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Proc. R. Soc. A (2008) 464, 2015–2035
doi:10.1098/rspa.2008.0071
Published online 8 April 2008
Velocity profile in the Knudsen layer
according to the Boltzmann equation
B Y C HARLES R. L ILLEY
AND
J OHN E. S ADER *
Department of Mathematics and Statistics, The University of Melbourne,
Parkville, Vic. 3010, Australia
Flow of a dilute gas near a solid surface exhibits non-continuum effects that are
manifested in the Knudsen layer. The non-Newtonian nature of the flow in this region
has been the subject of a number of recent studies suggesting that the so-called ‘effective
viscosity’ at a solid surface is half that of the standard dynamic viscosity. Using the
Boltzmann equation with a diffusely reflecting surface and hard sphere molecules,
Lilley & Sader discovered that the flow exhibits a striking power-law dependence on
distance from the solid surface where the velocity gradient is singular. Importantly, these
findings (i) contradict these recent claims and (ii) are not predicted by existing highorder hydrodynamic flow models. Here, we examine the applicability of these findings to
surfaces with arbitrary thermal accommodation and molecules that are more realistic
than hard spheres. This study demonstrates that the velocity gradient singularity and
power-law dependence arise naturally from the Boltzmann equation, regardless of the
degree of thermal accommodation. These results are expected to be of particular value in
the development of hydrodynamic models beyond the Boltzmann equation and in the
design and characterization of nanoscale flows.
Keywords: Knudsen layer; Boltzmann equation; rarefied gas dynamics
1. Introduction
Micro- and nanoscale systems are currently the subject of intense research and
development. Gas flows in such systems often exhibit rarefied flow effects, which
arise when the mean free path of gas molecules becomes significant relative to
some dimension that characterizes the system. Examples include gas flows in
microchannels, which are the basis for numerous micro-electro-mechanical
systems flow sensing and control applications (Ho & Tai 1998), thermal force
effects on microcantilevers (Gotsmann & Durig 2005) and the thermal
transpiration principle upon which the solid-state Knudsen compressor (Sone
2002; McNamara & Gianchandani 2005) is based.
Accurate simulations of such rarefied flow effects are vital for the efficient
design, optimization and future development of micro- and nanodevices.
Conventional Navier–Stokes solutions of rarefied flows are inaccurate because
rarefied flows are characterized by non-equilibrium distributions of molecular
* Author for correspondence ( [email protected]).
Received 18 February 2008
Accepted 14 March 2008
2015
This journal is q 2008 The Royal Society
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C. R. Lilley and J. E. Sader
velocities, thus violating the near-equilibrium assumption that is the underlying
basis of the Navier–Stokes description. Accurate rarefied flow modelling requires
solutions of the more fundamental Boltzmann equation. For rarefied flows of ‘real
world’ interest, the Boltzmann equation is usually solved numerically with Bird’s
direct simulation Monte Carlo (DSMC) method (Bird 1994). However, for many
applications of current interest, DSMC calculations can impose prohibitive
computational demands. This is exemplified by the work of Reese et al. (2003),
who report that a two-dimensional DSMC calculation of rarefied air flow around
a moving microcantilever required 24 hours on a parallel supercomputer with
3000 processors. Such intensive computational demands have motivated recent
interest in the application of high-order hydrodynamic models to simulate
rarefied flows (Reese et al. 2003; Guo et al. 2006; Gu & Emerson 2007; Mizzi et al.
2007; Struchtrup & Torrilhon 2007; Torrilhon & Struchtrup 2008), with the
expectation that such models may be employed in computational fluid dynamics
solvers to accurately capture rarefaction effects with much less computational
effort than comparable DSMC calculations.
An important consideration in studying gas flows in micro- and nanoscale
systems is the Knudsen layer, which is a rarefaction effect that extends to a
distance the order of one mean free path from a solid surface. Within the
Knudsen layer, molecules collide with the surface more frequently than they
collide with each other (Gallis et al. 2006). This produces a distribution of
molecular velocities that is perturbed significantly from the equilibrium
Maxwellian state, and results in two important rarefaction phenomena: first,
the gas at the surface has a finite velocity relative to the surface, known as the
slip velocity. Second, the gas near the surface exhibits non-Newtonian behaviour.
For micro- and nanoscale flows, which have characteristic dimensions of the
order of several mean free paths, the Knudsen layer occupies a large portion of
the flow and can therefore dominate the flow behaviour. A detailed knowledge of
the Knudsen layer structure is thus essential for modelling rarefied flows in
micro - and nanoscale systems.
The structure of the Knudsen layer has been studied extensively (Bardos et al.
1986; Cercignani 2000; Sone 2002; Gu & Emerson 2007; Mizzi et al. 2007;
Struchtrup & Torrilhon 2007; Torrilhon & Struchtrup 2008). Recently, Lilley &
Sader (2007) used existing solutions of the linearized Boltzmann equation (LBE)
and precise DSMC calculations to examine the structure of the Knudsen layer in
detail for shear flow past a solid wall. They discovered that the bulk gas velocity
u parallel to the surface is accurately described by the remarkably simple powerlaw behaviour
uKuð0Þf ya ;
ð1:1Þ
where y is the normal distance from the surface and az0.8. This result applies
for hard sphere molecules near a diffusely reflecting surface, which corresponds to
full thermal accommodation.
This power-law behaviour prevails to a distance of about one mean free path
from the surface, and thus describes the inner portion of the Knudsen layer.
Importantly, equation (1.1) establishes the existence of a velocity gradient
singularity at the surface. This prediction of a singularity is consistent with
the work of Willis (1962) and Sone (2002), who rigorously examined the
linearized (approximate) Bhatnagar–Gross–Krook (BGK) model equation
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Velocity profile in the Knudsen layer
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true velocity profile
Figure 1. Schematic of Kramers’ problem. Here, y is the normal distance from the stationary solid
surface at yZ0 and u(y) is the bulk gas velocity in the direction parallel to the surface. The
so-called macroscopic slip velocity a x is extrapolated from the linear velocity profile outside the
Knudsen layer where y/N, and differs from the true slip velocity u(0). x is called the slip length.
(Bhatnagar et al. 1954) and discovered a singularity of logarithmic form du/
dy fln y at the surface. The work of Lilley & Sader (2007) establishes that a
singularity also exists in the full Boltzmann equation for hard spheres. This
singularity is not captured by existing hydrodynamic models and contradicts
recent work (Fichman & Hetsroni 2005; Lockerby et al. 2005a) that predicts a finite
velocity gradient at the surface.
In this paper, we first examine the structure of the Knudsen layer for a hard
sphere gas near a surface with partial thermal accommodation using the
existing LBE solutions supported by precise DSMC calculations. These
investigations establish that both the above power-law description and velocity
gradient singularity are present in the Knudsen layer under these more general
conditions. We then harness the versatility of the DSMC method to investigate
the Knudsen layer for a more realistic gas model with partial thermal
accommodation, and again show that the power-law description is accurate
and that the singularity exists. These results corroborate and extend
the applicability of the power-law description (Lilley & Sader 2007) of the
Knudsen layer.
2. Background
In this paper, we study the Knudsen layer for Kramers’ problem (Kramers
1949), which is illustrated in figure 1. This problem considers the
unidirectional isothermal motion of a gas filling a half-space bounded by a
stationary planar solid surface. The Kramers’ problem is often considered in
fundamental studies of the Knudsen layer, and has been researched extensively
(e.g. Cercignani (2000) and Sone (2002) and references therein). The only
bulk flow gradient in the Kramers’ problem is du/dy, where u is the velocity
component parallel to the surface and y is the normal distance from the
surface. As y/N, du/dy tends to the constant value a. Consistent with
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C. R. Lilley and J. E. Sader
existing work on the shear-thinning nature of gases (Montanero et al. 2000;
Garzó & Santos 2003), we adopt the notion of an ‘effective viscosity’ to
describe the non-Newtonian behaviour inherent in the Kramers’ problem. This
effective viscosity m is defined by
t ;
mðyÞ Z du=dy where the shear stress t is constant in the Kramers’ problem. We emphasize
that the effective viscosity is a mathematical construct with no connection to
real gas properties, and its value will change with flow geometry
(Hadjiconstantinou 2006). It is adopted here for convenience, ease of
discussion and consistency with previous works.
We normalize the flow speed u according to
u~ Z u=ðalnom Þ:
The nominal mean free path lnom is given by
m½1 p 1=2
;
lnom Z
n 2mkT
where n and m are the molecular number density and mass; k is Boltzmann’s
constant; and T is the temperature. Here, m[1] is the first viscosity approximation
from the Chapman–Enskog solution of the Boltzmann equation (Chapman &
Cowling 1970), which
depends upon the molecular model. For hard spheres
pffiffiffi
lnom Z ð5p=16Þ=ð 2nAÞ, where A is the hard sphere cross section. We normalize
y according to
y~ Z y=lnom ;
and define the non-dimensional slip coefficient x~ by
x~ Z x=lnom :
Solutions of the Kramers’ problem must consider the non-equilibrium
distribution of molecular velocities in the Knudsen layer and hence must solve
the Boltzmann equation, as noted in §1. The Boltzmann equation provides a
rigorous description of a dilute gas and describes the gas behaviour in terms of
the temporal evolution and spatial variation of a general molecular velocity
distribution function f. For a steady flow in the absence of body forces, the
Boltzmann equation for a monatomic gas is
vf
vf
v$
:
Z
vx
vt coll
Here, f(x, v) is the distribution of molecular velocities v that depends upon the
position vector x. The collision term [vf/vt]coll is a nonlinear integral expression
that describes the change in f due to intermolecular collisions.
The nonlinear integro-differential form of the Boltzmann equation poses
formidable challenges to solution by analytical methods. Indeed, complete
closed-form solutions have not been found, even for the simple flows like
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Kramers’ problem. The major difficulties arise from the collision term. For
weakly non-equilibrium flows like Kramers’ problem, the collision term can be
linearized, yielding the LBE. The LBE retains the most important physical
characteristics of the full Boltzmann equation, yet is tractable for many problems
(Cercignani 1988). Computational tools offer the only practical means of solving
the full nonlinear Boltzmann equation (see Cercignani 2000, p. 114), the most
common being Bird’s DSMC method (Bird 1994). The DSMC method
captures macroscopic gas behaviour by modelling a set of simulator particles,
which represent the real gas molecules, as they undergo simulated intermolecular
collisions, interact with solid surfaces and move through physical space.
Proofs that the DSMC method solves the Boltzmann equation have been
provided by Wagner (1992) and Pulvirenti et al. (1994). Here, we consider the
existing LBE solutions of the Kramers’ problem in §3, and precise DSMC
solutions in §4.
An important aspect of the Kramers’ problem is the interaction between gas
molecules and the surface. Because the physical details of such gas–surface
interactions are complex, Maxwell’s simple boundary condition (Maxwell 1879)
is often used. Under this condition, a fraction s of molecules are reflected
diffusely from the surface, meaning that the reflected molecules have velocities
distributed according to the equilibrium distribution at the surface temperature
Twall. The remaining fraction 1Ks of molecules is reflected specularly, meaning
that their velocity components normal to the surface are simply reversed
upon reflection. Here, the parameter s is called the thermal accommodation
coefficient and ‘full’ and ‘partial’ thermal accommodation refer to cases with
sZ1 and s!1, respectively.
3. LBE solutions for hard sphere molecules
The LBE for Kramers’ problem can be written (Cercignani 1988) as
2avy ðvx KayÞ C vy vy h Z Lh;
where L denotes the linearized collision operator. The molecular velocity
vZ ðvx ; vy ; vz Þ has components vx parallel to u, vy oriented in the y-direction, and
vz orthogonal to vx and vy. The perturbation function h is given by
hðx; vÞ Z
f
K1;
f
ðjhj/ 1Þ;
where f is the absolute Maxwellian distribution.
Numerical solutions of the LBE for Kramers’ problem with hard sphere
molecules have been published by Loyalka & Hickey (1989, 1990), Ohwada et al.
(1989) and Siewert (2003). These solutions all consider full thermal accommodation (sZ1). Loyalka & Hickey (1990) and Siewert (2003) also provided
solutions for partial accommodation (s!1).
Lilley & Sader (2007) examined the LBE velocity solutions for sZ1.
Specifically, they studied these solutions in the asymptotic limit as y~/ 0, by
analysing the behaviour of y~ versus u~ K u~ð0Þ on a double logarithmic scale. This
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C. R. Lilley and J. E. Sader
10
1
external flow
Knudsen layer
0.1
0.01
0.1
1
10
Figure 2. Published LBE solutions of Kramers’ problem for sZ1 compared with the power-law
description of equation (3.1). This power-law description is fitted to the LBE solution of Ohwada
et al. (1989), and is indistinguishable from fits to the other three LBE solutions shown here. Circles,
Ohwada et al. data; solid line, power-law fit to Ohwada et al. data; triangles, Loyalka & Hickey
(1989) data; squares, Loyalka & Hickey (1990) data; asterisks, Siewert data.
plotting scheme reveals striking linearity, as shown in figure 2, and immediately
leads to the power-law velocity description of equation (1.1). In terms of the
normalized quantities u~ and y~, this power law can be written as
u~ Z u~ð0Þ C C y~a :
ð3:1Þ
The corresponding effective viscosity is then
m Z mN
y~1Ka
;
aC
where mN is the standard dynamic viscosity.
We emphasize that the Knudsen layer extends beyond y~Z 1, and there does
not exist a strict demarcation between the Knudsen layer and the outer
(external) flow region. Indeed, the Knudsen layer decays asymptotically into the
outer flow region. As such, the regions marked as ‘Knudsen layer’ and ‘external
flow’ around y~Z 1 in figures 2, 6 and 7 are given as a guide to the eye only.
Importantly, the power-law description is only valid in the region y~! 1, as is
clear from figure 2, which coincides with the inner part of the Knudsen layer.
For the hard sphere LBE solutions with sZ1, Lilley & Sader (2007) calculated
the fit parameters C and a by linear regression analysis of the log data for y~! 1.
These parameters are shown in table 1, together with u~ð0Þ, the slip coefficients
x~ and the sample correlation coefficients r. In every case, r is very close to unity,
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Velocity profile in the Knudsen layer
Table 1. Power-law parameters and slip coefficients for the LBE solutions. (Data for sZ1 differ
slightly from those of Lilley & Sader (2007) due to the different mean free paths used in data
reduction. Here r is the sample correlation coefficient for the linear regression analysis of the log y~
versus log ½~
u K u~ð0Þ data. Solutions: O, Ohwada et al. (1989); LH1, Loyalka & Hickey (1989); LH2,
Loyalka & Hickey (1990); S, Siewert (2003).)
solution
s
x~
u~ð0Þ
C
a
r
O
LH1
LH2
1
1
0.2
0.4
0.6
0.8
1
0.1
0.3
0.5
0.7
0.9
1
1.1320
1.1109
9.2140
4.1913
2.4987
1.6395
1.1144
19.2364
5.8739
3.1793
2.0096
1.3489
1.1141
0.8204
0.8147
8.6052
3.6636
2.0482
1.2625
0.8075
18.5867
5.3070
2.6913
1.5967
1.0076
0.8074
1.2423
1.2513
1.4920
1.4207
1.3537
1.2901
1.2320
1.5334
1.4624
1.3948
1.3311
1.2709
1.2421
0.8027
0.7978
0.7015
0.7229
0.7460
0.7707
0.7992
0.6946
0.7166
0.7397
0.7646
0.7918
0.8062
1–1.2!10K5
1–3.3!10K5
1–3.0!10K6
1–1.8!10K7
1–4.4!10K7
1–3.6!10K6
1–4.9!10K6
1–2.3!10K5
1–2.9!10K5
1–3.8!10K5
1–4.5!10K5
1–4.8!10K5
1–4.9!10K5
S
demonstrating the accuracy of the power-law description within the Knudsen
layer. Importantly, all LBE solutions have a distinctly less than unity. Since l is
the only natural length scale in a dilute gas flow, this power law establishes that
the velocity gradient is singular at the wall ð~
y Z 0Þ where the corresponding
effective viscosity of zero.
We repeated this analysis for the LBE solutions with partial thermal
accommodation (0.1%s%0.9) by Loyalka & Hickey (1990) and Siewert (2003),
and observed a similar power-law velocity behaviour in all cases. For these
solutions, values of u~ð0Þ, C, a, x~ and r are included in table 1. Again, the
correlation coefficients are small, demonstrating the accuracy of the power-law
description in the Knudsen layer. In all the cases, the power-law parameter
a is distinctly less than unity, as for the solutions with sZ1. Therefore, for
these LBE solutions with s!1, the power law also establishes the existence
of a velocity gradient singularity at the surface where the effective viscosity
is zero.
Figure 3 shows all the LBE solutions, plotted together with their
corresponding power-law velocity profiles, which further demonstrates the
accuracy of the power-law description within the Knudsen layer. To explore
the dependence of u~ð0Þ, C, a and x~ on the accommodation coefficient s, these
parameters are plotted versus s in figure 4. We discuss these figures in §5.
4. DSMC solutions for hard sphere and variable soft sphere molecules
We have two aims in simulating the Knudsen layer with the DSMC method.
First, we use precise DSMC calculations to confirm the accuracy of the hard
sphere LBE solutions and the ensuing power-law velocity description established
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C. R. Lilley and J. E. Sader
0.9 0.7
= 1.0 0.8 0.6 0.5 0.4
0
5
0.3
0.2
10
0.1
15
20
Figure 3. LBE solutions of Kramers’ problem for hard spheres at various s, published by Loyalka &
Hickey (1989, 1990), Ohwada et al. (1989) and Siewert (2003). All solutions include velocity
profiles at sZ1. Only Loyalka & Hickey (1990) and Siewert (2003) give profiles for s!1. Triangles,
Ohwada et al.; crosses, Loyalka & Hickey (1989); circles, Loyalka & Hickey (1990); squares,
Siewert; solid lines, power-law fits.
in §3. Since LBE is an approximation to the full nonlinear Boltzmann equation,
the resulting solutions must be validated against those of the full Boltzmann
equation, as afforded by precise DSMC calculations.
Second, we use DSMC calculations to probe the Knudsen layer structure for
a molecular model that is more realistic than the simple hard sphere, and has
not been studied with LBE. We use a major advantage of the DSMC method, in
that it can, in principle, incorporate the molecular models with any level of
complexity. Importantly, this provides a means of solving the Boltzmann
equation for realistic gas models that cannot be studied with LBE. Here, we
investigate the Knudsen layer structure for the variable soft sphere (VSS) model
(Koura & Matsumoto 1991). The VSS model approximates the molecules that
interact according to an intermolecular force that is inversely proportional to a
power of the molecular separation distance. Details of our hard sphere and VSS
models appear in appendix A.
A number of reports have studied the Knudsen layer with DSMC calculations
(Bird 1977; Lockerby et al. 2005b), with several appearing in the past year (Gu &
Emerson 2007; Lilley & Sader 2007; Mizzi et al. 2007; Struchtrup & Torrilhon
2007; Torrilhon & Struchtrup 2008). These studies employed Couette flow
solutions to capture Kramers’ problem, and we have followed suit using
the geometry illustrated in figure 5. Our DSMC code solved the full Couette
flow domain, and provided the mean flow velocities calculated by
u c ðy c ÞZ 1=2½u c ðyc ÞK u c ðKyc Þ. These mean velocities were transformed into the
reference frame of the Kramers’ problem for subsequent analysis (see appendix A).
While Bird (1977), Lockerby et al. (2005b), Gu & Emerson (2007), Mizzi et al.
(2007), Struchtrup & Torrilhon (2007) and Torrilhon & Struchtrup (2008) do not
report the power-law velocity structure of the Knudsen layer, the DSMC solution
of Lockerby et al. (2005b) does exhibit the power-law dependence with az0.8 for
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Velocity profile in the Knudsen layer
(a)
2023
(b)
50
1.6
20
1.5
10
5
1.4
2
1.3
1
1.2
(c) 0.85
(d) 50
0.80
20
10
0.75
5
0.70
2
0
0.2
0.4
0.6
0.8
accommodation coefficient,
1.0
1
0
0.2
0.4
0.6
0.8
accommodation coefficient,
1.0
~
Figure 4. Comparison of power-law parameters (a) u~ð0Þ, (b) C and (c) a, and (d ) slip coefficient x.
Error bars for C and a are shown for the DSMC solutions. These error bars represent 99% CIs from
the nonlinear regression analysis. Errors in u~ð0Þ and x~ are not visible on these axes. Curve fits
~ Open circles, Loyalka & Hickey; squares,
described in equation (5.3) of §5 are shown for u~ð0Þ and x.
Siewert; filled circles, DSMC (hard spheres); diamonds, DSMC (VSS); dot-dashed line, DSMC
curve fit; solid line, curve fit.
velocity profile
Figure 5. Geometry of DSMC calculations of Couette flow. u c is the bulk flow velocity parallel
to the wall and yc is the displacement from the midplane where ycZ0. The walls are at
ycZGH/2.
VSS molecules with diffusely reflecting walls (Lilley & Sader 2007). In Bird’s
study, the walls of the Couette flow simulation were close together, resulting in
interference between the Knudsen layers at each wall so that the Kramers’
problem was not captured accurately. This highlights the fact that the wall
separation H is a critical consideration in using Couette flow simulations to
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C. R. Lilley and J. E. Sader
total
Table 2. Summary of DSMC simulation parameters. (Ncolls
is the total number of intermolecular
collisions simulated during the entire calculation.)
no. of cells
initial number density
relative wall speed
wall temperature
Mach number
mean particles/cell
sample interval
velocity samples/cell
total particle moves
hard spheres
VSS molecules
NcellsZ1000
n 0Z1.507!1025 mK3
u wallZ30 m sK1
TwallZ273 K
0.097
w100
Dt
w109
1012
KnZ0.0589
DtZtnom/9.817
total
Z 5:01 !1010
Ncolls
KnZ0.06
DtZtnom/10
total
Z 6:31 !1010
Ncolls
capture the Kramers’ problem. The wall separation must be sufficiently large
such that the Knudsen layers near each wall do not interfere with each other.
Here, we specify H in terms of the Knudsen number Kn, using
H Z lnom =Kn:
Owing to the computational expense of the DSMC method in the nearcontinuum regime where Kn is small, H cannot be arbitrarily large. A suitable H
value must therefore be sufficiently large to avoid interference between the
Knudsen layers, and yet still small enough to permit solution with DSMC
calculations within a practical time period. To determine a suitable H, we first
performed a series of DSMC calculations at various Kn, using hard spheres and
sZ1. This analysis, presented in appendix B, shows that Knz0.06 is sufficiently
small to capture the Knudsen layers at each wall without interference.
As for any numerical technique, it is essential to test the numerical
convergence of DSMC solutions. Accordingly, we performed a detailed
convergence analysis using hard spheres with KnZ0.0589. This analysis is
presented in appendix C and demonstrates that our solution is converged. Our
DSMC simulation parameters are summarized in table 2. Appendix A also
contains details on the simulation time step Dt and the flow sampling interval
used. An important note on the pseudo-random number generator used in our
DSMC calculations appears in appendix D.
Using both the hard sphere and VSS models, we performed DSMC calculations of Couette flow with 0.05%s%1. Samples of the resulting velocity profiles
for sZ0.1, 0.6 and 1 are shown in figure 6. We used nonlinear regression to
calculate u~ð0Þ, C and a for our DSMC solutions, according to the power-law
description of equation (3.1). Our method for calculating x~ is given in appendix
B. These parameters are plotted versus s in figure 4.
The close agreement between the LBE and hard sphere DSMC solutions
is immediately apparent in figures 4 and 6, verifying that LBE accurately
approximates the full Boltzmann equation for the Kramers’ problem with
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Velocity profile in the Knudsen layer
(a)
(b)
10
external
flow
Knudsen
layer
external
flow
Knudsen
layer
1
0.1
0.01
0.1
1
(c)
10
0.1
1
10
10
external
flow
Knudsen
layer
1
0.1
0.01
0.1
1
10
Figure 6. DSMC solutions of Kramers’ problem compared with the published LBE solutions.
((a) sZ0.1, (b) sZ0.6, and (c) sZ1). In the reference frame of Kramers’ problem, the DSMC
solutions each contained 500 points. For clarity, some of these were removed for y~T 1:4. The
LBE solutions of Loyalka & Hickey (1989, 1990), Ohwada et al. (1989) and Siewert (2003) are
shown in (c). On this plot, power-law fits to these different LBE solutions are indistinguishable so only one is shown. Squares, Siewert solution; solid line, power-law fit to LBE solution and Siewert and Loyalka & Hickey LBE solutions; filled circles, DSMC (hard spheres);
crosses, DSMC ( VSS); open circles, Loyalka & Hickey (1990) solution; diamonds, published
LBE solutions.
hard spheres and s!1. Additionally, the distinct linear behaviour of the DSMC
velocity profiles in figure 6 demonstrates the accuracy of the power-law velocity
profile in the Knudsen layer for VSS molecules at various s. We again emphasize
that l is the only natural length scale in this dilute gas flow. Importantly, the
DSMC solutions all have a distinctly less than unity, which in turn confirms
the existence of a velocity gradient singularity at the surface where the
effective viscosity is zero. We note that our result of aZ0.83 for the VSS
model with sZ1 is consistent with the value of 0.8 estimated by Lilley & Sader
(2007) from the DSMC solution published by Lockerby et al. (2005b) using
VSS molecules.
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C. R. Lilley and J. E. Sader
5. Discussion
As noted in §1, two recent studies by Lockerby et al. (2005a) and Fichman &
Hetsroni (2005) predicted mzmN/2 at the surface in the Knudsen layer. This
result contrasts with the above analysis that establishes mZ0 at the surface. It is
therefore important to compare these previous models to the above power-law
description and investigate how well these models approximate a solution of the
Boltzmann equation.
In the first study by Lockerby et al. (2005a), the velocity profile within the Knudsen
layer was modelled with a so-called ‘wall function’. This wall function, given by
2
7
1
~
~
~
u ð~
y Þ Z y C xK
;
20 1 C y~
was based on a curve-fit approximation to an earlier LBE solution. The wall function
gives the velocity gradient ½d~
u =d~
yZ 17=10 at the surface, with the corresponding
effective viscosity mz0.59mN. An implicit assumption in formulating the wall
function was that the velocity field for Kramers’ problem is analytic at yZ0. Indeed,
such analytic behaviour is predicted by several high-order hydrodynamic models of
the Knudsen layer, which have the general form (Lockerby et al. 2005b)
uðyÞ Z k 1 C ay C k 2 expðGk 3 yÞ
ð5:1Þ
for Kramers’ problem, where the constants k 1,2,3 depend upon the model and a is the
velocity gradient as y/N (figure 1). Lockerby et al. (2005b) used the velocity
gradient from the wall function at y~Z 0 as a boundary condition in equation (5.1)
and obtained
7
u~ Z u~ð0Þ C y~ C
½1KexpðKK y~Þ;
ð5:2Þ
10K
to describe the velocity profile in the Knudsen layer and external flow. Here,
the constant Kpdepends
upon the hydrodynamic model for: the
BGK–Burnett
ffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffi
equations K Z p=2, the regularized
Burnett
equations
K
Z
5p=54
,pZhong’s
ffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffi
augmented Burnett equations K Z 3p and the R13 equations K Z 5p=18.
Figure 7 shows the results obtained from equation (5.2) for the regularized Burnett
and Zhong’s augmented equations. These two solutions form an envelope within
which the BGK–Burnett solution, the R13 solution and the wall function are
all contained.
In the second recent study, Fichman & Hetsroni (2005) proposed the effective viscosity
(
s=2 C ð1KsÞ~
y; y~! 1;
mð~
y; sÞ
Z
mN
1;
y~O 1;
giving the velocity gradient d~
u =d~
y Z 2=s at the surface which is finite for
sO0. The velocity profile obtained from this effective viscosity is also shown
in figure 7.
Importantly, figure 7 clearly shows that the wall function, the various
hydrodynamic models and the Fichman & Hetsroni model do not capture the
asymptotic form of the velocity profile in the Knudsen layer near the surface.
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2027
Velocity profile in the Knudsen layer
10
external flow
Knudsen layer
1
0.1
0.01
0.1
1
10
Figure 7. Hard sphere LBE solutions (Loyalka & Hickey 1989; Ohwada et al. 1989; Siewert 2003)
at sZ1 compared with the hydrodynamicpmodels
ffiffiffiffiffiffiffiffiffiffiffiffiffi described by equation (5.2). The solutions
of thepregularized
Burnett
equations
ðK
Z
5p=54Þ and Zhong’s augmented Burnett equations
ffiffiffiffiffiffi
ðK Z 3pÞ form an envelope containing the BGK–Burnett solution, the R13 solution and the wall
function. Circles, LBE solutions; solid line, power-law fit to LBE data; dashed line, regularized
Burnett; dot-dashed line, Zhong’s augmented Burnett; dotted line, Fichman & Hetsroni.
On a log–log scale, the true velocity distribution follows a distinct line with a
slope significantly less than unity, whereas the hydrodynamic models give
straight lines with slopes of unity. This failure of high-order hydrodynamic
models and the wall function to correctly predict the power-law structure may
explain why they cannot accurately capture the Knudsen layer, as concluded by
Lockerby et al. (2005b). Nonetheless, it is important to emphasize that the
power-law model predicts an effective viscosity m!mN/2 at distances less than
0.03 mean free paths away from the wall, which represents a small region of the
Knudsen layer. The effect of using such approximate hydrodynamic models as
opposed to the true velocity distribution in the Knudsen layer (possessing a
velocity gradient singularity at the wall) in full flow modelling (Zhang et al. 2006)
is unclear and requires further investigation.
Our analysis of the existing LBE solutions for Kramers’ problem, together
with our new DSMC calculations, clearly demonstrate that the power-law
description of the Knudsen layer given in equation (3.1) accurately describes the
velocity profile for both hard sphere and VSS molecules with full and partial
thermal accommodation, even at a distance of only wlnom/100 from the surface.
Since the power-law description is obtained from the LBE and DSMC solutions
of the Boltzmann equation, regardless of the degree of thermal accommodation at
the surface, this finding indicates that the velocity gradient singularity arises
naturally from the Boltzmann equation. This is supported by Willis (1962) and
Sone (2002), who proved the existence of the logarithmic singularity du/dy fln y
in the linearized (approx.) BGK equation.
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2028
C. R. Lilley and J. E. Sader
The discussion above clearly shows that the various high-order hydrodynamic
models considered by Lockerby et al. (2005b) do not accurately capture the
velocity structure of the Knudsen layer as predicted by the LBE solutions.
Importantly, these models do not provide a proper treatment of the boundary
conditions at the wall (Gu & Emerson 2007; Mizzi et al. 2007; Struchtrup &
Torrilhon 2007; Torrilhon & Struchtrup 2008), and indeed can only be formally
valid in the outer part of the Knudsen layer since they are derived from high-order
hydrodynamic treatments of the Boltzmann equation (Hadjiconstantinou 2006).
Thus, it is not surprising that deviations to predictions of the Boltzmann equation
exist for the inner part of the Knudsen layer, as shown above. The power-law
description, and the feature of a velocity gradient singularity at the surface, is
expected to motivate further research into hydrodynamic models of rarefied flow.
Our results show that the entire velocity profile for Kramers’ problem is
accurately represented by
(
u~ð0; sÞ C C ðsÞ~
y aðsÞ ; y~! 1;
u~ð~
y ; sÞ Z
~ C y~;
xðsÞ
y~O 1:
The corresponding effective viscosity for y~! 1 is
mð~
y ; sÞ Z mN
y~1KaðsÞ
;
aðsÞC ðsÞ
which is strongly non-Newtonian. The functional dependencies of u~ð0Þ, C, a and
x~ on the accommodation coefficient s were determined empirically using several
trial functions and nonlinear regression to yield
9
u~ð0; sÞ Z 2:01=sK1:39 C 0:19s; >
>
>
>
>
=
CðsÞ Z 1:58K0:33s;
ð5:3Þ
>
aðsÞ Z 0:69 C 0:13s;
>
>
>
>
~ Z 2:01=sK0:73K0:16s: ;
xðsÞ
~
The fits for u~ð0; sÞ and xðsÞ
are included in figure 4.
A striking feature of the data shown in figure 4, is that the power-law structure
of the Knudsen layer appears to be preserved in the asymptotic limit as s/0.
Indeed, the power-law exponent a(s) varies very little while going between the
limits of fully diffuse (aZ1) and fully specular (aZ0) reflection at the surface.
Furthermore, the product aC Z 1:090K0:022sK0:043s2 obtained from the
above formulae, increases by only approximately 6% as s decreases from unity
to zero. Given the scatter in the numerical data for both a and C (see figure 4),
we then conclude that the product aCz1.05 is a sound approximation for all s.
The reasons for this intriguing constant behaviour in aC, which appears directly
in the expression for the effective viscosity (see above), and the limited
variability in the power-law exponent, are unknown at present.
Interestingly, our DSMC calculations show that the structure of the Knudsen
layer for VSS molecules is very similar to that for hard spheres, indicating
that the power-law description is only weakly dependent on the molecular model.
This strongly suggests that the power-law description is a general physical
Proc. R. Soc. A (2008)
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Velocity profile in the Knudsen layer
2029
phenomenon, within the framework of the Boltzmann equation, which applies for
all pure monatomic gases. However, as noted by Lilley & Sader (2007), detailed
solutions of the Boltzmann equation using realistic intermolecular potentials
validated by accurate experimental measurements are necessary to make a
definitive general statement about the accuracy of the power-law behaviour in
real gases. Any such investigations must consider gas mixtures containing
molecules with rotational energy such as air, which are important in most
practical applications. Since the DSMC method offers the only practical means of
solving the Boltzmann equation for such mixtures, DSMC calculations will be
an essential component of future investigations into the power-law behaviour in
real gases.
6. Summary and conclusions
We have examined the structure of the velocity profile in the Knudsen layer
using LBE solutions of Kramers’ problem for hard sphere molecules with partial
thermal accommodation, according to Maxwell’s boundary condition (Maxwell
1879). Our study establishes that the velocity profile in the Knudsen layer, under
these conditions, also follows the power-law description originally found by
Lilley & Sader (2007) for hard spheres with full thermal accommodation. This in
turn shows that the velocity gradient is singular at the surface, i.e. the effective
viscosity is zero, under arbitrary thermal accommodation. These findings were
verified using precise DSMC calculations.
We also performed DSMC calculations to probe the structure of the Knudsen
layer for a gas composed of VSS molecules, which are more realistic than simple
hard spheres. These simulations also revealed the power-law velocity behaviour
within the Knudsen layer over a full range of accommodation coefficients. The
small difference we observed between the hard sphere and the VSS solutions
indicates that the power-law behaviour is only weakly dependent on the molecular
model, and suggests that it arises directly from the Boltzmann equation.
These results are expected to motivate future work into understanding the
origin of such behaviour by rigorous asymptotic analysis of the Boltzmann
equation. Given the importance of rarefied gas dynamics in small-scale flows, our
findings are thus expected to impact on the development and application of
nanoscale devices.
This research was supported by the Particulate Fluids Processing Centre, a special research centre
of the Australian Research Council and by the Australian Research Council Grants Scheme.
Appendix A. Details of DSMC calculations
The collision cross section A for VSS molecules is given by
A Z Ar ðgr =gÞ2y ;
ðA 1Þ
where g is the relative speed of the colliding molecules; and Ar, gr and y are
constants that depend upon the gas properties. For VSS molecules, scattering is
anisotropic in the centre-of-mass reference frame and depends upon the VSS
scattering parameter k. For a pure gas composed of VSS molecules, the first
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2030
C. R. Lilley and J. E. Sader
Chapman–Enskog viscosity approximation is (Bird 1994)
pffiffiffi
15 p
m
1
4kT ð1=2ÞCy
½1
;
m Z
16Sm Gð4KyÞ Ar gr2y
m
ðA 2Þ
where
Sm Z 6k½ðk C 1Þðk C 2Þ K1
is a ‘softness coefficient’ for VSS molecules (Koura & Matsumoto 1991). Our gas
model represented argon, for which mZ6.6335!10K26 kg. We used a reference
viscosity mr of 21.068 mPa s at a reference temperature Tr of 273 K (Kestin et al.
1984) to obtain Ar gr2y and hence A using equations (A 1) and (A 2). Our VSS
model used yZ0.31 and kZ1.4 (Bird 1994). Hard spheres are a special case of
VSS molecules with yZ0 and kZ1, giving AZ41.57 Å2.
We based our simulation time step Dt on the nominal mean time between
intermolecular collisions tnom, given by
1=2
pm
:
tnom Z lnom
8kTwall
The number of time steps between flow field samples was set to ½Dy=ðc DtÞC 1
where c Z ½pkTwall =ð2mÞ1=2 is a characteristic molecular velocity and DyZ H =Ncells
is the cell size.
We transformed the Couette flow velocity profiles uc ðyc Þ into the reference
frame of Kramers’ problem using
1 1 yc
1
1
uc
K
K
y~ Z
:
and u~ Z
Kn 2 H
gKn 2 u wall
Here g is a normalized midplane velocity gradient given by
H duc
:
gZ
u wall dyc ycZ0
We found g by performing linear regression on the velocity profile at distances
exceeding 5lnom from the walls. For cases where 5lnom exceeded H/2, we
performed linear regression on the velocity profile to a distance ycZGH/10 from
the midplane.
Appendix B. DSMC simulations to find suitable H
To find a sufficiently large wall separation H for our DSMC Couette flow
calculations, we ran a series of simulations at various Kn. These simulations all
used NcellsZ1000, arranged in a regular grid with an average of 100 particles per
cell and DtZtnom/9.817. We used u wallZ30 m sK1, giving a flow Mach number of
0.097. Despite its apparent simplicity, this problem is computationally
demanding for the DSMC method because it involves small mean flow velocities
relative to the thermal velocities, necessitating large sample sizes to reduce
statistical scatter. Accordingly, we sampled the flow 107 times, so that the final
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2031
Velocity profile in the Knudsen layer
(a) 0.90
0.85
range of
LBE solutions
(b) 1.35
range of
LBE solutions
1.30
1.25
0.80
1.20
0.75
1.15
0.70
1.10
0.65
1.05
(d ) 1.2
(c)
0.95
1.1
0.90
0.85
0.9
0.80
0.8
0.01
range of
LBE solutions
1.0
range of LBE solutions
0.1
1
0.7
0.01
0.1
1
Figure 8. Power-law parameters (a) u~ð0Þ, (b) C and (c) a, and (d ) slip coefficients x~ for DSMC
calculations at various Kn. Results shown are for hard sphere molecules with sZ1. In all cases
DtZtnom/9.817 and NcellsZ1000.
sample comprised approximately 109 velocities per cell. The absolute error in the
DSMC velocity solutions was G0.14 m sK1, using a 99% CI.
The power-law parameters u~ð0Þ, C and a for our Couette flow solutions at
various Kn were calculated with nonlinear regression with the power law of
equation (3.1). These are plotted in figure 8 together with slip coefficients x~
calculated using
1
1
~
xZ
K1 :
2Kn g
The error ranges in figure 8 represent 99% CIs for the nonlinear regression
calculations. Errors generally increase with decreasing Kn. This is because
the number of cells used to fit the power law within the Knudsen layer, given
by KnNcells, decreases with decreasing Kn, leading to more error in the
regression calculation.
Nevertheless, within the errors shown in figure 8, it appears that u~ð0Þ, C, a
and x~ are constant for Kn(0.1. Hence, we conclude that Couette flow with
Knz0.06 gives a sufficiently large H to capture the Kramers’ problem without
interference between the Knudsen layers near each wall. As shown by Lilley &
Sader (2007), these simulation parameters provide a DSMC solution in close
agreement with the published LBE solutions (Loyalka & Hickey 1989, 1990;
Ohwada et al. 1989; Siewert 2003) for hard sphere molecules and sZ1. We note
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2032
C. R. Lilley and J. E. Sader
parameter value
(a) 1.4
1.3
(b)
1.2
1.1
1.0
0.9
0.8
0.7
1
10
100
1
10
100
(c) 1.4
parameter value
1.3
1.2
1.1
1.0
0.9
0.8
0.7
100
1000
Figure 9. Convergence analysis for DSMC solution of Couette flow at KnZ0.0589 using hard
spheres and sZ1. Error bars represent 99% CIs from the nonlinear regression calculation. For
clarity, errors are shown for u~ð0Þ data only. Similar errors apply for C, a and x~ data. (a) NcellsZ
1000 and DtZtnom/9.817, varying u wall; (b) NcellsZ1000 and u wallZ30 m sK1, varying Dt/tnom; and
(c) u wallZ30 m sK1 and DtZtnom/9.817, varying Ncells. Crosses, u~ð0Þ; squares, C; down triangles, a;
~
pluses, x.
that this Kn value is considerably lower than the values of approximately 0.2 and
0.13 used previously (Bird 1977; Lockerby et al. 2005b).
Appendix C. Convergence analysis
We tested the convergence of our hard sphere DSMC solution at KnZ0.0589,
sZ1, u wallZ30 m sK1, DtZtnom/9.817 and NcellsZ1000 by systematically
varying u wall, Dt and Ncells. The resulting power-law parameters and slip
coefficient x~ are shown in figure 9. These plots show that the Couette flow
simulation with Knz0.06 and u wallZ30 m sK1 using NcellsZ1000 and Dtztnom/
10 provides a converged solution.
The low subsonic Mach number of 0.097 in our Couette flow calculations
ensured that the flow conditions were very nearly incompressible and isothermal,
as necessary for Kramers’ problem. This is demonstrated in figure 10, which
shows the number density and temperature profiles for a DSMC solution at KnZ
0.0589. Variations in these quantities were less than approximately 0.05%,
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Velocity profile in the Knudsen layer
2033
0.5
0
– 0.5
– 0.0002
0
0.0002 0.0004
normalized quantity
0.0006
Figure 10. Profiles of normalized number density n/n 0K1 and normalized temperature T/TwallK1
for DSMC Couette flow solution at KnZ0.0589 with u wallZ30 m sK1, NcellsZ1000 and
DtZtnom/9.817.
confirming that the incompressible and isothermal flow conditions necessary
to capture the Kramers’ problem were achieved with excellent accuracy. The
temperature jump at the surface is only approximately 0.05 K.
Appendix D. A note on the pseudo-random number generator
Initially, we used the random( ) function, supplied with the Linux Fedora Core 4
operating system, to generate pseudo-random numbers. This function uses a nonlinear
additive feedback algorithm to generate successive pseudo-random numbers in
the range [0,231K1], and has the approximate period of 16(231K1)z34.4!106.
However, this generator resulted in Couette flow velocity profiles that did not pass
through the point (u c, yc)Z(0, 0). Subsequent calculations employed the Mersenne
Twister generator (Matsumoto & Nishimura 1998) to generate pseudo-random
numbers. This generator has excellent properties, including an extremely long period
of 219 937K1z4.3!106001. The velocity profile obtained using the Mersenne Twister
showed a clear improvement over that obtained with the random( ) generator,
in that the velocity profile passed very close to the point (u c, yc)Z(0, 0). All DSMC
results reported here were obtained using the Mersenne Twister generator. In
our implementation, this generator had only 62% of the CPU demand of the
random( ) generator.
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