slip boundary us ¼ ½uk þ ð1 rv Þuk þ rv uw ð19Þ
2
condition at the wall in two dimensions becomes
(Lockerby et al. 2004): where uk is the tangential component of gas velocity one
MFP away from the surface. The boundary condition
ous oun predicts accurate wall slip velocity when Kn \ 0.5,
us uw ¼ rp k þ ð16Þ
on oxt S although resulting in poor mass flow rate prediction
where un is the gas velocity normal to the wall and xt is the (Karniadakis et al. 2005). Using a Taylor series expansion
coordinate tangential to the wall. The additional derivative of uk about us (Beskok and Karniadakis 1999), yields:
q un/q xt should also be considered for microfluidic devices
ous k2 o2 us k3 o3 us
with significant roughness inducing two components of the us uw ¼ rp k þ þ þ
on S 2 on2 S 3! on3 S
velocity close to the wall (Colin 2012). However, the first
term at the right hand side is the order of OðKnÞ, while the ð20Þ
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Attempts to implement the above slip boundary where . The combination of rv = 0 and rn = 0 corre-
condition using numerical simulation methods are rather sponds to the specular reflection and the combination of
difficult. Second-order and higher derivatives of velocity rv = 1 and rn = 1 refers to the diffuse scattering.
cannot be computed and simulated accurately near the wall In addition, Lord (1991, 1995) presented a transforma-
(Gad-el-Hak 1999). tion of the CL model with the DSMC method and extended
Zhang et al. (2010a, b) considered the effect of relative it widely to rarefied gas microflows. The Maxwell model
position of the slip surface in the KL on the slip boundary and the Cercignani, Lampis and Lord (CLL) model are the
condition and developed a new slip model in the form: most common gas-surface interaction models used with the
2
DSMC method.
1 ð1 CZ Þrv ous k2 o us
us uw ¼ k þ ð1 CZ Þ
rv on S 2 on2 S
4.2 Gas-surface molecular interaction: surface
ð21Þ
adsorption theory
Compared with the classical second-order slip boundary
condition proposed by Beskok and Karniadakis (1999), the A physical approach can be developed to describe the slip
coefficient CZ ¼ ns =k (CZ 2 ½0; 1), in which ns is the effect by considering the interfacial interaction between the
distance between the wall and the slip surface, is proposed gas molecules and surface (Langmuir 1933; Myong et al.
in the corrected second-order boundary condition. The 2005). In this approach, the gas molecules are assumed to
corrected second-order slip boundary condition was used to interact with the surface of the solid via a long range force,
solve the N–S equations for confined fluids at the and consequently the gas molecules can be adsorbed onto
microscale and nanoscale (Zhang et al. 2012a, b). the surface (Langmuir 1933). The Langmuir slip model
based on the surface chemistry theory can be explained by
4.1.2 Cercignani–Lampis (CL) model surface adsorption isotherm.
Using the Langmuir adsorption isotherm, the Langmuir
To provide a more physical description of the gas-surface slip model had been developed by Eu et al. (1987) and
interaction, the CL model was presented to satisfy the Myong (2001, 2004a, 2005), for the gas-surface molecular
fundamental scattering kernel principles and distinguish interaction and the velocity slip can be expressed as:
the momentum and energy accommodation coefficients us ¼ aM uw þ ð1 aM Þuloc ð24Þ
(Albertoni et al.1963; Cercignani and Lampis 1971; Loy-
alka et al. 1975; Ohwada et al. 1989; Garcia and Siewert where the subscript loc denotes the local value adjacent to
2010). the wall. aM is the fraction of the surface covered by
Klinc and Kuscer (1972) presented the variation result adsorbed atoms at thermal equilibrium and aM ¼ bM p=ð1þ
pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi
for the slip coefficient rp for the CL gas-surface model as: bM pÞ and aM ¼ bM p=ð1 þ bM pÞ for monatomic and
2 pffiffiffi diatomic gas molecules, respectively (Myong 2004b). The
lf p 2 r25 2r22 ð2 r55 Þ r25
rp;K ¼ 1þ parameter bMis a function of the interfacial interaction
l 2 r22 pð2 r25 Þ 2
parameters KM and the wall temperature T and can be
ð22Þ expressed as bM = KM/kBT. bM plays a crucial role on the
where r22 = r25 = rv and r55 is the coefficient depending reaction constant for gas-surface molecule interaction
on the energy accommodation coefficient rn , lf is defined (Choi and Lee 2008) and it has the simplest expression as:
to be the first-order approximation to the viscosity l as 1
computed by Chapman and Cowling (1970), and lf/l = 1 bM ¼ ð25Þ
4xM Kn
for Maxwellian molecules and lf/l = 0.984219 for rigid-
sphere gas interactions (McCormick 2005). This expres- where xM is a function of the interaction parameters and
sion agrees identically with the analytic equation for the shows very similar to the slip coefficient rp in Maxwell
velocity slip coefficient derived from a different variational model (Myong 2004a). Myong et al. (2005) pointed out that
approach by Cercignani and Lampis (1989). the slip coefficient xM in the Langmuir model is a physical
McCormick (2005) deduced the relationship between parameter of heat adsorption while the accommodation
r55 and rn, and gave the CL gas-surface interaction model coefficient rv in the Maxwell model is a free parameter from
and rigid-sphere gas interaction including rv and rnas: the concept of diffusive reflection. Comparisons of the slip
pffiffiffi flows between the Maxwell model and Langmuir model
p 2 rv KM ð1 rv Þð1 rn Þ using the LBM were reported by Kim et al. (2007) and Chen
rp;M ¼ 0:9687 1 þ 0:1366rv
2 rv 2 rv and Tian (2010). Zhang (2011) reviewed that the slip
ð23Þ models were specified in the boundary treatments as the
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input to match empirical or analytical descriptions. The (1975) considered the KL effect and calculated the slip
boundary slip observed from LBM simulations is more coefficient using the Bhatnagar–Gross–Krook (BGK)
phenomenal than physical as those from other methods, kinetic model and a Maxwell diffuse-specular scattering
such as numerical results from the BE and DSMC method kernel. A simple modified expression of the slip coefficient
(Guo and Zheng 2008; Zhang 2011). S with the accommodation coefficients rv was proposed by:
2 rv
Sðrv Þ ¼ ðSð1Þ 0:1211ð1 rv ÞÞ ð26Þ
rv
5 Slip coefficient and model
where S(1) is the slip coefficient for rv = 1 and equals to
The no-slip boundary condition is assumed to apply at the 1.016, i.e., the value obtained theoretically by Albertoni
solid–fluid interface under normal conditions. However, it et al. (1963). Porodnov et al. (1974) provided the experi-
is well known that at higher Knudsen number this condi- mental data of the slip coefficient and the corresponding
tion is violated and the gas slips at the wall (Dongari et al. values of rv for some gases and showed that the slip
2007). Maxwell (1879) first proposed a first-order slip coefficient is higher than unity for light gases, such as
model to calculate the slip velocity at the wall for atomi- helium and neon.
cally smooth walls. Later many other heuristic extended Gabis et al. (1996) presented a spinning rotor gauge
slip models have been proposed even for atomically rough model to describe the torque that an unbounded gas of rigid
walls and are comprehensively summarized by Karniadakis sphere molecules inducing on a macroscopic sphere and
et al. (2005). Slip models have been proposed to improve introduced an estimation of the slip coefficient by:
Various kinetic models have been reported to calculate the where KM = 1 refers to the rigid-sphere gas interaction.
slip coefficient at the gas–solid interaction. Loyalka et al. When rv = 1, the slip coefficient SM = SRS = 0.97576
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and SCL = 0.97577 for the Maxwell model, rigid-sphere nonlinear least-squares method (Maurer et al. 2003), that
model and CL model, respectively, while SS = 0.98733 yields:
obtained by Siewert (2003).
S ¼ 1 þ Aexp exp
j Kn þ Bj Kn
2
ð31Þ
Sharipov (2011) compared the results corresponding to
the CL scattering law with the data obtained by applying where Aexp
j and Bexp
are the coefficients obtained by the
j
the BE and presented the following expression: nonlinear least-squares Marquard–Levenberg algorithm, in
1:772 which j denotes the order of the polynomial (Ewart et al.
SS ¼ 0:754 ð30Þ 2007b).
rv
Aubert and Colin (2001) considered a pressure-driven
For a single gas under the assumption of diffuse flow in the rectangular microducts and calculated the sec-
scattering, Sharipov and Seleznev (1998) and Sharipov ond-order model for slip flow between parallel plates using
(2011) summarized that the values of the slip coefficient the boundary conditions from Deissler (1964), the poly-
based on all kinds of models vary in the range of nomial expression is given by:
0:968 rp 1:03. Sharipov and Seleznev (1998), Siewert
and Sharipov (2002), and Sharipov (2011) also presented KnO Kn2
S ¼ 1 þ aAC1 þ aAC2 2 O InðPÞ ð32Þ
and reviewed the data on the slip coefficient based on Pþ1 P 1
diffuse-specular scattering kernel for the Maxwell boundary where aAC1 and aAC2 are the coefficients depending on the
condition. Table 3 presents the numerical results based on ratio of the cross-section of the microchannel and
various models, including the BGK model (Loyalka et al. momentum accommodation coefficient.
1975; Sharipov 2011), the LBE (Wakabayashi et al. 1996; Roohi and Darbandi (2009) presented an IP (information
Siewert 2003), the S-model (Siewert and Sharipov 2002) preservation)-based slip coefficient model as:
and the MC-model (McCormick 2005). It can be seen that
KnO Kn2O
the values of the slip coefficient obtained from the LBE S ¼ 1 þ aR1 þ ð51InðPÞ þ 34:07InðaR1 ÞÞ
P þ 1 1 P2
(Wakabayashi et al. 1996; Siewert 2003) are slightly
smaller than unity. The BGK models (Loyalka et al. ð33Þ
1975; Sharipov 2011) provide the value slightly larger where aR1 and aR2 are defined as aR1 ¼ 11:72 þ
than unity. Therefore, the analytical solutions (26) and 1þ0:89Kn þ4:7Kn2
42:253
(29a, 29b) provide values close to those numerical results 1þð0:21þ0:47=KnO Þð0:21þ0:47P=KnO Þ and aR2 ¼ P2 þ0:89KnOPþ4:7Kn
O
2.
O O
and can be successfully used in practical predictions and For the general form of the second-order boundary
calculations. condition, the coefficient is deduced using the IP scheme as:
KnO Kn2
5.1.2 Polynomial expansion approach S ¼ 1 þ 12C1 þ 12C2 2 O InðPÞ ð34Þ
Pþ1 P 1
In slip flow and near transition regimes, the experimental where KnO is the Knudsen number outlet, P is the ratio of
mass flow rate data can be fitted with first- and second- the inlet and outlet pressures, and C1 = (2 - rv)/rv and
power polynomial forms of the Knudsen number using a C2 = 9/8 (Aubert and Colin 2001). Arkilic et al. (1997)
Table 3 Comparison of the slip coefficient for the Maxwell boundary condition
Sðrv Þ
rv BGK LBE (Wakabayashi S-model (Siewert and LBE MC-model BGK
(Loyalka et al. 1975) et al. 1996) Sharipov 2002) (Siewert 2003) (McCormick 2005) (Sharipov 2011)
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proposed an experimental investigation on the gaseous slip improved slip model, called the stress-density ratio model,
flow in long microchannels for accurately measuring the applicable especially to transition regime by the DSMC
mass flow and obtained the first-order slip coefficient method for the ultra-thin film gas lubrication. The shear
KnO stress on the wall is produced by the definition in DSMC
S ¼ 1 þ 12C1 Pþ1 .
as:
5.1.3 DSMC method sN ¼ ng ðmuN1 uN2 Þjwall ð40Þ
From the definition of slip flow, the slip coefficient should where uN1 and uN2 are the components of molecular
be affected not only by the macro-parameters (temperature velocity. The slip coefficient can be obtained from DSMC
and speed of solid wall), but also the micro-parameters (the results as:
mass, diameter and number density of gas molecules) (Pan us q
S¼ ð41Þ
et al. 1999). During the past decades, a series of test cases sN
were performed using the DSMC method to study the slip
coefficient. 5.1.4 Linearized Boltzmann method
Pan et al. (1999) synthesized Bird’s conclusion (Bird
1994) and discussion from Beskok and Karniadakis (1994) There are considerable successes in extending the N–S
and expressed a general slip coefficient in the form: equations with high-order slip boundary condition into the
transition regime. In order to provide analytical expressions
S ¼ SðUW ; kB T; m; d; n; Lc Þ ð35Þ for the first- and second-order velocity slip coefficients,
where Lc is the local characteristic length, and kBT is the Cercignani and Lorenzani (2010) and Lorenzani (2011)
average kinetic energy parameter. The slip coefficient is considered the LBE for HS molecules and used the CL
easy to determine from the numerical calculations using the scattering kernel to describe the gas-wall interaction. The
DSMC method (Bird 1994) by: BE can be linearized about a Maxwellian fM0 by:
16 2
experiential parameter. 9 pat BL CL DL , in which the relative parameter was
Mcnenly et al. (2005) selected the N–S solution from the reported by Lorenzani (2011). The first-order and second-
pffiffiffi
family that best fits the DSMC data by performing a linear order coefficients can be written as C1 ¼ rL0 rL1 =ð3 pÞ and
least-squares fit of the DSMC velocity profile with: C2 ¼ rL0 rL2 =ð3pÞ according to the solution of the N–S
Suy equations obtained by S ¼ 1 þ 6C1 Kn þ 12C2 Kn2 . To check
G c ¼ c ð38Þ the reliability of the analysis approach, Lorenzani (2011)
Syy c
compared the first-order slip coefficient with the highly
where Suy and Syy are the relative position and velocity accurate numerical results from Siewert (2003), and found that
parameters in the DSMC cell. The slip coefficient can be the BGK first-order slip coefficients are similar to those
determined to best capture the non-equilibrium flow as: determined by the solution of the BE for HS molecules. The
second-order slip coefficients were found to be significantly
DU Gc
S¼ ð39Þ dependent on the interaction models, such as the Maxwell
2ðð2 rV Þ=rV ÞGc Kn
kernel and the CL model. In the case of a fully diffusive
where DU is the velocity drop between the lower and upper boundary rv = 1, Cercignani and Lorenzani (2010) predicted
wall velocities. Ng and Liu (2002) devoted to develop an the model performs well even further into the transition
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regime, while Hadjiconstantinou’s second-order slip model where NK is the index of the fluid lattices. The problem of
(Hadjiconstantinou 2003) and Lockerby’s Maxwell–Burnett discrete effects in the kinetic boundary condition was also
slip model (Lockerby et al. 2004) only capture the flow addressed by Guo et al. (2007a, b). Watari (2009) con-
accurately up to Kn \ 0.4 and Kn B 1.60, respectively. ducted the velocity slip simulations in the slip flow regime
using a multispeed finite-difference lattice Boltzmann
5.1.5 Lattice Boltzmann method (LBM) method (FDLBM).
Modeling of microscale and nanoscale flows has been an 5.1.6 Experimental measurement
active application area for the research of LBM (Zheng
et al. 2006; Li and Kwok 2003). Cornubert et al. (1991) Kuhlthau (1949) presented the experimental setup con-
presented the first analysis of the slip velocity using the sisting of a rotating inner cylinder with radius R1 and sta-
LBM, in which the slip velocity was demonstrated ana- tionary outer cylinder with radius R2 with a low pressure
lytically and numerically for the bounce-back and specular gas in the gap. Agrawal and Prabhu (2008a) examined the
reflection boundary conditions. This method has proved to experimental data of Kuhlthau (1949) and deduced the slip
be effective when dealing with microflow of moderate coefficient in the form:
Knudsen number and has some success at predicting the
1
KL (Aidun and Clausen 2010). When the gas microflows dTr 1 dTr 1
S¼ þ 4plLr CF ð2kÞ 3
beyond the slip flow regime, a higher-order LBM needs to dxr R22 R21 dxr R1 þ R32
provide a quantitative prediction as well as to reproduce the ð46Þ
presence of the KL (Shan et al. 2006; Ansumali et al.
2007). Recently, Zhang (2011) reviewed and discussed the where Tr is the torque, xr is the angle velocity, Lr is the
models and applications of the LBM for microfluidics. length of the inner cylinder, and CF is the correction factor
Kim et al. (2008) presented an analytic solution of the and CF = 1.91. Agrawal and Prabhu (2008a) suggested
D2Q9 LBE for Poiseuille flow. To obtain a correct value of from the analysis on Kuhlthau’s data that S = 0.13 for
the slip coefficient, the effective diffuse scattering condi- Kn \ 0.1 and S = 1.70 for 0.1 \ Kn \ 8.3.
tion was introduced by combining the diffuse scattering Maurer et al. (2003) performed gas flow experiments in
boundary condition and the bounce-back scheme. The slip a shallow microchannel and presented new sets of accurate
coefficient in the general form of the second-order measurements for a well-resolved range of Knudsen
boundary condition can be given by: number. The slip factor was defined as:
rffiffiffi 12Qv lPO L
6 1 rK 4 S¼ ð47Þ
S¼1þ Kn þ f ðrq Þ Kn2 ð44Þ DPPm wb3
p 1 þ rK p
where Qv is the volumetric flow rate, PO is the outlet
where rK is the fraction of gas particles reflected with the pressure, DP is the pressure drop, Pm is the average pres-
bounce-back rule and it influences only the first-order slip sure, w, b, and L are the width, thickness and length of the
coefficient when f ðrq Þ ¼ 1. Sbragaglia and Succi (2005) channel, respectively.
suggested that the construction of the body force in the A development of slip factor was proposed in the form
LBM should be modified to adjust the second-order slip
coefficient. When the energy flux term is taken into S 1 þ 6C1 Kn þ 12C2 Kn2 ð48Þ
account, f ðrq Þ satisfies f ðrq Þ ¼ 1=rq , in which rq is set to Agrawal and Prabhu (2008b) summarized the
0.9 for matching the mass flow rate of the LBE to the experimental measurements of the values of slip
second-order of Knudsen number. A general lack of coefficients as reported by Sreekanth (1969), Ewart et al.
agreement exists in the definition of Kn and k in the LBMs, (2007a, b), Maurer et al. (2003), and Yamaguchi et al.
which should be considered to allow comparison with (2011), as listed in Table 4. The principle of measurement
existing experimental and analytical results (Guo et al. applied by Ewart et al. (2007b) is similar to that of Maurer
2006; Aidun and Clausen 2010). et al. (2003). Since the experimental measurements had
For the modified LBE with the effective diffuse scat- been proposed to certain geometric microdevice or
tering boundary condition (Kim et al. 2008), the normal- material, the differences among them are unavoidable.
ized slip velocity with discrete lattice effects can be written For instance, Ewart et al. (2007a) performed experimental
as: measurements in silica microtube with diameter of
rffiffiffi
6 1 rK 41 2 1 25.2 lm, which is different from Yamaguchi et al. (2011)
S¼1þ Kn þ Kn ð45Þ with diameter of 320 lm.
p 1 þ rK p rq NK
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5.2 First-order slip model Lilley and Sader (2008) had presented the modifications of
Maxwell’s argument. Then, the second-order or higher-
Velocity field in the slip flow regime can be determined order slip coefficient is believed to play an important role
from the N–S equations subject to the velocity slip in simulating rarefied gas microflows at larger Knudsen
boundary condition. Neglecting the thermal creep effects, numbers. Solutions of the BE for the slip coefficients were
the first-order slip velocity boundary condition is given by originally obtained for the significantly simpler BGK
(Maxwell 1879): model. Early work by Cercignani (1988) and recent results
for the HS gas show that the first-order coefficients are
oU
Us Uw ¼ C1 Kn ð49Þ fairly insensitive to the gas models (e.g., HS, BGK)
on S
(Hadjiconstantinou 2006).
where the value of slip coefficient, originally derived by Figure 6 shows the comparison of experimental data of
Maxwell, is C1 = 1. Maurer et al. (2003) with those derived from other slip
Bird (1994) used the DSMC method to simulate the models in the literature. It can be seen that Ewart’s model
Couette flow using the VHS molecular model and con- (Ewart et al. 2007b) and Maurer’s model (Maurer et al.
cluded that the slip velocity is very close to koU=on. The 2003) agree well with the scattered experimental data
conclusion has a very strong attraction for researchers who (Maurer et al. 2003) in the entire range of investigated
prefer to use the N–S equations with velocity slip boundary Knudsen numbers. Porodnov’s model (Porodnov et al.
conditions to investigate the slip flow characteristics (Pan 1974), Colin’s model (Colin et al. 2004) and Pan’s model
et al. 1999). More accurate values for the slip coefficient (Pan et al. 1999) underestimate the slip coefficients due to
have been determined using BE, DSMC, MD simulations, their lacking of the second-order term compared to the
and experimental measurements (Maurer et al. 2003; experimental data (Maurer et al. 2003). Colin’s model
Bahukudumbi et al. 2003; Bahukudumbi and Beskok 2003; (Colin et al. 2004) with the smallest first-order coefficient
Agrawal and Prabhu 2008a). Table 5 provides the values (C1 = 1.02) performs the worst prediction. Aubert and
and modified formulations for the slip coefficient in the Colin’s model (Aubert and Colin 2001) underestimates the
literature. The above research represents only a fraction of slip coefficient at smaller Knudsen number and performs
the proposed models, yet almost all of them can be con- well when Kn [ 0.8. The theoretical analyses proposed by
sidered an extension of Maxwell’s original model. Fichman and Hetsroni (2005) agree very well with the
Lockerby et al. (2005a) pointed out that the kinetic experiments in the range of low Knudsen numbers
theory and molecular simulations indicate Maxwell’s first- (Kn \ 0.2). The reason might be, for higher Knudsen
order slip coefficient C1 = 1 is sometimes not quite numbers, the interaction of molecules with the opposite
accurate and overestimates the amount of microscopic wall should be taken into account as well. Figure 6 also
actual slip velocity (Li et al. 2011). By using an approxi- illustrates that the experimental and analytical slip coeffi-
mate method in the kinetic theory, Loyalka (1971), cients increase much faster than the prediction obtained by
Loyalka et al. (1975), Bahukudumbi et al. (2003) and the first-order slip theory (Porodnov et al. 1974; Pan et al.
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Table 5 Some values and modified formulations for the first-order slip coefficients
References Value/formulation Approach Limited range
1999; Colin et al. 2004) in the transition regime, which They often validated their second-order slip models by
indicates that the contribution of second-order slip should demonstrating the capability of predicting an accurate flow
be taken into account. rate (Li et al. 2011).
So far, there is no general agreement on the values of the
slip coefficients. Slip coefficient in the slip models usually
5.3 Second-order slip model
is investigated in two different means taking rv into
account. One way is to fix the value of rv. For the case of
Various slip models have been proposed to calculate the
rv = 1, Table 6 presents a comparison of the values of the
slip velocity at higher Knudsen number in the literature. A
second-order slip coefficients that have been proposed in
simple extension of Maxwell’s model with the second-
the literature. In Hadjiconstantinou’s model (Hadjicon-
order term can be written as:
stantinou 2003), C2 = 0.61 is used for local velocity dis-
2 rv oU Kn2 o2 U tribution and C2 = 0.31 for mean velocity and friction
Us Uw ¼ Kn þ ð50Þ
rv on s 2 on2 s factor. For the case of the lubrication microbearings, the
slip model and coefficient are chosen differently according
For isothermal flows, the slip velocity of all the second-
to their configurations, gas film characteristic length and
order slip models can be expressed in a general form when
surface effect (Ng and Liu 2002; Zhang et al. 2009, 2010a,
rv = 1, yields
b, 2011; Chen and Bogy 2010).
2
In Table 6, Lockerby et al. (2004) proposed that the
oU o U
Us Uw ¼ C1 Kn C2 Kn2 ð51Þ value of the second-order slip coefficient should be in the
on s on2 s
range of 0.145–0.19, which depends on the Prandtl number
Inclusive reviews of the slip boundary condition choices of the gas flow but irrespective of the values of the
were also provided by Colin (2005), Barber and Emerson accommodation coefficient. The estimation approach pro-
(2006), Dongari et al. (2007), Tang et al. (2007a, b), Weng posed by Lorenzani (2011) and Cercignani and Lorenzani
and Chen (2008), Cao et al. (2009), Chen and Bogy (2010). (2010) to obtain the second-order slip coefficients seems
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pffiffiffi
0:003
p oU
Us Uw ¼ 0:49 Kn
2 on s
p 0:5335 o2 U
oU 0:65 2 o U
Us Uw ¼ 1:15Kn 0:25Kn Kn ð55Þ
on s on2 s
Fig. 7 Comparison of the non-dimensional flow rate QN as a function Beskok and Karniadakis (1996) and Beskok and
of the Knudsen number Kn for various slip models Karniadakis (1999) also provided the non-dimensional
boundary condition from Eq. (20) in the form
new model to describe the gas flow behavior in microtubes
2 rv oU Kn2 o2 U
avoiding time-consuming calculations and compared with Us Uw ¼ Kn þ þ
rv on s 2 on2 s
the results obtained by Loyalka (1969). The high-order slip
flow boundary condition can be written as: ð56Þ
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where ðo=onÞrepresents the gradients normal to the wall meaning is the velocity flux into the surface divided by the
surface. velocity of flow field on the surface. Beskok and Karni-
Based on the asymptotic analysis, Beskok and Karni- adakis (1999) determined the value bBK and provided some
adakis (1999) and Karniadakis and Beskok (2002) devel- results for transition and free-molecular regimes.
oped a physics-based empirical flow model and proposed Xue and Fan (2000) put a step forward and presented a
the following general velocity slip boundary condition for high-order slip expression with replacing Kn by tanhðKnÞ
the compressible flow: as:
2 rv Kn oU 2 rv oU
Us Uw ¼ ð57Þ Us Uw ¼ tanhðKnÞ ð58Þ
rv 1 bBK Kn on s rv on s
where bBK is a generalized slip coefficient, which is an This statement involves only the first derivation of the
empirical parameter to be determined either experimentally velocity, which leads to results close to those calculated by
or from LBE or DSMC data. Moreover, its physical DSMC method even in relatively high Knudsen number.
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5.5 Hybrid slip model 1 8Cð4:5 xm Þlref ðkB Tref =pÞxm 0:5
kðVHSÞ ¼ pffiffiffiffiffiffiffiffiffiffi
2ng p 15ðmg =pÞ1=2 ðkB Tref Þxm
5.5.1 Effective mean free path (MFP)
xm 0:5
Tref
ð64Þ
(1) Various definitions The MFP k of gas molecules is an T
average distance traveled by a molecule before colliding
where lref and Tref are the reference conditions, xm is a
with another one in the equilibrium state and also can be
constant which determined by the type of the gas and can
defined as:
be obtained from experimental data.
c To describe the actual transport properties, Koura and
k¼ ð59Þ
t Matsumoto (1992) introduced the variable soft sphere
where t is the collision frequency. (VSS) model as:
Peng et al. (2004) incorporated Bird’s definition (Bird 1 16Cð4:5 xm Þlref ðkB Tref =pÞxm 0:5
1994) and the approximation result from Chapmann and kðVHSÞ ¼ pffiffiffiffiffiffiffiffiffiffi
2ng p 5ð#m þ 1Þð#m þ 2Þðmg =pÞ1=2 ðkB Tref Þxm
Enskog (Chapman and Cowling 1970) by taking the spe-
xm 0:5
cific gas constant RP ¼ kB =m and obtained the following Tref
ð65Þ
formulation: T
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi where #m is also a value which can be determined by the
2pRP T
k¼l ð60Þ same method for xm . For air, in the VSS model #m ¼ 1:5775
2p
and xm ¼ 0:7, while in the VHS model #m ¼ 1:0 and
Using the Chapman and Enskog method, the MFP k can xm ¼ 0:7. When #m ¼ 1:0 and xm ¼ 0:5, the expression of
be rewritten regarding of the viscosity as: the VSS model reduces to the HS model (Sun et al. 2002).
rffiffiffiffiffiffiffiffiffiffiffi Bird (1994) and Koura and Matsumoto (1992) compared
16 p
k¼ vg ð61Þ the VHS model and VSS model with the HS model and
p 2RP T
derived the expressions for the effective MFP as:
where vg is the kinematic viscosity. Cð4:5 xm Þ
When collisions due to the intermolecular interaction kVHS ¼ ð66Þ
6pxm 0:5
are not well defined, Cercignani (1988) proposed to use the
viscosity-based MFP and
rffiffiffiffiffiffiffiffiffiffiffi #m Cð4:5 xm Þ
p kVSS ¼ ð67Þ
k¼ vg ð62Þ ð#m þ 1Þð#m þ 2Þpxm 0:5
2RP T
Sun et al. (2002) presented new analytical slip models
The viscosity-based MFP is very close to the exact result
incorporating the VHS and VSS molecular effects and
for HS molecules (Kim et al. 2008). The MFP for the BGK
obtained the slip coefficients for the VHS model with
molecules can be defined as:
C1 = 0.62228 and C2 = 0.3872 and VSS model with
rffiffiffiffiffiffiffiffiffiffiffi
pRP T C1 = 0.63875 and C2 = 0.408, respectively.
k¼ sg ð63Þ (3) Wall-function approach For an isothermal, incom-
2
pressible flow, Zheng et al. (2006) incorporated the wall-
where sg is the relaxation time in the Boltzmann–BGK function approach into a D2Q9 LBE model and presented
equation. the effective MFP expression as:
In the description of experimental results, the viscosity-
k
based MFP is widely used to define the Knudsen number. keffðZÞ ¼ ð68Þ
However, considering the nature of MFP, the definition of 1 þ 0:7eCL y=k
MFP can be different for various studies, except for HS This approach can be applied to more complex
molecules (Shan et al. 2006). geometries by assuming that the influence of overlapping
(2) Molecular model Although the HS model is widely KLs is additive. Outside the KL, the effective MPF
used for its simplicity, the rate of effective cross-section approaches the MFP in the bulk flow, while, at the wall
decreases directly related to the change of the coefficient of (y = 0), the effective mean free path is 1.7 times smaller
viscosity with temperature. Bird (1994) proposed the than in the bulk flow (Zheng et al. 2006).
modified MFP for the variable hard sphere (VHS) (4) Matthiessen rule The Matthiessen rule had been
model as: widely used to consider the boundary scattering effects on
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electron and phonon transport (Ascroft and Mermin 1976). keffðGÞ ¼ kUðKnÞ
To explain the boundary scattering, the MFP can be eval-
1
uated using this rule as: ¼ k 1 þ ða 1Þea þ ðb 1Þeb
2
1 1 1
¼ þ ð69Þ a2 Ei ðaÞ b2 Ei ðbÞ ð72Þ
keffðMÞ ks kb
where ks is the MFP from molecular scattering and can be where a ¼ y=k, b ¼ ðH yÞ=k, Ei ðxÞ denotes the expo-
pffiffiffi R1
simply calculated as ks ¼ k= 3, and kb is the MFP due to nential integral function defined by Ei ðxÞ ¼ 1 t1 ext dt,
boundary scattering. in which H is the distance between the two parallel plates,
Shen et al. (2007) took the film thickness H/2 for kb to and U is a monotonous function of Kn and satisfying
account for the fact that there are two overlapping KLs and lim UðKnÞ ¼ 1. Guo et al. (2006) mentioned that the
calculated the effective MFP using the Matthiessen rule Kn !0
(Ascroft and Mermin 1976) as: function UðKnÞ derived by Stops is very complicated and is
difficult for practical applications. The reason may be that
1 1
keffðSÞ ¼ pffiffiffi þ ð70Þ it contains an exponential integral function Ei ðxÞ, which
k= 3 H=2 needs a numerical integration and leads to considerable
The effective MFP should be modified with different computations (Li et al. 2011). Moreover, heuristic
geometries. Chen and Bogy (2010) remarked that Shen’s expressions of effective viscosity should be proposed to
slip model (Shen et al. 2007) is also a second-order type enable the computation efficient and the implementation
without considering the effective MFP and the rule is ease.
seldom used for rarefied gas dynamics. The application of Guo et al. (2006) presented the expression of the
the Matthiessen rule to rarefied gases is not supported by effective MFP with an approximation to replace Stops’
the BE (Chen and Bogy 2010). expression as follows:
pffiffiffi
(5) Probability distribution function approach The 2 3=4
influence of a solid wall on the MFP of the gas molecules keffðGZÞ ¼ kUðKnÞ ¼ arc tan 2Kn k ð73Þ
p
can be analyzed by considering the probability of the free
path of a gas molecule. The idea of using transport Integrating the density function wðrÞ, Arlemark et al.
R
parameters that are influenced by an effective MFP was (2010) applied a probability function UðrÞ ¼ wðrÞdr and
presented by Stops (1970). Many researchers (Stops 1970; developed a three-dimensional probability function-based
Peng et al. 2004; Guo et al. 2007a, b, 2008; Arlemark et al. effective MFP keffðAÞ as:
2010; Dongari et al. 2011b) had paid more attention on ( "
deducing the effective MFP for various gas flows in 1 a X7
a
keffðAÞ ¼k 1 e þ eb þ 4 ecos½ð2i1Þp=28
MEMS/NEMS. 82 i¼1
Figure 9 shows the distribution of the molecule free path
X
7 b X
6
a
in terms of the molecule traveling a distance r. Stops þ4 ecos½ð2i1Þp=28 þ 2 ecosðpi=14Þ
(1970) presented the free path of a molecule following a i¼1 i¼1
#)
probability distribution function X
6 b
1 r þ2 ecosðpi=14Þ ð74Þ
wðrÞ ¼ exp ð71Þ i¼1
k k
Using this probability distribution function, Guo et al. Comparison of both effective MFP models (Stops 1970;
(2007a, b) and Guo et al. (2008) derived a geometry- Arlemark et al. 2010) with MD simulation data (Dongari
dependent effective MFP for the two parallel plates, i.e., et al. 2011a) shows that both models are useful only up to
Knudsen numbers of about 0.2 (Dongari et al. 2011b).
y Moreover, the Stops’ probability distribution is only valid
under equilibrium conditions (Dongari et al. 2011a, b).
To extend the N–S–Fourier (N–S–F) equations used for
θ+ the gas flows at microscales and nanoscales in the transi-
r+
tion regime, Dongari et al. (2011b) proposed a power-law-
θ r H
r− based effective MFP model. For non-equilibrium gas, a
θ− power-law form of the distribution function with diverging
O higher-order moments was hypothesized and expressed as:
wðrÞ ¼ CD ðaD þ r Þnn ð75Þ
Fig. 9 A molecule confined between two planar walls with spacing H
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where aD and CD are constants with positive values, which MFP and viscosity are two interactive parameters and
are determined through the zero and first moments, and many research works pay more attention on their combina-
exponent nn can be obtained by making one of the higher- tions for gas microflows with different boundary conditions.
order moments divergent. Their relationship can be expressed as (Dongari et al. 2010)
The effective MFP based on the power-law distribution l
keff ¼ eff k ð79Þ
function can be given by (Dongari et al. 2011b): l
8 2
1n 39
8
>
>
> y 1n H y 1n X y >
>
>
> 6 1þ þ 1þ þ4 1þ 7>>
>
>
> 6 a a a cos½ð2i 1Þp=32 7 >
>
> 6 i¼1 7>>
>
>
< 6
7 >
=
6
1 6 X8
Hy 1n X7
y 1n 7
keffðAÞ ¼ k 1 6 þ4 1þ þ2 1þ 7 ð76Þ
> 96 6 a cos½ð2i 1Þp=32 a cosðpi=16Þ 7>
>
> i¼1 i¼1 7>>
>
> 6
1n 7 >
>
>
> 6 X7 7 >
>
>
> 4 H y 5 >
>
>
: þ2 1þ >
;
i¼1
a cosðpi=16Þ
The power-law-based effective MFP (Dongari et al. To distinguish the differences between some heuristic
2011b) was validated against MD simulation data (Dongari effective MFP and viscosity models for gas microflows
et al. 2011a) up to Kn = 1, and also compared with the from different viewpoints, the effective viscosity models
theoretical models from Stops (1970) and Arlemark et al. are reviewed in this section.
(2010). (1) Various definitions For HS molecules, the coefficient
To study the unidirectional flow of the rarefied gas near of viscosity l can be obtained by using the Chapman and
the boundary region, Peng et al. (2004) presented a nano- Enskog method (Chapman and Cowling 1970) as:
scale effect function keffðPÞ =k to describe the gas behavior pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
5 pmkB T
between two parallel plates based on the kinetic theory. l¼ ð80Þ
16 pdg2
The probability density of the direction distribution of
molecule velocity is: The bulk viscosity of dilute gases for the HS model can
HðhP ; bP Þ ¼ sin hP =4p ð77Þ also be derived from the Chapman–Enskog theory and
given by:
where hP, bP are random variables uniformly distributed in
5p
the whole space, and the integration of HðhP ; bP Þ in the l¼ cqk ð81Þ
32
whole integration space is clearly equal to 1. The effective pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
MFP considering the nanoscale boundary effects can be Where c ¼ 8kB T=pmg and q ¼ p=ðRp TÞ.
expressed as (Peng et al. 2004): A simple kinetic theory based result proposed by Max-
k
well for the bulk viscosity is (Pollard and Present 1948):
k 1 4h ; ðh kÞ
keffðPÞ ¼ ð78Þ
k 3h h
In h
k ; ðh kÞ
1
4k 2k l ¼ cqk ð82Þ
3
Chen and Bogy (2010) argued that the modified MFP is
not necessary for Fukui and Kaneko’s (FK) model (Fukui Alexander et al. (1998) used the Green–Kubo theory to
and Kaneko 1988) in which the MFP is characterized by evaluate the transport coefficients in DSMC and derived
the BGK gas molecules in the equilibrium state. from the dilute gas Enskog values for the viscosity as:
rffiffiffiffiffiffiffiffiffiffiffiffi
5 mkB T 16 L2x
5.5.2 Effective viscosity l¼ 1 þ ð83Þ
16dc2 p 45p k2
Gas viscosity is an important property to account for the where dc is the collision diameter, Lx is the width of the
momentum exchange between gas molecules. The effective cell.
viscosity is a mathematical construct with no connection to Hadjiconstantinou (2000) considered the convergence
real gas properties, and its value will change with flow with respect to a finite time step when the cell size is
geometry (Lilley and Sader 2008). negligible and examined the effects of the discretization
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error in DSMC calculations of the viscosity using the almost independent of the surface accommodation aR, and
Green–Kubo theory. The resulting expression of the vis- DR = -0.293 and ER = 0.531 for HS model, and DR =
cosity becomes: -0.328 and ER = 0.612 for the BGK model, respectively
rffiffiffiffiffiffiffiffiffiffiffiffi" # (Reese et al. 2007). Moreover, the effective viscosity does
5 mkB T 16 ðcm DtÞ2
l¼ 1þ ð84Þ not generate artificial stresses in the KL.
16 p 75p k2 (3) Karniadakis-style model Some heuristic effective
viscosity models had been proposed in previous studies for
where cm is the most probable speed of the gas molecules.
gas microflows from different viewpoints (Beskok and
(2) KL-based model Considering the effect of KL, Lilley
Karniadakis 1999; Sun and Chan 2004; Roohi and Dar-
and Sader (2008) investigated gas flow for small Knudsen
bandi 2009; Michalis et al. 2010), which can also be
numbers and suggested a power-law dependence of vis-
expressed in the form:
cosity on the dimensionless distance from the solid surface
as: leff ¼ lWðKnÞ ð89Þ
y~1dy~ where WðKnÞ has different form. The researchers (Zheng
leffðLSÞ ¼ l ð85Þ
Cy~dy~ et al. 2006; Guo et al. 2006) found that the viscosity cor-
rections can improve the numerical accuracy to some
where Cy~ and dy~ can be obtained from the LBE and the extent, but still cannot give satisfactory results for the gas
empirical determinations of the functional dependencies of flows at a higher Knudsen number (Li et al. 2011).
Cy~ and dy~ on the thermal accommodation coefficient rT are Karniadakis et al. (2005) considered the rarefaction
Cy~ðrT Þ ¼ 1:58 0:33rT and dy~ðrT Þ ¼ 0:69 þ 0:13rT effects and proposed a hybrid formula for the viscosity
(Lilley and Sader 2008). The DSMC calculations showed coefficient as follows:
that the KL for VSS molecules is very similar to that for
1
the HS ones and the power-law description is weakly leffðKÞ ¼ l ð90Þ
dependent on the molecular model (Lilley and Sader 2007). 1 þ aK Kn
To evaluate the isothermal microflows, Lockerby et al. where aK is a coefficient and should be adjusted with a
(2005a) proposed a wall-function type of viscosity in the complicated inverse hyperbolic-tangent function. Beskok
KL derived from a curve fit to the KL velocity profile and Karniadakis (1999) first suggested an expression for
originally derived by Cercignani (1990) as: the viscosity in the transition regime and conducted
h i1 numerical computations of flow in cylinders and channels
leffðLÞ l 1 þ 0:7ð1 þ y~Þ3 ð86Þ
using the N–S formulation complemented with a slip
boundary condition at aK ¼ 2:2. Sun and Chan (2004)
When a part of the molecules is reflected diffusively and
reported that they found good agreement of their model
another one is reflected specularly, Fichman and Hetsroni
with DSMC and with the LBE results at aK ¼ 2.
(2005) presented the effective viscosity as:
r Michalis et al. (2010) investigated the rarefaction effect
l 2v þ ð1 rv Þ~
y ð~ y\1Þ on gas viscosity via DSMC modeling of rarefied channel
leffðFHÞ ¼ ð87Þ
l ð~
y [ 1Þ flows and also found such an expression with a Bosanquet-
Reese et al. (2007) reviewed the Fichman and Hetsroni type approximation:
model (Fichman and Hetsroni 2005) that it cannot capture 1 1 1
¼ þ ð91Þ
the asymptotic form of the velocity profile in the KL near leffðMÞ l l1
the surface and provided the expression for the effective
viscosity required to reproduce the KL structure within an They confirmed this expression through a direct
N–S–F model as: calculation of the gas viscosity from its shear-stress-
h i1 based definition and the rarefaction factor was found to be
leffðRÞ ¼ l 1 AR ðDR aR þ ER Þð1 þ y~ÞAR 1 ð88Þ aK 2 in the transition flow regime. The result is same as
that presented by Sun and Chan (2004).
where AR , DR and ER are the curve-fitting coefficients. If y~ It can be seen from Stops’ expression that the local
becomes large outside the KL, the effective viscosity tends effective MFP is a function of the distance from the wall.
to be of the actual viscosity and the scaling effect does not However, the Bosanquet-type effective viscosity is inde-
work on it. Reese et al. (2007) obtained the coefficients for pendent of the distance due to its value averaging over the
two different gas molecular models, AR = - 2.719 (HS) cross section. The overall rarefaction effect on the gas
and AR = - 2.025 (BGK), with aR = 1 from the data in the viscosity should be taken into account with the Bosanquet-
literature (Loyalka and Hickey 1989a, b; Wakabayashi type effective viscosity (Guo et al. 2006; Li et al. 2011).
et al. 1996). They also found that the coefficient AR is Michalis et al. (2010) also confirmed that a Bosanquet-type
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expression of effective viscosity describes satisfactorily the the normalized viscosity coefficient decreases as the
dependence of gas viscosity on the Knudsen number in the Knudsen number increases. Differences among the varia-
transition regime. tional solution (Cercignani 1969), empirical model (Veijola
(4) Shear stress model Bahukudumbi et al. (2003) and Turowski 2001), the DSMC results (Bahukudumbi
derived an empirical shear model, which is uniformly valid et al. 2003) and linearized Boltzmann solution (Sone et al.
in the entire Knudsen regime for steady and quasi-steady 1990) are almost invisible and the maximum deviation is
oscillatory Couette flows with an effective viscosity, which less than 1 %. The NS-based viscosity coefficient is close
is given by: to the IP-based model at a lower Knudsen number. NS-
based and IP-based models are similar to the DSMC pre-
1
leff ¼ l þ C1 Kn f ðKnÞ ð92Þ dictions for Kn\0:1, and gradually underestimate the
2
viscosity at higher Knudsen number ranges. The Karni-
where f ðKnÞ can be regarded as a generalized formulation adakis model (Karniadakis et al. 2005) with aK ¼ 2:2
as a function of the Knudsen number from shear-stress agrees with the Sun and Chan model and is close to the IP-
models (Cercignani 1969; Sone et al. 1990; Veijola and based model when Kn [ 0:2. The viscosity coefficient is
Turowski 2001; Bahukudumbi et al. 2003). Table 8 gives not the sole parameter in determining the mass flow and
several expresses of the generalized formulation of f ðKnÞ. should be combined with the slip boundary condition
Considering the IP wall shear stress, Roohi and Dar- (Roohi and Darbandi 2009). Although the various heuristic
bandi (2009) derived an expression for the viscosity coef- effective viscosity models are proposed from different
ficient as a function of Knudsen number in the form: viewpoints, they can contribute to N–S equations for cap-
sw;IP ðxÞ turing the high Knudsen number effects in the transition
leffðRDÞ ¼ ð93Þ regime (Guo et al. 2006).
oVt =on
where Vt is the particle information velocity, sw,IP is the 5.5.3 Gaseous mixture
ðNSÞ ðBÞ ðABÞ
wall shear stress and sw;IP ¼ sw þ sw þ sw þ , in
which the superscripts NS, B, and AB denote the N–S, the Though in practice one meets mixtures more often than a
Burnett, and the augmented Burnett equations, respectively. single gas, there are very few investigations on the slip
The NS-based and IP-based viscosity coefficient coefficient for gaseous mixtures (Sharipov and Kalempa
expressions are expressed as (Roohi and Darbandi 2009): 2003). The slip boundary condition for a gaseous mixture is
h i more complicated due to the slip coefficient for a gaseous
rv þ 6Kn 6Kn2 mixture differing from that for a single gas. The concen-
leffðRDÞ ¼l ð94Þ
NS rv þ 6Kn þ 13:5Kn2 tration gradient near a solid surface causes a slip of the
h i rv þ 0:89Kn þ 4:70Kn2 mixture along the surface. Some slip coefficients were
leffðRDÞ ¼l ð95Þ provided for a mixture obtained by the moment method
IP rv þ 0:75Kn þ 19:98Kn2
applied to BE (Ivchenko et al. 1997; Sharipov and Kal-
where the subscript NS and IP refer to the fact that there empa 2003; Garcia and Siewert 2007). Naris et al. (2004)
have been derived from the NS-based mass flow rate provided an efficient methodology to solve internal flows
relation and the IP simulation data, respectively.Moreover, of binary gaseous mixtures through microchannels over the
Bahukudumbi et al. (2003) pointed that it is not possible to whole range of the Knudsen number. Pitakarnnop et al.
construct a shear stress model from N–S-level constitutive (2010) also verified that the implementation of the linear-
equations that are uniformly valid in the entire Knudsen ized BGK and McCormack models for solving rarefied
number regime (Park et al. 2004). Therefore, in a rarefied flows through microchannels is valid for the gaseous
gas flow system confined by solid walls, the path of gas mixture.
molecule colliding with the walls will be shorter than the The definition of the slip coefficient for a mixture is very
MFP defined in unbounded systems (Tang et al. 2008; Guo similar to that for a single gas. Equation (15) can be
et al. 2006, 2008; Li et al. 2011). As mentioned in the rewritten as (Sharipov and Kalempa 2003)
above two sections, some modifications or corrections on
lv0 ous
the MFP and the viscosity have been developed to reflect us uw ¼ rp ð96Þ
the effect of gas molecule/wall interactions. p on S
Figure 10 shows comparison of the normalized viscosity where v0 is the characteristic molecular velocity of the
coefficients predicted by various methods, such as the HS
mixture and v0 ¼ ð2kB T=mÞ1=2 , in which the mean
DSMC results (Bahukudumbi et al. 2003), linearized
molecular mass of the mixture is defined as:
Boltzmann solution of Sone et al. (1990), and IP simula-
tions by Roohi and Darbandi (2009). It can be found that m ¼ C0 m1 þ ð1 C0 Þm2 ð97Þ
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Bahukudumbi et al. (2003) 0:52969Knþ1:20597 Least-square fitting to the linearized Boltzmann solution of Sone et al. (1990)
0:52969Kn2 þ1:627666Knþ0:602985
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Table 9 Slip coefficients rp of the mixtures (He–Xe and He–Ar) as a function of molar concentration C0 for various models
C0 rp
He–Xe He–Ar
Rigid spheres McCormack model Lennard–Jones (Sharipov LBE (Garcia and McCormack model (Siewert
(Ivchenko et al. 1997) (Garcia and Siewert 2007) and Kalempa 2003) Siewert 2007) and Valougeorgis 2004)
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microflows, some researchers represented the surface multiscale self-affine roughness in a rectangular micro-
roughness as periodic distributions, including sinusoidal, channel. Several probabilistic models, such as the mixed
rectangular, triangular, elliptical and trapezoidal surface model (Fukui and Kaneko 1993), the model with flow
roughness, for two-dimensional simulation (Sun and Faghri factor (Chen et al. 2004) and the striated rough surface
2003; Wang et al. 2005; Ji et al. 2006; Cao et al. 2006; (White 2010) had also been proposed to describe the
Duan and Muzychka 2008; Zhang and Meng 2009; Xiong roughness effects.
and Chung 2010). Sun and Faghri (2003) modeled the In addition, some studies have attempted to extend the
roughness by an array of rectangular modules using DSMC slip models to flows over the curved or rough surfaces
method. Zhang and Meng (2009) analyzed the flow char- (Myong et al. 2005). Table 11 presented a simple overview
acteristics of the microbearing considering the coupled of the roughness model and slip model for several typical
rarefaction and roughness effects with a new second-order microfluidic devices reported in the literature. Although
slip model (Wu 2008). The roughness was described by the some researchers have paid more attention to the roughness
simple sinusoidal waves. For three-dimensional simula- effect on gas flows in different microfluidic devices, the
tions, rectangular prism elements or conical elements were slip boundary condition has been described with a simple
used to express the roughness effect on the gas microflow form (Cao et al. 2006; Duan and Muzychka 2008; Khadem
depending on the roughness element geometry (Hu et al. et al. 2009) or even without consideration (Lilly et al. 2007;
2003; Rawool et al. 2006; Baviere et al. 2006; Lilly et al. Xiong and Chung 2010; Ozalp 2011). Very few experi-
2007; Kunert and Harting 2007). Hu et al. (2003) devel- mental investigations have concentrated on the effect of
oped a three-dimensional finite volume based numerical surface roughness on boundary slip. Neto et al. (2005)
model to simulate the flow in microchannels with rectan- summarized some challenges, including difficult to pro-
gular prism rough elements on the surface. Kunert and duce suitable surfaces of controlled roughness, additional
Harting (2007) used a three-dimensional LB model to undesired changes at the interface, uncertainly associated
simulate the flow in rough microchannels with periodic with roughness and lack of appropriate theoretical
surfaces, including cosines, squares, and triangles. description of the realistic surface roughness, in predicting
However, the rough surface topography is a non- the roughness effect. Therefore, it is necessary to get a
stationary random process (Chen et al. 2009; Xiong and better understanding of the coupled effects of surface
Chung 2010). Some researchers carried out more efforts to roughness and velocity slip on gas microflows. The suitable
model the random roughness even if there are many diffi- and efficient models must be developed and applied to
culties. Croce and Agaro (2004) explicitly modeled the estimate the roughness and velocity slip effects in the
surface roughness through a set of random generated peaks practical engineering design of microfluidic devices in
along an ideal smooth surface in the microchannels. Li MEMS.
et al. (2002) investigated the effects of surface roughness
on the slip flow in long microtubes. The rough surface was 5.6 Tangential momentum accommodation coefficient
represented as a porous film based on the Brinkman- (TMAC)
extended Darcy model. However, Blanchard and Ligrani
(2007) mentioned that slip is independent of the surface As one of the most important parameters for determining
roughness magnitude and mostly due to rarefaction. Bah- the degree of the slip in all of the slip models, the TMAC
rami et al. (2006) developed a model to predict the flow in can be used to characterize the tangential momentum
rough microtubes with a Gaussian isotropic distribution. transport between the gas and wall (Cao et al. 2009) and
Xiong and Chung (2010) combined a bi-cubic Coons patch should be predicted in slip and transition flow regimes for
with Gaussian distributed roughness heights and presented microflows and nanoflows (Agrawal and Prabhu 2008b;
a three-dimensional random surface roughness model to Veltzke and Thoming 2012). Most of the analytical,
investigate the laminar flow in microtubes. Cao et al. numerical, and experimental investigations concerning
(2006) investigated the effect of surface roughness on slip rarefaction effects on gas microflows relate to the TMAC.
flow in microchannels using non-equilibrium MD simula- The TMAC rv is defined as the fraction of gas mole-
tion method. The surface roughness can be modeled by cules reflected diffusively from a solid surface in rarefied
triangular, rectangular, sinusoidal, and randomly triangular gas microflows (Pitakarnnop et al. 2010), and can be
waves. The roughness effect is indicated to be significant expressed as:
for gas microflows at the small Knudsen number. The
si sr
power-law behavior obtained from AFM images for rv ¼ ð101Þ
si
MEMS surfaces was similar to fractal Weierstrass–Man-
delbrot (W–M) surface results (Bora et al. 2005). Chen where si and sr are the average incident and reflected gas
et al. (2009) used W–M function to characterize the molecules, respectively. Since the incident and reflected
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Table 11 Overview of roughness model and slip model for different microfluidic devices
Component Roughness model Slip model References
Microchannel Triangular, rectangular, sinusoidal and random Maxwell’s first-order model (Maxwell 1879) Cao et al. (2006)
triangular wave models
Triangular and random triangular wave models Maxwell’s first-order model (Maxwell 1879) Khadem et al. (2009)
Two-dimensional W–M function-based model No-slip model Chen et al. (2009)
Conical model No-slip model Lilly et al. (2007)
Fractal geometry model Hadjiconstantinou’s second-order model Liu and Ni (2008)
(Hadjiconstantinou 2003)
Microtube Coons surface model No-slip model Xiong and Chung
(2010)
Sinusoidal corrugation model First-order model (Barber and Emerson 2006) Duan and Muzychka
(2008, 2010)
Porous flow model High-order model (Weng et al. 1999) Li et al. (2002)
Microbearing Sinusoidal wave model Wu’s second-order model (Wu and Bogy 2003) Zhang and Meng (2009)
Striated rough surface model No-slip model White (2010)
Three-dimensional W–M function-based model Wu model (Wu 2008) Zhang et al. (2012a, b)
Micropipe Triangular wave model No-slip model Ozalp (2008, 2011)
flows (Albertoni et al. 1963; Loyalka et al. 1975; 5.6.1 Experimental measurement
Wakabayashi et al. 1996), and explained that:
2 The experimental measurement of the TMAC is very
as ¼ 1:016191 pffiffiffi 1:1466 ð103Þ important and valuable. Agrawal and Prabhu (2008b)
p
reviewed different measurement approaches to determine
The value 1.016191 was obtained numerically by the TMAC with various gas–surface combinations and
Loyalka et al. (1975) from the kinetic equation BGK conditions and clearly described the mechanisms of accu-
model under the assumption of a full accommodation of the rate experimental measurements. Finger et al. (2007)
molecules at the wall, whereas Wakabayashi et al. (1996) summarized that the TMACs obtained from most of the
obtained a value of 0.98737 by solving the LBE. Young experimental measurements are in the range of 0.85–1.06,
(2011) obtained the values for the linearized Grad 13- while some researchers had experimentally observed
moment (LG13) and linearized Regularized 13-moment TMACs to be between 0.2 and 1.0 (Gad-el-Hak 1999;
(LR13) BGK models are 0.886 and 0.919, respectively. Karniadakis and Beskok 2002).
Ewart et al. (2007b) obtained the value of as ¼ 0:933
Generally, the determination of the TMAC depends on
0:003 for helium using the first-order fitting, while Graur the mass flow measurement. Table 13 provides several
et al. (2009) obtained the most pertinent values of a1st s ¼ typical experimental measurement methods to obtain the
2nd TMACs for helium from the literature. Even for the same
0:889
0004 and as ¼ 0:956
0005 for the first-order
and second-order treatments, respectively, for nitrogen gas of helium, the TMAC values have differences for dif-
using the same criteria as those performed by Ewart et al. ferent measurement methods and at various Knudsen ran-
(2007b). Table 7 also provides various values and ges. Although many experiments are performed to
formulations of slip coefficients concerning as obtained determine the TMAC, there has no methodology, which
from different approaches. Barber and Emerson (2002) can predict the TMAC for a given set of conditions. The
concluded that the different solutions between the BE and difference and uncertainty in the measurements unavoid-
Maxwell estimation of as lead to considerable confusion ably lead to widely varying TMACs. Therefore, the results
and prevent the clear understanding of gas microflows. of these previous works on the measurements of the
As listed in Table 12, it can be observed that the obvious TMACs cannot be directly used in a quantitative way for
differences among the various experimental and data-fit- micro- and nanoflows.
ting approaches. For the first- and second-order polynomial To verify the differences from several methods, Yam-
fitting approaches, the TMAC of second order is closer to aguchi et al. (2011) employed the constant-volume method
unity than that of first order. The different values of rp and presented by Arkilic et al. (2001) and Ewart et al. (2007b)
rv obtained for various gases indicate that nature of the gas to measure the mass flow rate through a single microtube
plays a significant role on the slip characteristics in the gas and deduced the TMAC from the slip coefficients using the
microflows. Maxwell model, Loyalka model (Loyalka et al. 1975) and
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Table 12 Experimental coefficients obtained from the first-order and second-order polynomial fitting reported by Maurer et al. (2003), Ewart
et al. (2007a, b) and Graur et al. (2009)
Gas r1st
p r2nd
p r1st
v r2nd
v
Knudsen range References
Sharipov model (Sharipov 2004). As listed in Table 14, the 5.6.2 Analytical model
TMACs are smaller than unity, and rLv and rSv are more
accurate than rM v from the viewpoint of the kinetic theory.
Since it is currently impossible to make a direct measure-
The results show that the differences in gas species are ment of the TMAC, the TMAC value should be inferred by
small and a little difference in two geometry conditions other methods. Barber and Emerson (2002) suggested that
(Yamaguchi et al. 2011). the TMAC varies with the Knudsen number based on an
Barber and Emerson (2002) verified that the TMAC is analytical model from the experimental observations
substantially a function of molecular weight of the gas, (Arkilic et al. 2001; Maurer et al. 2003). The analytical
energy of the incoming molecules, wall material, and sur- model to calculate TMAC should consider the consistent
face roughness. The surface roughness plays a significant relationship between the slip coefficients and slip models
effect on the TMAC. The TMAC values can vary about (Agrawal and Prabhu 2008b; Cao et al. 2009).
15 % with the surface roughness effect even for the same From the determination of the first-order slip coefficient
surface material (Thomas and Lord 1974; Jang and C1 reviewed in Sect. 4, one can deduce the TMAC using
Wereley 2006). However, there are two kinds of contro- the following formulation:
vertible results. On the one hand, Blanchard and Ligrani 2
rv ¼ ð104Þ
(2007) designed and performed experiments to measure the C1 þ 1
TMAC on the walls with different rough surfaces, and
found that a smaller value of TMAC is obtained for the Maurer et al. (2003) presented a formulation of the
rougher surface. A strong dependence of the TMAC on the TMAC based on the regression analysis method (Arkilic
surface roughness can be seen from Table 15 for both air et al. 2001) and gave the determination expression as:
and helium. Turner et al. (2004) also obtained that the 12Kn
rv ¼ ð105Þ
effects of surface roughness on friction factors, accom- S þ 6Kn 1
modation coefficients, and slip velocity are generally where the slip coefficient S can be referred to the mass flow
insignificant. On the other hand, Chew (2009) summarized rate. Maurer et al. (2003) suggested that it allows reducing
the comparison of TMAC measurements from the litera- the uncertainty on the determination of the TMAC by
ture, as listed in Table 16. In the comment, it can be taking the second-order term into account. Moreover,
observed obviously that most of the TMACs are signifi- Eq. (30) is also an appropriate expression for calculating
cantly below unity, but the TMAC increases above the TMAC from the slip coefficient definition (Sharipov
unity for rough surfaces. Sun and Li (2008) verified that 2011). However, the validity of the existing models should
the surface roughness often causes the gas molecules col- be verified with further works.
liding with the wall more frequently so that the TMAC Agrawal and Prabhu (2008b) presented the correlation
increases with the increase of the roughness. Neither the between the TMAC and Knudsen number based on the
diffusive proportion nor the mass flow rate is influenced available data by the following expression:
by the surface topology (Veltzke and Thoming 2012).
Therefore, it is necessary to find a universal measurement rv ¼ 1 logð1 þ Kn0:7 Þ ð106Þ
method to determine the TMAC with surface roughness The equation was chosen to be rv ! 1 as Kn ! 0 and
effects. the explanation of the decrease of the TMAC with the
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Table 13 The values of TMAC for helium from various experimental measurement methods
Measurement method rv Knudsen range References
Table 14 Comparison of the TMACs using the Maxwell model, ðrv ÞM ¼ 3:6603ð0:7268
2 3
Loyalka model (Loyalka et al. 1975) and Sharipov model (Sharipov !1=2
2004) adapted from Yamaguchi et al. (2011) 1:0928
þ 1:648SÞ4 1 þ 15
Gas rM rLv rSv ð0:7268 þ 1:648SÞ2
v
Gas Smooth disk Medium rough Rough disk To explain the effect of Knudsen number on the TMAC,
disk Fig. 12 shows variation of the TMAC as a function of
Ra (nm) TMAC Ra (nm) TMAC Ra (nm) TMAC Knudsen number from different experiment data and
analytical results. It can be found that the TMAC varies
Air 10 0.885 404 0.346 770 0.145 irregularly and fluctuates near the unity with the change of
Helium 10 0.915 404 0.357 1,100 0.253 Knudsen number in the slip and transition flow regimes.
Most of the TMAC values are less than unity in the transition
flow regime, especially for the analytical results. The TMAC
value decreases monotonically with the increasing of
Table 16 Summary of TMAC measurements between smooth and
rough silicon reported by Chew (2009) Knudsen number from the analytical models presented by
Maurer et al. (2003) and Agrawal and Prabhu (2008a, b).
Gas Smooth silicon Rough silicon
(10–100 nm) (10 lm) However, the TMAC has larger difference between these
two methods in the slip flow regime. From the experimental
Air 0.95 1.04 measurements, Gabis et al. (1996) and Maurer et al. (2003)
Helium 0.99 1.00 predicted that the TMAC decreases generally with the
Hydrogen 1.02 1.04 increase of Knudsen number for helium. Yamamoto et al.
Water vapor 0.99 1.02 (2006) also found the same result for nitrogen. However, the
Nitrogen 0.99 1.01 TMAC does not decrease monotonically with increasing
Knudsen number for argon (Gabis et al. 1996). Even for the
same gas such as air, the TMAC displays different varying
increase of Knudsen number was not reported (Agrawal trends in the slip and transition flow regimes. Whatever the
and Prabhu 2008b). Knudsen number range and whatever the analytical
McCormick (2005) estimated the accommodation approach, it can be found that the values of TMAC
coefficient as an inverse problem from the experimental decrease with the increase of molecular weights of the gas
data (Siewert 2003) with an iterative method. The (Graur et al. 2009). The effect of Knudsen number on the
accommodation coefficients for the Maxwell one-parame- TMAC is still not clarified from the literature and should be
ter model and CL two-parameter model are estimated as: predicted from the experimental measurements.
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Acknowledgments This work was supported by the National 27:3–12
Science Foundation of China under Grant No. 11072147 and the Barber RW, Sun Y, Gu XJ, Emerson DR (2004) Isothermal slip flow
Specialized Research Fund for State Key Laboratory of Mechanical over curved surfaces. Vaccum 76:73–81
System and Vibration under Grant No. MSVMS201106, and spon- Barisik M, Beskok A (2011) Molecular dynamics simulations of
sored by Shanghai Rising-Star Program under Grant No. shear-driven gas flows in nano-channels. Microfluid Nanofluid
11QA1403400. 11:611–622
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