of 53
J. Fluid Mech.
(2012),
vol
. 708,
pp.
197–249.
c
©
Cambridge University Press 2012
197
doi:10.1017/jfm.2012.302
Asymptotic analysis of the Boltzmann–BGK
equation for oscillatory flows
Jason Nassios
1
and John E. Sader
1,2
1
Department of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia
2
Kavli Nanoscience Institute and Department of Physics, California Institute of Technology, Pasadena,
CA 91125, USA
(Received 12 January 2012; revised 2 May 2012; accepted 13 June 2012;
first published online 10 August 2012)
Kinetic theory provides a rigorous foundation for calculating the dynamics of gas flow
at arbitrary degrees of rarefaction, with solutions of the Boltzmann equation requiring
numerical methods in many cases of practical interest. Importantly, the near-continuum
regime can be examined analytically using asymptotic techniques. These asymptotic
analyses often assume steady flow, for which analytical slip models have been derived.
Recently, developments in nanoscale fabrication have stimulated research into the
study of oscillatory non-equilibrium flows, drawing into question the applicability
of the steady flow assumption. In this article, we present a formal asymptotic
analysis of the unsteady linearized Boltzmann–BGK equation, generalizing existing
theory to the oscillatory (time-varying) case. We consider the near-continuum limit
where the mean free path and oscillation frequency are small. The complete set of
hydrodynamic equations and associated boundary conditions are derived for arbitrary
Stokes number and to second order in the Knudsen number. The first-order steady
boundary conditions for the velocity and temperature are found to be unaffected by
oscillatory flow. In contrast, the second-order boundary conditions are modified relative
to the steady case, except for the velocity component tangential to the solid wall.
Application of this general asymptotic theory is explored for the oscillatory thermal
creep problem, for which unsteady effects manifest themselves at leading order.
Key words:
kinetic theory, MEMS/NEMS, non-continuum effects
1. Introduction
The Navier–Stokes equations together with the no-slip boundary condition provide
a rigorous framework for studying many flow phenomena, including turbulence
(Reynolds 1895; Batchelor 1953; Ashurst
et al.
1987; Clercx & van Heijst 2009),
oscillatory flows (Stokes 1851; Pozrikidis 1992), boundary layer effects (Prandtl 1904;
Schlichting 1960), and pipe and channel flows (Reynolds 1883; Orszag & Kells 1980).
Applicability of the Navier–Stokes equations and no-slip condition is contingent on
validity of the continuum approximation. Miniaturization or operation at low gas
densities can lead to violation of this fundamental tenet. The degree of gas rarefaction
can be captured by the Knudsen number,
Kn
, defined as the ratio of the mean free
† Email address for correspondence: jsader@unimelb.edu.au
https://doi.org/10.1017/jfm.2012.302
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198
J. Nassios and J. E. Sader
path
λ
to the characteristic length scale of the flow
L
c
:
Kn
=
λ
L
c
.
(1.1)
Maxwell (1878) showed that rarefaction effects in a gas could be modelled by
allowing for velocity slip at the boundaries (assumed to be solid walls), and presented
the first of many slip models for rarefied flows (Weng & Chen 2008). This area has
received significant attention of late, motivated by the need to model flow phenomena
in micro- and nanoscale devices, e.g. those encountered in nanoelectromechanical
systems (NEMS). These modern devices have many applications, including ultra-
fine scale mass measurement (Craighead 2000; Cleland 2002; Yang
et al.
2006;
Bargatin, Kozinsky & Roukes 2007; Burg
et al.
2007), fluid property detection
(Sader 1998; Boskovic
et al.
2002; Motamedi & Wood-Adams 2010), sensing of
environmental conditions (Berger
et al.
1997; Lavrik, Sepaniak & Datskos 2004),
and atomic resolution imaging (Binnig, Quate & Gerber 1986; Giessibl 2003). The
flows generated by these small devices are often oscillatory and some lie outside the
realm of standard continuum theory. For example, these devices can exhibit oscillation
frequencies in the microwave range and length scales comparable to the gas mean free
path, rendering invalid the fundamental assumptions of this classical theory.
Solutions to non-equilibrium gas flows have been widely reported, and are primarily
based on an analysis of the Boltzmann equation (Boltzmann 1872) or an associated
model equation, the Boltzmann–BGK equation (Bhatnagar, Gross & Krook 1954;
Welander 1954). The derivation of these equations utilizes particle conservation
principles, ensuring their general applicability over all oscillation frequencies and
length scales. Direct analysis of the Boltzmann equation is complicated by the
nature of the collision integral (Cercignani 2000). The BGK approximation retains
the qualitative properties of the collision operator and simplifies the analysis of
many significant gas flows (Cercignani 2000). The validity of the approximation is
contingent on the assumption of a uniform collision frequency, which is valid provided
temperature perturbations in the gas are small. A well-documented shortcoming of
the Boltzmann–BGK equation is that the correct Prandtl number for monatomic
gases is not recovered (Cercignani 1988; Sone 2000). Despite these issues, the BGK
approximation is widely applied and has proved to be an important tool in many
theoretical investigations (see Cercignani 2000 and Sone 2000). As such, this article
focuses on the Boltzmann–BGK equation.
Asymptotic analyses of both the Boltzmann and Boltzmann–BGK equations have
been undertaken to probe the near-continuum regime. Pioneering methods include the
Hilbert and Chapman–Enskog expansions (Hilbert 1900, 1912; Chapman 1916; Enskog
1917). We explore a generalization of the former procedure in this article. Grad’s
moment method can also be used to generate hydrodynamic equations that characterize
non-equilibrium flows (Grad 1949, 1958). Numerical schemes in use today primarily
implement Monte Carlo methods, in particular the direct simulation Monte Carlo
(DSMC) method (Bird 1963). More recently, the lattice Boltzmann (LB) method has
been investigated (McNamara & Zanetti 1988; Higuera & Jim
́
enez 1989).
The DSMC method was originally developed by Anderson and Lord Kelvin
(Bird 1978). The method simulates gas flows using the collisional dynamics of
the constituent particles (Bird 1998). In this regard, DSMC has found widespread
application in analysing flows for a variety of intermolecular potentials. However,
at the low Mach numbers intrinsic to nanoscale systems, statistical noise can
dominate these numerical solutions (Hadjiconstantinou
et al.
2003). Techniques have
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Kinetic theory of oscillatory flows in rarefied gas
199
also been proposed to minimize noise at low Mach numbers (Fan & Shen 2001;
Baker & Hadjiconstantinou 2005; Chun & Koch 2005; Baker & Hadjiconstantinou
2008; Ramanathan & Koch 2009; Ramanathan, Koch & Bhiladvala 2010; Radtke,
Hadjiconstantinou & Wagner 2011).
The origins of the LB method are more recent, evolving from the lattice gas cellular
automata (LGCA) model (Frisch, Hasslacher & Pomeau 1986). The first formal LB
algorithm appeared just over two decades ago, and was later simplified through the
use of the Boltzmann–BGK equation (McNamara & Zanetti 1988; Higuera & Jim
́
enez
1989; Chen
et al.
1991; Qian, D’Humi
`
eres & Lallemand 1992). The LB method is
now used to study a diverse range of flow problems (Chen
et al.
2003; Yu, Girimaji
& Luo 2005; Shi & Sader 2010). Most recently, its applicability to rarefied gas flows
has also been explored (Colosqui
et al.
2010; Shi
et al.
2011; de Izarra, Rouet & Izrar
2011).
The Chapman–Enskog method provides a link between kinetic theory and gas
hydrodynamics (Chapman 1916; Enskog 1917; Chapman & Cowling 1960; Cercignani
2000; Sone 2000). In this method, a formal asymptotic expansion of the Boltzmann
equation is sought for rarefied flows in the continuum limit, via an expansion of the
distribution function and the streaming operator. The result is a series of hydrodynamic
equations at successive orders of Knudsen number. One recovers the Euler equations
to leading order and the Navier–Stokes equations of continuum flow to first order.
At subsequent orders, the hydrodynamic equations are the Burnett and super-Burnett
equations (Burnett 1935; Chapman & Cowling 1960; Shavaliyev 1993; Agarwal, Yun
& Balakrishnan 2001).
Issues regarding these higher-order Burnett equations remain. The instability of
the Burnett and super-Burnett equations has been the focus of considerable research
(Bobylev 1982; Struchtrup & Torrilhon 2003; Struchtrup 2005). While methods have
been discussed to regularize or stabilize these equations (Zhong, MacCormack &
Chapman 1993; Jin & Slemrod 2001; Soderholm 2007), their applicability in a general
context has not been established. Second, these higher-order differential equations
require additional boundary conditions to determine their unique solution (Lee 1994;
Agarwal
et al.
2001). Numerous methods and models have been proposed to resolve
this issue (Maurer
et al.
2003; Shen
et al.
2007; Gu & Emerson 2007; Torrilhon &
Struchtrup 2008). While the Chapman–Enskog procedure provides us with a means
to derive higher-order bulk flow equations, there remains no consensus on a systemic
approach to solve the associated boundary value problem.
Grad’s moment method (Grad 1949, 1958) bypasses the solution for the mass
distribution function by taking moments of the Boltzmann or Boltzmann–BGK
equation (Torrilhon & Struchtrup 2008). However, the resulting set of moment
equations are coupled, and appropriate closures and boundary conditions must be
derived. These issues have been addressed by Struchtrup & Torrilhon (2003), Torrilhon
& Struchtrup (2008) and Groth & McDonald (2009).
Hilbert proposed a method for solving the Boltzmann equation using a formal
asymptotic expansion in the Knudsen number (Hilbert 1900, 1912). To leading order
in the Knudsen number, the method recovers the Euler equations for inviscid flow as
the governing hydrodynamic equations. At higher order, the Euler equations reappear,
but now with forcing terms related to lower-order stresses and heat flows (Sone 2000).
Interestingly, the classical (nonlinear) Navier–Stokes equations never arise (Sone 2000).
The relevant boundary conditions to apply at each order in the Knudsen number
remained unresolved at the time of Hilbert (Hilbert 1900, 1912; Cercignani 2000; Sone
2000).
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200
J. Nassios and J. E. Sader
Cercignani addressed this issue and derived tangential velocity boundary conditions
directly from the linearized Boltzmann–BGK equation for the first- and second-order
(steady) hydrodynamic equations (Cercignani 1962, 1964). Sone elaborated on this
approach and derived a general set of boundary conditions for linearized steady flow
over simple (i.e. rigid) solid walls (Sone 1969, 1974). Incident particles were assumed
to undergo diffuse reflection from these walls with zero net mass flux. The shape or
curvature of the walls was considered arbitrary, the only restriction being that they
were smooth. The bulk flow equations (away from any surface) were derived using
the classical Hilbert expansion, which recovered the Stokes equations for creeping
flow at each order in the Knudsen number – non-equilibrium effects did not alter
the macroscopic hydrodynamic equations. To account for the Knudsen layer near the
surface, a matched asymptotic expansion was performed for small Knudsen number,
with the surface geometry characterized by the method of moving frame (Cartan
1977; Sone 2000). The anisotropy of space within the boundary layer resulted in an
infinite set of integral equations, which were solved simultaneously using numerical
methods up to second order in the Knudsen number (Sone 1964, 1965; Sone & Onishi
1973). This asymptotic formulation recovered the no-slip condition as the unique
boundary condition in the continuum limit, and the slip models derived by Cercignani
at first and second order (Sone 1968
b
, 1969). Furthermore, the set of higher-order
slip conditions contained thermal gradient terms whose presence was later verified
experimentally (see Sone 2000). Corrective terms to account for curvature effects of
the solid walls also appeared, and were subsequently confirmed for microscale flows
(Tibbs, Baras & Garcia 1997). This resolved the boundary value problem originally
encountered by Hilbert (Hilbert 1900, 1912; Cercignani 2000). The theory assumed
steady flow at low Mach and Reynolds number.
While many of the assumptions underlying Sone’s original theory have since
been relaxed, e.g. the low Mach and Reynolds number restrictions, the effect of
unsteadiness is yet to be fully explored (Sone & Onishi 1978; Onishi & Sone 1979;
Ohwada, Sone & Aoki 1989
a
,
b
; Sone, Ohwada & Aoki 1989; Ohwada & Sone 1992).
Some canonical unsteady flows have been considered, such as Stokes’ second problem
and the linearized Rayleigh problem (Sone 1964, 1965, 1968
a
). The influence of time-
dependence on the bulk flow has also been investigated for the hard-sphere Boltzmann
equation (Bardos, Golse & Levermore 1993, 1998; Sone 2007). Interestingly, unsteady
effects have been shown to not affect the existing steady boundary conditions and
Knudsen layer corrections to first order in the Knudsen number (Sone 2007). The
effect of unsteadiness on the full hydrodynamic equation/boundary condition set, up
to and including second order in the Knudsen number, is yet to be investigated.
Second-order models have received considerable attention recently, particularly in
the context of oscillatory flows (Hadjiconstantinou 2003, 2005
b
; Cao
et al.
2009;
Pitakarnnop
et al.
2009). Interestingly, Hadjiconstantinou’s numerical studies have
suggested that unsteady motion does not affect the second-order tangential slip model
(Hadjiconstantinou 2005
b
). A mathematical proof of this suggestion is yet to appear.
More recently, Takata
et al.
(2012) presented a formal asymptotic analysis of the
unsteady (linear in time,
t
) heating of two parallel plane walls. They show that
unsteady heating of this form results in a modification of the classical second-order
steady temperature slip model derived by Sone (1974).
In this article, we present a formal asymptotic analysis of the Boltzmann–BGK
equation for oscillatory flows in a slightly rarefied gas, i.e. small Knudsen numbers.
In so doing, we elucidate the effect of unsteadiness on the bulk flow hydrodynamic
equations and their associated boundary conditions. This is performed up to and
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Kinetic theory of oscillatory flows in rarefied gas
201
including equations second-order in the Knudsen number. Our analysis is restricted
to low Mach and Reynolds number flow; all perturbations in the macroscopic
quantities about their equilibrium values are thus small. This permits linearization
of the governing equation and boundary conditions, and justifies the assumption
of uniform gas collision frequency implicit in the BGK approximation. Numerical
techniques have been proposed and applied to analyse solutions of the linearized
Boltzmann–BGK equation for some canonical oscillatory rarefied gas flows (Sharipov
& Kalempa 2007, 2008; Manela & Hadjiconstantinou 2010; Yap & Sader 2012).
While these analyses provide high-accuracy data sets for linearized problems of
immense practical importance, analytical expressions provide additional insight into
the underlying physics driving the gas flows. The work we present in this article
provides the essential generalization to the original steady asymptotic theory of Sone
(1969, 1974).
We prove that oscillatory motion leads to compressible bulk flow at first order in
the Knudsen number; non-isothermal flows are also shown to evolve according to
a modified Navier–Stokes equation. In contrast, both isothermal and non-isothermal
flows obey a (different) modified Navier–Stokes equation at second order in the
Knudsen number. This feature is distinct from the well-known steady flow asymptotic
theory (Cercignani 1962, 1964; Sone 1969, 1974), where the bulk flow equations are
identical to the Stokes equations at all orders in the Knudsen number.
Tangential velocity slip models, up to and including second order in the Knudsen
number, are found to be identical to those for steady flow. This validates the
observation of Hadjiconstantinou (2005
b
), who suggested that the conventional
(steady) second-order tangential slip model can be used to analyse oscillatory flows.
Even so, other second-order components of the boundary conditions are modified by
unsteady effects. In particular, we show that oscillatory (time-varying) heating results
in a modification to the second-order temperature slip model that is equivalent to
that presented in Takata
et al.
(2012). Our analysis is supplemented by additional
terms, which account for non-zero boundary curvature, wall-normal velocity and gas
compressibility effects, as we shall discuss. The leading-order effects of unsteadiness
on gas rarefaction thus manifest themselves through the bulk flow equations, rather
than their associated boundary conditions.
To demonstrate application of this theory, we study flows generated by oscillatory
temperature gradients applied to two adjacent solid walls, i.e. thermal creep and
transpiration. This generalizes the original work of Sone (1966), who considered the
classical thermal creep problem of steady flow in an infinite half-space, due to a time-
independent temperature gradient imposed along the wall (Reynolds 1879; Maxwell
1879; Kennard 1938).
This phenomenon has found applications in a wide range of contexts. For example,
thermal transpiration in a circular pipe was considered by Sone (1968
b
) and Ohwada
et al.
(1989
b
), and subsequently applied to study of the Knudsen compressor (Knudsen
1909
a
,
b
; Loyalka 1971; Loyalka, Petrellis & Storvick 1979; Vargo
et al.
1999). More
recently, thermal creep has been proposed as a mechanism for motion of (volatile)
Leidenfrost drops along a ratchet surface (Lagubeau
et al.
2011; W
̈
urger 2011). In
addition to its practical significance, the leading-order flow generated in the thermal
creep problem occurs at
O
(
Kn
)
. As discussed, the leading-order effect of unsteadiness
in a rarefied flow also appears at
O
(
Kn
)
for non-isothermal flows. Consequently,
we present a detailed analysis of the oscillatory (time-varying) thermal creep
problem.
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202
J. Nassios and J. E. Sader
While several canonical shear-driven gas flows, e.g, oscillatory Couette flow, have
been examined using a variety of numerical and analytical methods, these flows are
isothermal in the linear limit (Park, Bahukudumbi & Beskok 2004; Hadjiconstantinou
2005
a
; Sharipov & Kalempa 2007, 2008; Tang
et al.
2008). The leading-order effect
of unsteadiness in these flows occurs via a modification to the classical Navier–Stokes
description at second order in the Knudsen number, i.e.
O
(
Kn
2
)
: this does not alter the
leading-order shear flow. For this reason, we restrict ourselves to a detailed exposition
of the thermal creep problem in this article.
Section 2 begins with a brief outline of the linearized Boltzmann–BGK equation
and its derivation. Details of the matched asymptotic expansion, relevant curvature
equations, and a discussion of the scaling analysis are then given. In § 3, we present
a detailed exposition of the required asymptotic results for the bulk flow equations
and boundary conditions up to second order in the Knudsen number. Key findings
and formulas from the bulk flow analysis are summarized in § 3.1. The complete set
of hydrodynamic equations, up to second order in the Knudsen number, are given
in table 1. A discussion of the Knudsen layer analysis, up to first order in the
Knudsen number, is presented in § 3.2.1, with the complete of associated Knudsen
layer corrections and slip boundary conditions in table 2. Corresponding discussion
and summary of the formulas at second order in the Knudsen number are given in
§ 3.2.2 and table 3, respectively. The Knudsen layer integral equations, Knudsen layer
corrections, and slip coefficients are relegated to the Appendices. We conclude in § 4
by exploring the application of this theory to oscillatory thermal creep between two
adjacent walls.
2. Theoretical framework
The mass distribution function of the gas,
F
(
x
,
v
,
t
)
, is assumed to obey the
Boltzmann–BGK equation (Bhatnagar
et al.
1954; Welander 1954; Vincenti & Kruger
1965),
F
t
+
v
i
F
x
i
+
a
i
F
∂v
i
=
ν
(
ρ
(
x
,
t
)
f
0
(
v
)
F
)
,
(2.1)
where the equilibrium velocity distribution function at the local temperature,
T
(
x
,
t
)
,
and mean velocity,
̄
v
(
x
,
t
)
, is
f
0
(
v
)
=
(
1
π
v
mp
(
T
)
)
3
exp
(
[
v
i
− ̄
v
i
v
mp
(
T
)
]
2
)
,
(2.2)
and
t
is time,
ν
is the particle collision frequency,
k
B
is Boltzmann’s constant, and
x
,
v
and
a
are the particle position, particle velocity and body force, respectively. The most
probable collision speed of gas molecules of mass
m
at temperature
T
(
x
,
t
)
is defined
as
v
mp
(
T
)
=
2
k
B
T
m
,
(2.3)
and the local density is
ρ(
x
,
t
)
. The local density
ρ
, mean gas velocity
̄
v
and
temperature
T
are given by the following moments
of the mass distribution
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Kinetic theory of oscillatory flows in rarefied gas
203
function,
ρ
=
−∞
F
d
v
,
(2.4
a
)
̄
v
=
1
ρ
−∞
v
F
d
v
,
(2.4
b
)
3
k
B
T
m
=
1
ρ
−∞
(
v
̄
v
)
2
F
d
v
,
(2.4
c
)
p
ρ
=
k
B
T
m
,
(2.4
d
)
where the equation of state is the ideal gas law, and
p
(
x
,
t
)
is the local pressure.
Mass, momentum and energy are conserved quantities over intermolecular collisions
(Vincenti & Kruger 1965):
0
0
0
=
−∞
m
m
v
1
2
m
v
2
[
ρ(
x
,
t
)
f
0
(
v
)
F
]
)
d
v
.
(2.5)
We restrict our analysis to low Mach and Reynolds number flows. This permits
linearization of the governing equation for
F
, its moments, and the collisional invariant
relations about their respective equilibrium solutions. To proceed, we thus define
ρ
=
ρ
0
(
1
+
σ(
x
,
t
)
)
,
(2.6
a
)
T
=
T
0
(
1
+
τ(
x
,
t
)
)
,
(2.6
b
)
p
=
p
0
(
1
+
P
(
x
,
t
)
)
,
(2.6
c
)
F
=
ρ
0
E
0
(
1
+
φ(
x
,
v
,
t
)
)
,
(2.6
d
)
where
ρ
0
,
T
0
,
p
0
and
ρ
0
E
0
are the equilibrium density, temperature, pressure and
mass distribution, respectively; the functions
σ
,
τ
,
P
and
φ
are perturbations to these
equilibrium values. The function
E
0
in (2.6) is given by
E
0
=
(
1
π
v
mp
(
T
0
)
)
3
exp
(
[
v
i
v
mp
(
T
0
)
]
2
)
.
(2.7)
We then substitute (2.6) into (2.1), (2.4) and (2.5) and linearize the resulting system.
This allows all time-varying functions to be expressed in terms of the explicit time-
dependence,
α(
x
,
v
,
t
)
= ̃
α(
x
,
v
)
exp
(
i
ω
t
)
,
(2.8)
where i is the usual imaginary unit,
ω
is the radial frequency of oscillation, and
α
represents any of: (i) the perturbations in (2.6); (ii) the mean gas velocities
̄
v
i
;
or (iii) the body force
a
i
. The body force is thus regarded in general as oscillatory
(time-varying). For simplicity we omit the ‘
̃
’ notation used in (2.8); henceforth, all
dependent functions are thus frequency-dependent expressions. This immediately leads
to the required linearized Boltzmann–BGK equation for oscillatory flow,
i
ωφ
+
v
i
∂φ
x
i
2
v
2
mp
(
T
0
)
v
i
a
i
=
ν
(
σ
φ
+
2
v
2
mp
(
T
0
)
̄
v
i
v
i
+
[
(
v
i
v
mp
(
T
0
)
)
2
3
2
]
τ
)
,
(2.9)
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204
J. Nassios and J. E. Sader
where the moment equations are
σ
=
−∞
φ
E
0
d
v
,
(2.10
a
)
̄
v
i
=
−∞
v
i
φ
E
0
d
v
,
(2.10
b
)
3
2
τ
=
−∞
(
(
v
i
v
mp
(
T
0
)
)
2
3
2
)
φ
E
0
d
v
,
(2.10
c
)
P
=
σ
+
τ,
(2.10
d
)
with the collisional invariant relations in (2.4) taking the form
0
0
0
=
−∞
1
v
v
2
(
σ
φ
+
2
v
2
mp
(
T
0
)
̄
v
i
v
i
+
[
(
v
i
v
mp
(
T
0
)
)
2
3
2
]
τ
)
E
0
d
v
.
(2.11)
2.1.
Scaling and generalized coordinates
The following scales are chosen for the particle velocity
v
i
, mean velocity
̄
v
i
and bulk
acceleration
a
i
,
v
s
=
v
mp
(
T
0
),
̄
v
s
=
v
mp
(
T
0
),
a
s
=
v
mp
(
T
0
)ω,
(2.12)
where the subscript
s
denotes a scale. The scaling for
a
i
is appropriate for an
oscillatory body force. Two length scales exist: the mean free path of the gas
λ
,
and the geometric length scale of the solid
L
c
, which is assumed to be much larger
than the mean free path. This separation of length scales results in a local flow
near the solid surface, i.e. within the Knudsen layer, and a bulk flow away from the
surface. The flows in these complementary regions will be solved for using a matched
asymptotic expansion in the Knudsen number,
Kn

1.
The bulk flow is analysed via a classical Hilbert expansion, with isotropic length
scale
L
c
(Hilbert 1900, 1912; Sone 1969, 1974; Cercignani 2000). Within the Knudsen
layer, the Boltzmann–BGK equation is scaled in the normal direction to the walls by
the mean free path
λ
, while the tangential directions are scaled by the geometric
length scale
L
c
. Thus, all perturbations to the mass distribution function and its
respective moments (represented by
α
), take the form
α
=
α
H
+
α
K
.
(2.13)
This yields two physically distinct equations to solve. The bulk flow quantities are
represented by a subscript
H
, while the Knudsen layer corrections are denoted by a
subscript
K
.
The surface geometry within the Knudsen layer is specified using the method of
moving frame (Cartan 1977; Sone 2000). A local orthonormal coordinate system
is chosen comprising two (principal) tangent vectors
t
1
i
and
t
2
i
(parametrized by
χ
1
and
χ
2
respectively) and an associated outward wall-normal
n
i
. The coordinate
normal to a solid wall is denoted
η
, and is scaled by the mean free path
λ
. The
wall curvature is chosen to be negative when its centre of curvature lies within the
gas (Sone 1969, 1974). The surface thus satisfies the following geometric relations
https://doi.org/10.1017/jfm.2012.302
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Kinetic theory of oscillatory flows in rarefied gas
205
(see Cartan 1977; Sone 2000; Ando 2011):
t
q
j
t
q
i
x
j
=−
κ
q
n
i
(
1
)
q
g
q
t
3
q
i
,
(2.14
a
)
t
q
j
t
3
q
i
x
j
=
(
1
)
q
g
q
t
q
i
,
(2.14
b
)
t
q
j
n
i
x
j
=
κ
q
t
q
i
,
(2.14
c
)
t
q
j
∂κ
q
x
j
=−
g
q
(
κ
1
κ
2
)
,
(2.14
d
)
κ
ij
=
κ
1
t
1
i
t
1
j
+
κ
2
t
2
i
t
2
j
,
(2.14
e
)
̄
κ
=
κ
1
+
κ
2
2
,
(2.14
f
)
where the index
q
takes the values 1
,
2, and the normal and geodesic curvatures are
κ
q
and
g
q
, respectively. The curvature tensor
κ
ij
is defined on the surface in terms of the
normal curvatures and local tangents, whereas the mean curvature is
̄
κ
.
2.2.
Boundary conditions
Solution to (2.9) is sought subject to classic diffuse reflection from rigid walls of
arbitrary and smooth shape. The functional form of the mass distribution function for
reflected particles is
F
b
=
ρ
b
(
1
π
v
mp
(
T
b
)
)
3
exp
(
[
v
i
V
i
v
mp
(
T
b
)
]
2
)
,
(2.15)
where the subscript ‘
b
’ denotes values at the solid walls, and
V
i
is the velocity of the
solid walls.
Zero net mass flux at the solid walls is also required. This immediately leads to the
following results for the density of particles re-emitted from the walls and the usual
no-penetration condition:
ρ
b
=−
2
π
v
mp
(
T
b
)
(
v
i
V
i
)
n
i
<
0
(
v
i
V
i
)
n
i
F
d
v
,
(2.16
a
)
V
i
n
i
= ̄
v
i
n
i
.
(2.16
b
)
Linearizing and scaling (2.15) and (2.16) then gives the required set of boundary
conditions for particles reflected from the solid wall:
φ
b
=
σ
b
+
2
V
i
v
i
+
(
v
2
i
3
2
)
τ
b
,
(2.17
a
)
σ
b
=
π
V
i
n
i
1
2
τ
b
2
π
−∞
−∞
0
−∞
ξφ
E
(v
i
t
1
i
,v
i
t
2
i
,ξ)
d
ξ
d
(
v
i
t
1
i
)
d
(
v
i
t
2
i
)
,
(2.17
b
)
V
i
n
i
= ̄
v
i
n
i
,
(2.17
c
)
ξ
=
v
i
n
i
V
i
n
i
.
(2.17
d
)
We have defined the normalized Gaussian
E
:
E
(v
1
,v
2
,v
3
)
=
π
3
/
2
exp
(
v
2
1
v
2
2
v
2
3
)
.
(2.18)
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206
J. Nassios and J. E. Sader
From (2.17), it then follows that outgoing particles from the solid walls satisfy the
condition
φ
K
>
0
η
=
0
=
φ
b
φ
H
.
(2.19)
In the outer region of the Knudsen layer, i.e.
η
→∞
, we also require that the
Knudsen layer correction decays to zero,
lim
η
→∞
φ
K
,ξ<
0
=
0
.
(2.20)
Finally, all solid walls are considered rigid, as formalized by the condition of zero
rate-of-strain,
V
i
x
j
+
V
j
x
i
=
0
.
(2.21)
This ensures the curvatures
κ
s
and
g
s
are constant and well-defined over the course of
the motion.
2.3.
System of equations and their solution
The mass distribution function is written as the sum of a bulk flow component and
a Knudsen layer correction, as per (2.13). This leads to the following set of scaled
governing equations:
i
2
β
k
2
φ
H
+
k
v
i
∂φ
H
x
i
β
k
2
a
i
v
i
=
σ
H
φ
H
+
2
̄
v
H
|
i
v
i
+
(
v
2
i
3
2
)
τ
H
,
(2.22
a
)
i
2
β
k
2
φ
K
+
k
(
v
i
t
1
i
t
1
j
∂φ
K
x
j
+
v
i
t
2
i
t
2
j
∂φ
K
x
j
)
=
σ
K
φ
K
v
i
n
i
∂φ
K
∂η
+
2
̄
v
K
|
i
v
i
+
(
v
2
i
3
2
)
τ
K
.
(2.22
b
)
Here, a vertical line ‘
|
’ in the subscript is used to delineate between the indices of the
tensor and other subscripts; this notation shall be used henceforth.
The corresponding boundary conditions are given in (2.19) and (2.20), and the
scaled collisional invariants become
0
0
0
=
−∞
1
v
v
2
(
σ
A
φ
A
+
2
̄
v
A
|
i
v
i
+
(
v
2
i
3
2
)
τ
A
)
E
(
v
)
d
v
,
(2.23)
while the moment equations are
σ
A
=
−∞
φ
A
E
(
v
)
d
v
,
(2.24
a
)
̄
v
A
|
i
=
−∞
v
i
φ
A
E
(
v
)
d
v
,
(2.24
b
)
3
2
τ
A
=
−∞
(
v
2
i
3
2
)
φ
A
E
(
v
)
d
v
,
(2.24
c
)
and the pressure is
P
A
=
σ
A
+
τ
A
.
(2.25)
We emphasize that all variables in (2.22)–(2.25) represent their scaled quantities. The
subscript
A
in (2.23)–(2.25) takes the values
H
or
K
.
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Kinetic theory of oscillatory flows in rarefied gas
207
The scaled Knudsen number,
k
, is defined by
k
=
π
2
Kn
,
(2.26)
and the Stokes number,
β
, is
β
=
ω
L
2
c
ν
kin
(
T
0
)
.
(2.27)
The scaled Knudsen number
k
and frequency ratio,
ω/ν
, are small and are related by
ω
ν
=
1
2
β
k
2
.
(2.28)
The scaled Knudsen number
k
simplifies the resulting analytical expressions, and is
used henceforth. The kinematic viscosity,
ν
kin
(
T
0
)
, at temperature
T
0
is related to the
most probable speed and mean free path by
ν
kin
(
T
0
)
=
π
4
v
mp
(
T
0
)
λ.
(2.29)
3. Asymptotic formulae
In this section, we derive the required asymptotic formulae for the bulk flow
and Knudsen layer corrections, in the limit of small scaled Knudsen number, i.e.
k

1. With the Stokes number
β
being a natural parameter of the problem, the
decomposition in (2.28) ensures that the frequency ratio is always small in the
asymptotic limit of infinitesimal scaled Knudsen number.
The mass distribution function and its moments appearing in the linearized
Boltzmann–BGK equation (2.22), the collisional invariants (2.23), the moment
equations (2.24), and the diffuse reflection, zero net mass flux and decay conditions
(equations (2.17), (2.19) and (2.20)), are expanded asymptotically in the small
parameter
k
, i.e.
α
=
n
=
0
α
(
n
)
k
n
,
(3.1)
where
α
represents any of these dependent functions, and
α
(
n
)
is the
n
th component.
Substituting (3.1) into the bulk flow equation in (2.22), and equating powers of
k
,
produces the required set of relations for
φ
(
n
)
H
(we remind the reader that
φ
(
n
)
H
is
the
n
th-order term in the
k
-expansion of
φ
H
); see (3.1). Enforcing the collisional
conservation laws (equation (2.23)) produces a set of hydrodynamic equations at each
order in
k
. These are analysed up to second order (i.e.
n
=
2) in § 3.1.
The corresponding boundary conditions at each order,
n
, are derived by analysis of
the Knudsen layer equation in (2.22). This results in a set of first-order differential
equations for
φ
(
n
)
K
, i.e. the
n
th-order term in the
k
-expansion of
φ
K
. Substituting the
solutions to these differential equations into the moment equations (2.24), yields a set
of simultaneous integral equations. The integral equations at second order (i.e.
n
=
2)
are given in appendix A – integral equations at lower order are identical to steady
flow (Sone 1969), as we shall discuss. Solutions to the integral equations for
n
6
2
are obtained using numerical techniques and give the required bulk flow boundary
conditions and Knudsen layer corrections.
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208
J. Nassios and J. E. Sader
3.1.
Hilbert expansion and bulk flow hydrodynamic equations
Proceeding as outlined above, we arrive at the following results for
φ
(
n
)
H
:
φ
(
0
)
H
=
σ
(
0
)
H
+
2
̄
v
(
0
)
H
|
i
v
i
+
(
v
2
i
3
2
)
τ
(
0
)
H
,
(3.2
a
)
φ
(
1
)
H
=
σ
(
1
)
H
+
2
̄
v
(
1
)
H
|
i
v
i
+
(
v
2
i
3
2
)
τ
(
1
)
H
v
j
∂φ
(
0
)
H
x
j
,
(3.2
b
)
φ
(
2
)
H
=
σ
(
2
)
H
+
2
̄
v
(
2
)
H
|
i
v
i
+
(
v
2
i
3
2
)
τ
(
2
)
H
v
j
∂φ
(
1
)
H
x
j
+
i
2
βφ
(
0
)
H
+
β
a
i
v
i
,
(3.2
c
)
φ
(
n
)
H
=
σ
(
n
)
H
+
2
̄
v
(
n
)
H
|
i
v
i
+
(
v
2
i
3
2
)
τ
(
n
)
H
v
j
∂φ
(
n
1
)
H
x
j
+
i
2
βφ
(
n
2
)
H
,
n
>
3
.
(3.2
d
)
Substituting (3.2) into the collision invariants in (2.23), and after some algebra, we
obtain the required set of bulk flow hydrodynamic equations listed in table 1. Note that
the order
n
=
3 continuity equation is required to close the order
n
=
2 hydrodynamic
system (not shown).
The results in table 1 contrast with the steady formulation (
β
=
0) of Sone (1969),
where it was found that the incompressible Stokes equations were recovered to all
orders in the scaled Knudsen number
k
. Indeed, Sone’s steady results are recovered
from table 1 for
β
=
0. For the general unsteady case,
β >
0, gas compressibility
affects the equations of motion for
n
>
1. Specifically, the linearized Navier–Stokes
equations are recovered at leading order, with compressibility effects and temperature
corrections modifying the bulk flow equations at higher order; the first-order equations
were discussed in Sone (2000). Even for isothermal and unidirectional flows, unsteady
effects lead to departures from the classical linearized Navier–Stokes equation at
higher order (
n
>
2); see table 1.
3.2.
Knudsen layer corrections and boundary conditions
Performing a similar analysis for the Knudsen layer leads to the following set of
first-order governing equations for
φ
(
n
)
K
:
v
i
n
i
∂φ
(
n
)
K
∂η
+
φ
(
n
)
K
=
Q
(
n
)
,
(3.3)
where
Q
(
0
)
=
σ
(
0
)
K
+
2
̄
v
(
0
)
K
|
i
v
i
+
(
v
2
i
3
2
)
τ
(
0
)
K
,
(3.4
a
)
Q
(
1
)
=
σ
(
1
)
K
+
2
̄
v
(
1
)
K
|
i
v
i
+
(
v
2
i
3
2
)
τ
(
1
)
K
v
i
(
∂χ
1
x
i
∂φ
(
0
)
K
∂χ
1
+
∂χ
2
x
i
∂φ
(
0
)
K
∂χ
2
)
,
(3.4
b
)
Q
(
n
)
=
σ
(
n
)
K
+
2
̄
v
(
n
)
K
|
i
v
i
+
(
v
2
i
3
2
)
τ
(
n
)
K
v
i
(
∂χ
1
x
i
∂φ
(
n
1
)
K
∂χ
1
+
∂χ
2
x
i
∂φ
(
n
1
)
K
∂χ
2
)
+
i
2
βφ
(
n
2
)
K
,
n
>
2
.
(3.4
c
)
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Kinetic theory of oscillatory flows in rarefied gas
209
n
=
0
0
=
P
(
0
)
H
x
i
0
=
̄
v
(
0
)
H
|
i
x
i
i
β
̄
v
(
0
)
H
|
i
=−
P
(
1
)
H
x
i
+
2
̄
v
(
0
)
H
|
i
x
2
j
+
β
a
i
3i
5
β
(
τ
(
0
)
H
2
3
σ
(
0
)
H
)
=
2
τ
(
0
)
H
x
2
i
n
=
1
i
2
βσ
(
0
)
H
=
̄
v
(
1
)
H
|
i
x
i
i
β
̄
v
(
1
)
H
|
i
=−
P
(
2
)
H
x
i
+
2
̄
v
(
1
)
H
|
i
x
2
j
+
1
3
2
̄
v
(
1
)
H
|
j
x
i
x
j
+
2i
β
3
∂τ
(
0
)
H
x
i
3i
5
β
(
τ
(
1
)
H
2
3
σ
(
1
)
H
)
=
2
τ
(
1
)
H
x
2
i
n
=
2
i
2
βσ
(
1
)
H
=
̄
v
(
2
)
H
|
i
x
i
i
β
̄
v
(
2
)
H
|
i
+
β
2
2
̄
v
(
0
)
H
|
i
= −
P
(
3
)
H
x
i
+
2
̄
v
(
2
)
H
|
i
x
2
j
+
1
3
2
̄
v
(
2
)
H
|
j
x
i
x
j
+
2i
β
3
∂τ
(
1
)
H
x
i
23i
β
30
P
(
1
)
H
x
i
i
2
β
2
a
i
β
(
2
a
i
x
2
j
+
2
a
j
x
i
x
j
)
3i
5
β
(
τ
(
2
)
H
2
3
σ
(
2
)
H
)
29i
20
β
(
2
τ
(
0
)
H
x
2
i
)
=
2
τ
(
2
)
H
x
2
i
+
19
5
2
x
2
j
(
2
τ
(
0
)
H
x
2
i
)
T
ABLE
1. Bulk flow hydrodynamic equations up to and including second order.
The associated boundary conditions at order
n
are (equations (2.17), (2.19) and (2.20))
φ
(
n
)
b
=
σ
(
n
)
b
+
2
V
(
n
)
i
v
i
+
(
v
2
i
3
2
)
τ
(
n
)
b
,
(3.5
a
)
σ
(
n
)
b
=
π
V
(
n
)
i
n
i
1
2
τ
(
n
)
b
2
π
−∞
−∞
0
−∞
ξφ
(
n
)
E
(v
i
t
1
i
,v
i
t
2
i
,ξ)
d
ξ
d
(
v
i
t
1
i
)
d
(
v
i
t
2
i
)
,
(3.5
b
)
V
(
n
)
i
n
i
= ̄
v
(
n
)
i
n
i
.
(3.5
c
)
https://doi.org/10.1017/jfm.2012.302
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