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A phenomenological and extended continuum approach for modelling non-equilibrium flows

TL;DR: In this paper, a new technique that combines Grad's 13-moment equations (G13) with a phenomenological approach to rarefied gas flows is presented, which does not require extra boundary conditions explicitly; Grad equations for viscous stress and heat flux are used as constitutive relations for the conservation equations instead of being solved as equations of transport.
Abstract: This paper presents a new technique that combines Grad’s 13-moment equations (G13) with a phenomenological approach to rarefied gas flows. This combination and the proposed solution technique capture some important non-equilibrium phenomena that appear in the early continuum-transition flow regime. In contrast to the fully coupled 13-moment equation set, a significant advantage of the present solution technique is that it does not require extra boundary conditions explicitly; Grad’s equations for viscous stress and heat flux are used as constitutive relations for the conservation equations instead of being solved as equations of transport. The relative computational cost of this novel technique is low in comparison to other methods, such as fully coupled solutions involving many moments or discrete methods. In this study, the proposed numerical procedure is tested on a planar Couette flow case, and the results are compared to predictions obtained from the direct simulation Monte Carlo method. This test case highlights the presence of normal viscous stresses and tangential heat fluxes that arise from non-equilibrium phenomena, which cannot be captured by the Navier–Stokes–Fourier constitutive equations or phenomenological modifications.

Summary (2 min read)

1 Introduction

  • Micro-electro-mechanical systems (MEMS) have found many applications in industrial and process systems, biomedical devices, environmental control devices, micro-processor cooling, and high precision printing.
  • Rapid progress in micro-engineering has not been matched by an increased understanding of the fundamental physics occurring in such small-scale domains.
  • The Navier–Stokes–Fourier (NSF) fluid dynamic equations with conventional no-slip boundary conditions are no longer valid when the system characteristic length scale approaches the molecular mean free path of the gas [5].
  • In discrete molecular methods, the fluid is modelled using a microscopic formalism, i.e., as a collection of moving molecules which interact through collisions or very close proximity potentials.
  • Extended gas dynamic continuum models can be derived by either perturbation methods, commonly known as the Chapman–Enskog expansion [4], or moment methods [6,9].

2 Extended hydrodynamics: the moment method

  • The angular brackets <> indicate the traceless part of a tensor.
  • Two options exist to derive relationships for these higher-order moments; (i) derive their transport equations, or (ii) obtain a constitutive relationship in the form of a closure approximation in terms of the five lower-order moments, i.e., ρ, vi and T .
  • −Pr qi τc , (3) where the collision frequency is defined as τc = µ/p. Higher-order moments ρ<i jk>, ρrr<ik> and ρrrss appear in the viscous stress and heat flux transport equations, Eqs. (2) and (3).
  • A closure for the first 13 moments yields the Grad 13-moment equations (G13), whereby ρ<i jk> = 0, ρrr<ik> = 7RT τik and ρrrss = 15p2/ρ. The NSF equations can be derived from both moment equation sets described here.
  • A fully coupled solution of any moment equation set larger than five requires additional boundary conditions, which may be derived from kinetic schemes [18].

3 A method of differential iteration using moment equations

  • The specific structure of the moment equations lends itself to being decoupled into two sub-systems and solved without the need for any additional boundary conditions.
  • System I is equivalent to the conservation laws together with constitutive relations for the viscous stress and the heat flux.
  • It is assumed that the non-equilibrium fluid property fields are continuous up to the boundaries.
  • The solution procedure switches between the two sub-systems until the L2 error norm, defined for the uncontrollable boundary data in system II over successive iterations, is considered to be sufficiently small [12].
  • The mathematical details of this iterative procedure have been presented in [10–12].

4 Phenomenological variants of simple continuum models

  • In continuum models, phenomenological techniques are often used to capture a particular physical behavior rather than modelling it from first principles.
  • The subscript ‘wall’ indicates wall boundary conditions.
  • Nevertheless, modifying such boundary conditions is not sufficient to fully capture many non-equilibrium effects in the transition regime.
  • Modifying the coefficients α1, α2 and β1 in the velocity slip and temperature jump conditions of Eqs. (12) and (13) only changes the magnitude of the wall boundary values.

5 Capturing non-equilibrium effects: a combined technique

  • Restricting the modelling of rarefied gas flows to the conservation equations and modifications thereof might hamper the possibility of capturing the correct physics occurring in the transition regime.
  • The combination of conservation equations and scaling methods alone is not enough to capture all non-equilibrium flow effects.
  • Nevertheless, this method can capture some of the stress/strain rate non-linearities occurring in the proximity of solid boundaries.
  • We, therefore, propose a solution of the moment equations in conjunction with constitutive scaling methods within the Knudsen layer.
  • Hence the same scaling function will be assumed for the momentum flux and thermal heat flux.

7 Results

  • The G13 equations and their modifications have been solved using differential iteration, in which system I is the governing and constitutive equation set resolving ρ, v1, T , p, τ12 and q2.
  • The production terms of the uncontrollable boundary data, i.e., τ11, τ22 and q1, are computed in system II and weighted values thereof appear as source terms in system I in the next iteration.
  • In both these cases, the Mach number, defined as Ma = viwall/ √ 2RTwall, was approximately 0.3.
  • The results are shown in half-space for clarity, and compare the DSMC predictions to the NSF solution with conventional slip/jump boundary conditions and the new solution to the G13 equations for both test cases.

8 Discussion and conclusions

  • Figures 2 and 3 show that the combination of the G13 equations with scaling functions achieves a better representation of the Knudsen layer in the near wall region than the standard NSF slip-flow solution.
  • As shown in Fig. 3f, this is partly due to the extended constitutive terms in the G13 equations and further improved by the wall scaling function.
  • (4) The proposed method captures some of the non-equilibrium effects which are otherwise absent from NSF solutions.
  • Additionally, the trend for tangential heat flux, Q1 is better represented in the proposed method.
  • Additionally, the authors would like to thank Y. Zheng and L. O’Hare for the informative discussions.

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Continuum Mech. Thermodyn. (2007) 19: 273–283
DOI 10.1007/s00161-007-0054-9
ORIGINAL ARTICLE
S. Mizzi · R. W. Barber · D. R. Emerson · J. M. Reese ·
S. K. Stefanov
A phenomenological and extended continuum approach
for modelling non-equilibrium flows
Received: 26 February 2007 / Accepted: 28 June 2007 / Published online: 31 July 2007
© Springer-Verlag 2007
Abstract This paper presents a new technique that combines Grad’s 13-moment equations (G13) with a
phenomenological approach to rarefied gas flows. This combination and the proposed solution technique cap-
ture some important non-equilibrium phenomena that appear in the early continuum-transition flow regime. In
contrast to the fully coupled 13-moment equation set, a significant advantage of the present solution technique
is that it does not require extra boundary conditions explicitly; Grad’s equations for viscous stress and heat
flux are used as constitutive relations for the conservation equations instead of being solved as equations of
transport. The relative computational cost of this novel technique is low in comparison to other methods, such
as fully coupled solutions involving many moments or discrete methods. In this study, the proposed numer-
ical procedure is tested on a planar Couette flow case, and the results are compared to predictions obtained
from the direct simulation Monte Carlo method. This test case highlights the presence of normal viscous
stresses and tangential heat fluxes that arise from non-equilibrium phenomena, which cannot be captured by
the Navier–Stokes–Fourier constitutive equations or phenomenological modifications.
Keywords Flows in micro-electromechanical systems (MEMS) and nano-electromechanical systems
(NEMS) · Micro- and nano- scale flow phenomena · Rarefied gas dynamics · Non-continuum effects ·
Non-equilibrium gas dynamics
PACS 47.61.Fg, 47.61.-k, 47.45.-n, 47.61.Cb, 47.70.Nd
1 Introduction
Micro-electro-mechanical systems (MEMS) have found many applications in industrial and process systems,
biomedical devices, environmental control devices, micro-processor cooling, and high precision printing.
Micro-ducts, micro-heat-exchangers, micro-pumps, and micro-air-vehicles are now commonly used terms.
Rapid progress in micro-engineering has not been matched by an increased understanding of the fundamental
physics occurring in such small-scale domains. One particularly active research area of emerging importance
Communicated by M. Slemrod
S. Mizzi (
B
) · R. W. Barber · D. R. Emerson
Centre for Microfluidics and Microsystems Modelling, Computational Science and Engineering Department,
CCLRC Daresbury Laboratory, Keckwick Lane, Daresbury, WA4 4AD Warrington, UK
E-mail: s.mizzi@dl.ac.uk
J. M. Reese
Mechanical Engineering Department, University of Strathclyde, James Weir Building, G1 1XJ Glasgow, UK
S. K. Stefanov
Institute of Mechanics, Bulgarian Academy of Sciences, Block 4, Acad. G. Bonchev Street, 1113 Sofia, Bulgaria

274 S. Mizzi et al.
is understanding the gas dynamics occurring in these miniaturized domains. This is because the general char-
acteristics observed in macro-scale flows are not always applicable to micro-sized domains. For example, the
Navier–Stokes–Fourier (NSF) fluid dynamic equations with conventional no-slip boundary conditions are no
longer valid when the system characteristic length scale approaches the molecular mean free path of the gas [5].
The inadequacy of the NSF equations to represent gas dynamics in micro-domains stems from the fact that
they are only able to describe flows which are close to equilibrium. Molecular collisions are the mechanism
by which gas molecules equilibrate energy and momentum. Hence, if a gas is too rarefied, or is confined to
micro-geometries, the number of collisions is reduced considerably relative to the bulk flow in the system,
inducing non-equilibrium phenomena. If λ is the mean free path, i.e., the average distance travelled by a gas
molecule between successive collisions, and H is the system characteristic length, then Kn = λ/H,isa
dimensionless parameter called the Knudsen number that gauges the degree of non-equilibrium.
The NSF equations can be obtained from the Boltzmann equation,
f
t
+ c
i
f
x
i
+ a
i
f
c
i
=
f
t
C
, (1)
through approximations of f , the particle distribution function (PDF). Equation(1) is considered to be the
fundamental governing equation of any dilute gas characterized by binary collisions. However, its solution
is a non-trivial task due to the dimensional complexity of the collisional operator on the right hand side of
Eq. (1). Various methods have been used to obtain a simpler approximation: each method attempting to retain
acceptable accuracy in order to capture the fundamental gas dynamics. There are two main approaches.
In discrete molecular methods, the fluid is modelled using a microscopic formalism, i.e., as a collection
of moving molecules which interact through collisions or very close proximity potentials. Such modelling
can be performed using either statistical ensemble averages, e.g., direct simulation Monte Carlo (DSMC) [2],
or deterministic methods, e.g., molecular dynamics (MD) [17]. Although such methods achieve a realistic
picture, their application has been restricted to simple flows due to their computationally intensive nature.
Continuum modelling is another approach. The fluid is assumed continuous and infinitely divisible so
it is possible to define velocity, density, pressure, and other properties at any point in space and time. This
modelling approach can be sub-divided further. Extended gas dynamic continuum models can be derived by
either perturbation methods, commonly known as the Chapman–Enskog expansion [4], or moment methods
[6,9]. The usual conservation equations with the NSF constitutive relations are derived in each case [18].
The other category of continuum models is the use of simple phenomenological extensions to the governing
conservation equations in order to capture non-equilibrium effects. The application of velocity slip [15] and
temperature jump [19] boundary conditions as well as constitutive model scaling in the form of wall function
methods [13] have been used in continuum-based approaches.
It has previously been shown that Grad’s 13-moment equations are unable to capture the non-linear stress/
strain relationship in the near-wall region [14]. In this paper we focus on continuum-based methods with
a unique combination of moment and phenomenological extensions in order to achieve a better description
of non-equilibrium effects that are typical of gaseous flows in micron-sized domains. Combining the two
methods not only improves the description of non-equilibrium phenomena but also retains the relatively low
computational cost—a desirable feature for modelling tools used in engineering design applications.
2 Extended hydrodynamics: the moment method
The governing fluid conservation laws for mass, momentum and energy contain moments in the form of stress,
σ
ij
= pδ
ij
+ σ
<ij>
= pδ
ij
+ τ
ij
and heat flux, q
i
. The angular brackets <> indicate the traceless part of
a tensor. Two options exist to derive relationships for these higher-order moments; (i) derive their transport
equations, or (ii) obtain a constitutive relationship in the form of a closure approximation in terms of the five
lower-order moments, i.e., ρ,v
i
and T .
One form of closure approximation is by considering a Hermite polynomial expansion of the PDF [6]. The
resulting closure approximation for the first five moments are termed the Euler equations with the resulting
constitutive relations being τ
ij
= 0andq
j
= 0. When transport relations (governing equations) for the viscous
stress, τ
ij
, and heat flux, q
i
, are derived, the moment equations are then of the form:
∂τ
ij
t
+
∂τ
ij
v
k
x
k
+
4
5
q
<i
x
j>
+
∂ρ
<ijk>
x
k
+ 2p
∂v
<i
x
j>
+ 2τ
k<i
∂v
j>
x
k
=−
τ
ij
τ
c
(2)

A phenomenological and extended continuum approach for modelling non-equilibrium flows 275
and
q
i
t
+
q
i
v
k
x
k
+
1
2
∂ρ
rr<ik>
x
k
+
1
6
∂ρ
rrss
x
i
5
2
p
ρ
p
x
i
5
2
p
ρ
∂τ
ik
x
k
τ
ik
ρ
p
x
k
τ
ij
ρ
∂τ
jk
x
k
+
7
5
q
k
∂v
i
x
k
+
2
5
q
k
∂v
k
x
i
+
2
5
q
i
∂v
k
x
k
+ ρ
<ijk>
∂v
j
x
k
=−Pr
q
i
τ
c
, (3)
where the collision frequency is defined as τ
c
= µ/ p.
Higher-order moments ρ
<ijk>
, ρ
rr<ik>
and ρ
rrss
appear in the viscous stress and heat flux transport equa-
tions, Eqs. (2) and (3). Again, there are two options to obtain relationships for these higher-order moments:
either using a closure approximation, or formulating further transport equations recursively until a closure is
defined. A closure for the first 13 moments yields the Grad 13-moment equations (G13), whereby ρ
<ijk>
= 0,
ρ
rr<ik>
= 7RTτ
ik
and ρ
rrss
= 15p
2
. The NSF equations can be derived from both moment equation sets
described here. Regularization of the Euler equations yields the NSF expressions [18].
A fully coupled solution of any moment equation set larger than five requires additional boundary con-
ditions, which may be derived from kinetic schemes [18]. The derivation of such boundary conditions is
a non-trivial task and becomes more complex as more moments are considered [7]. In addition, a coupled
solution of systems with a large number of moments incurs a computational cost that is comparable to that
of discrete methods, although the consideration of more moments should yield a better approximation to the
Boltzmann equation.
3 A method of differential iteration using moment equations
The specific structure of the moment equations lends itself to being decoupled into two sub-systems and
solvedwithout the need for any additional boundary conditions. The Maxwellian iteration, best illustrated in [8],
decouples the conserved variables, i.e., ρ,v
i
and T from all the higher-order moments τ
ij
, q
j
, ρ
<ijk>
, ρ
rr<ik>
,
ρ
rrss
, etc. The first Maxwellian iteration of the G13 moment equations yields the NSF equations. Liu proposed
a decoupling method along these lines with the iterations being weighted [11] and the method has been shown
to be mathematically consistent and convergent [12]. In order to illustrate this approach, the G13 moment set
will be used as an example.
We denote the two sub-systems as I and II. System I is equivalent to the conservation laws together with
constitutive relations for the viscous stress and the heat flux. System II consists of all the transport equations
of the non-equilibrium variables which will be used as modified constitutive relations in system I, i.e., the
viscous stresses, heat fluxes, and all higher-order moments for which the boundary conditions are unknown. It
is assumed that the non-equilibrium fluid property fields are continuous up to the boundaries. Such an assump-
tion is plausible even though discontinuities might exist at the wall boundaries in a similar way to velocity slip
and temperature jump. The value on the gas side is the value that ultimately is of significance to the solution
and this value should be continuous with the rest of the field. The unknown boundary data are extrapolated
after having solved system II.
For the G13 moment set, system I can be written as
∂ρ
(n)
t
+
∂ρ
(n)
v
(n)
i
x
i
= 0, (4)
∂ρ
(n)
v
(n)
i
t
+
ρ
(n)
v
(n)
i
v
(n)
j
+ σ
(n)
ij
x
j
= 0, (5)
3R
2
∂ρ
(n)
T
(n)
t
+
∂ρ
(n)
v
(n)
j
T
(n)
x
j
+ σ
(n)
ij
∂v
(n)
i
x
j
+
q
(n)
j
x
j
= 0, (6)
τ
(n)
ij
=−τ
c
∂τ
(n1)
ij
t
+
∂τ
(n1)
ij
v
(n)
k
x
k
+
4
5
q
(n1)
<i
x
j>
+ 2p
(n)
∂v
(n1)
<i
x
j>
+ 2τ
(n1)
k<i
∂v
(n)
j>
x
k
(7)

276 S. Mizzi et al.
and
q
(n)
i
=−
τ
c
Pr
q
(n1)
i
t
+
q
(n1)
i
v
(n)
k
x
k
+
1
2
∂ρ
(n1)
rr<ik>
x
k
+
1
6
∂ρ
(n)
rrss
x
i
5
2
p
(n)
ρ
(n)
p
(n)
x
i
5
2
p
(n)
ρ
(n)
∂τ
(n1)
ik
x
k
τ
(n1)
ik
ρ
(n)
p
(n)
x
k
τ
(n1)
ij
ρ
(n)
∂τ
(n1)
jk
x
k
+
7
5
q
(n1)
k
∂v
(n)
i
x
k
+
2
5
q
(n1)
k
∂v
(n)
k
x
i
+
2
5
q
(n1)
i
∂v
(n)
k
x
k
, (8)
whereas system II can be expressed as follows:
τ
(n)
ij
=−τ
c
w
n
∂τ
(n)
ij
t
+
∂τ
(n)
ij
v
(n)
k
x
k
+
4
5
q
(n)
<i
x
j>
+ 2p
(n)
∂v
(n)
<i
x
j>
+ 2τ
(n)
k<i
∂v
(n)
j>
x
k
+
(
1 w
n
)
τ
(n1)
ij
(9)
and
q
(n)
i
=−
τ
c
w
n
Pr
q
(n)
i
t
+
q
(n)
i
v
(n)
k
x
k
+
1
2
∂ρ
(n)
rr<ik>
x
k
+
1
6
∂ρ
(n)
rrss
x
i
5
2
p
(n)
ρ
(n)
p
(n)
x
i
5
2
p
(n)
ρ
(n)
∂τ
(n)
ik
x
k
τ
(n)
ik
ρ
(n)
p
(n)
x
k
τ
(n)
ij
ρ
(n)
∂τ
(n)
jk
x
k
+
7
5
q
(n)
k
∂v
(n)
i
x
k
+
2
5
q
(n)
k
∂v
(n)
k
x
i
+
2
5
q
(n)
i
∂v
(n)
k
x
k
+
(
1 w
n
)
q
(n1)
i
, (10)
where the superscript n indicates the iteration index and w
n
is a monotonically decreasing weight, such that
w
n
= 1/n. One whole iteration comprises a complete solution of both sub-systems.
The iterative procedure is initiated at n = 1 by solving for equilibrium—that is, by setting all non-equi-
librium variables in system I to zero, i.e., τ
(0)
ij
= 0andq
(0)
j
= 0. At n = 1, it can be shown that system I is
equivalent to the NSF equations. Having solved system I, we substitute all variables into system II containing
the non-equilibrium components, which are assumed to be continuous up to the boundary. The solution proce-
dure switches between the two sub-systems until the L
2
error norm, defined for the uncontrollable boundary
data in system II over successive iterations, is considered to be sufficiently small [12].
The mathematical details of this iterative procedure have been presented in [10–12]. However, it should be
noted that their solutions do not take into account the velocity slip or temperature jump boundary conditions.
In this paper, the same method has been used with the incorporation of velocity slip and temperature jump
into the solution, together with other modifications that better represent the dynamics of non-equilibrium gas
flows.
4 Phenomenological variants of simple continuum models
In continuum models, phenomenological techniques are often used to capture a particular physical behavior
rather than modelling it from first principles. Such an approach can be adopted to model features in non-equi-
librium micro-scale gas flows. Velocity slip, temperature jump, and strain-rate scaling through the Knudsen
layer are phenomenological models that will be briefly described here.
4.1 Velocity slip and temperature jump boundary conditions
The velocity slip and temperature jump boundary conditions are simplified phenomenological approaches to
represent both non-equilibrium and gas-surface interaction effects occurring near solid walls. Such boundary
conditions were first suggested by [15] and [19], respectively. Using Grad’s closure approximation for the PDF,
we find that the boundary conditions accounting for velocity slip and temperature jump are of the form [18]:
p
α
= p +
1
2
τ
jk
n
j
n
k
, (11)
(v
i
)
slip
= (v
i
)
gas
(v
i
)
wall
=−
1
5
(
α
2
q
i
n
i
q
k
n
k
)
p
α
±
2 σ
σ
π RT
2
1
2
α
1
τ
ij
n
j
n
i
τ
jk
n
j
n
k
p
α
(12)

A phenomenological and extended continuum approach for modelling non-equilibrium flows 277
and
T
wall
T
gas
1
2 σ
T
σ
T
π
2RT
1
2
1
2 p
α
β
1
q
k
n
k
+
1
4 p
α
τ
jk
n
j
n
k
(v
i
)
slip
(v
i
)
slip
4RT
, (13)
respectively. The subscript wall’ indicates wall boundary conditions.
Both boundary conditions are extensively used in the slip-flow regime, i.e., 0.001 < Kn < 0.1[5]incom-
bination with the NSF equations. Various modifications are also presented in the literature in order to extend
their validity into the early transition regime, i.e., 0.1 < Kn < 1.0 [1]. Nevertheless, modifying such bound-
ary conditions is not sufficient to fully capture many non-equilibrium effects in the transition regime. Further
modifications to the governing or constitutive equations are required in order to capture effects such as the
non-linear stress/strain-rate and the non-linear heat-flux/temperature-gradient relationships in the proximity
of solid walls.
4.2 The Knudsen layer and constitutive scaling
Much of the drive behind the extension of slip and jump boundary conditions has been to include the effects of
the Knudsen layer—the local region of non-equilibrium flow extending a few molecular mean free paths from
solid walls. Modifying the coefficients α
1
, α
2
and β
1
in the velocity slip and temperature jump conditions of
Eqs. (12) and (13) only changes the magnitude of the wall boundary values. While this may provide a better
solution in the bulk flow, it fails to model the effects of the Knudsen layer in the vicinity of the wall.
One possible solution that has been proposed to represent this non-equilibrium region is to introduce a
scaling function, ψ, in the viscous stress/strain-rate relationship which is dependent on the distance, ¯x
j
, from
a solid boundary. Any such scaling function should have a large effect within a few mean free paths from the
wall and scale to unity in the bulk flow. Cercignani [3] solved the linearized Boltzmann equation and showed
that the one-dimensional velocity profile through the Knudsen layer was of the form:
v
i
¯x
j
=−
τ
ij
µ

¯x
j
+ ξ λI
¯x
j
λ

, (14)
where ξ is a constant and I
¯x
j
is a correction function for the Knudsen layer. Lockerby et al. [13] identified
one such correction function for the momentum Knudsen layer:
I
¯x
j
λ
7
20
1 +
¯x
j
λ
2
. (15)
For an essentially one-dimensional flow, the stress tensor then has the scaled form:
τ
12
=−
µ
ψ
dv
1
dx
2
=−
1 +
7
10
1 +
¯x
2
λ
3
1
µ
dv
1
dx
2
. (16)
This representation is similar to other scaling methods used in micro-liquid transport models to describe
non-linearities close to the wall [16].
5 Capturing non-equilibrium effects: a combined technique
Restricting the modelling of rarefied gas flows to the conservation equations and modifications thereof might
hamper the possibility of capturing the correct physics occurring in the transition regime. The combination of
conservation equations and scaling methods alone is not enough to capture all non-equilibrium flow effects.
Nevertheless, this method can capture some of the stress/strain rate non-linearities occurring in the proximity
of solid boundaries.
We, therefore, propose a solution of the moment equations in conjunction with constitutive scaling meth-
ods within the Knudsen layer. In a similar manner to the approach described in [13], we introduce a viscosity
scaling function in the G13 constitutive relations in order to model non-linearities in the flow close to surfaces.
To retain the definition of the Prandtl number for a monatomic gas, Pr = 2/3, we shall assume that the
thickness of the momentum and thermal Knudsen layers are equivalent. Hence the same scaling function will

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TL;DR: 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, and is 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.
Abstract: 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 high-order 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.

68 citations


Cites background from "A phenomenological and extended con..."

  • ...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)....

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  • ...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…...

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  • ...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)....

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  • ...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…...

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  • ...…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…...

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Journal ArticleDOI
TL;DR: In this paper, flow and heat transfer in a bottom-heated square cavity in a moderately rarefied gas was investigated using the R13 equations and the Navier-Stokes-Fourier equations.
Abstract: Flow and heat transfer in a bottom-heated square cavity in a moderately rarefied gas is investigated using the R13 equations and the Navier–Stokes–Fourier equations. The results obtained are compared with those from the direct simulation Monte Carlo (DSMC) method with emphasis on understanding thermal flow characteristics from the slip flow to the early transition regime. The R13 theory gives satisfying results—including flow patterns in fair agreement with DSMC—in the transition regime, which the conventional Navier–Stokes–Fourier equations are not able to capture.

43 citations

Journal ArticleDOI
TL;DR: In this paper, a model of thermal conductivity of nanofluids based on extended irreversible thermodynamics is proposed with emphasis on the role of several coupled heat transfer mechanisms: liquid interfacial layering between nanoparticles and base fluid, particles agglomeration and Brownian motion.
Abstract: A modelling of the thermal conductivity of nanofluids based on extended irreversible thermodynamics is proposed with emphasis on the role of several coupled heat transfer mechanisms: liquid interfacial layering between nanoparticles and base fluid, particles agglomeration and Brownian motion. The relative importance of each specific mechanism on the enhancement of the effective thermal conductivity is examined. It is shown that the size of the nanoparticles and the liquid boundary layer around the particles play a determining role. For nanoparticles close to molecular range, the Brownian effect is important. At nanoparticles of the order of 1–100 nm, both agglomeration and liquid layering are influent. Agglomeration becomes the most important mechanism at nanoparticle sizes of the order of 100 nm and higher. The theoretical considerations are illustrated by three case studies: suspensions of alumina rigid spherical nanoparticles in water, ethylene glycol and a 50/50w% water/ethylene glycol mixture, respectively, good agreement with experimental data is observed.

35 citations

Journal ArticleDOI
TL;DR: In this article, rarefied gas flow between two parallel moving plates maintained at the same uniform temperature is simulated using the direct simulation Monte Carlo (DSMC) method, and the effects of the important molecular structural parameters such as molecular diameter, mass, degrees of freedom and viscosity-temperature index on the macroscopic behavior of gases are investigated.

35 citations

Journal ArticleDOI
TL;DR: The model accurately reproduces MD free-energy calculations of hydration asymmetries for monatomic ions, titratable amino acids in both their protonated and unprotonated states, and the Mobley "bracelet" and "rod" test problems.
Abstract: We show that charge-sign-dependent asymmetric hydration can be modeled accurately using linear Poisson theory after replacing the standard electric-displacement boundary condition with a simple nonlinear boundary condition. Using a single multiplicative scaling factor to determine atomic radii from molecular dynamics Lennard-Jones parameters, the new model accurately reproduces MD free-energy calculations of hydration asymmetries for: (i) monatomic ions, (ii) titratable amino acids in both their protonated and unprotonated states, and (iii) the Mobley “bracelet” and “rod” test problems [D. L. Mobley, A. E. Barber II, C. J. Fennell, and K. A. Dill, “Charge asymmetries in hydration of polar solutes,” J. Phys. Chem. B 112, 2405–2414 (2008)]. Remarkably, the model also justifies the use of linear response expressions for charging free energies. Our boundary-element method implementation demonstrates the ease with which other continuum-electrostatic solvers can be extended to include asymmetry.

33 citations

References
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01 Jan 1991

6,395 citations


"A phenomenological and extended con..." refers methods in this paper

  • ...Extended gas dynamic continuum models can be derived by either perturbation methods, commonly known as the Chapman–Enskog expansion [ 4 ], or moment methods [6,9]....

    [...]

Book
16 Jun 1994
TL;DR: The direct simulation Monte Carlo (or DSMC) method has, in recent years, become widely used in engineering and scientific studies of gas flows that involve low densities or very small physical dimensions as mentioned in this paper.
Abstract: The direct simulation Monte Carlo (or DSMC) method has, in recent years, become widely used in engineering and scientific studies of gas flows that involve low densities or very small physical dimensions. This method is a direct physical simulation of the motion of representative molecules, rather than a numerical solution of the equations that provide a mathematical model of the flow. These computations are no longer expensive and the period since the 1976 publication of the original Molecular Gas Dynamics has seen enormous improvements in the molecular models, the procedures, and the implementation strategies for the DSMC method. The molecular theory of gas flows is developed from first principles and is extended to cover the new models and procedures. Note: The disk that originally came with this book is no longer available. However, the same information is available from the author's website (http://gab.com.au/)

5,311 citations


"A phenomenological and extended con..." refers methods in this paper

  • ...Such modelling can be performed using either statistical ensemble averages, e.g., direct simulation Monte Carlo (DSMC) [ 2 ], or deterministic methods, e.g., molecular dynamics (MD) [17]....

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Book
28 Apr 1997
TL;DR: This book describes the extremely powerful technique of molecular dynamics simulation, which involves solving the classical many-body problem in contexts relevant to the study of matter at the atomic level.
Abstract: From the Publisher: This book describes the extremely powerful technique of molecular dynamics simulation, which involves solving the classical many-body problem in contexts relevant to the study of matter at the atomic level. The method allows the prediction of the static and dynamic properties of substances directly from the underlying interactions between the molecules. Because there is no alternative approach capable of handling such a broad range of problems at the required level of detail, molecular dynamics methods have proved themselves indispensable in both pure and applied research.

3,124 citations


"A phenomenological and extended con..." refers methods in this paper

  • ...Such modelling can be performed using either statistical ensemble averages, e.g., direct simulation Monte Carlo (DSMC) [2], or deterministic methods, e.g., molecular dynamics (MD) [ 17 ]....

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Book
01 Jan 1969
TL;DR: In this paper, the authors present a simulation of a free jet expansion of a high-energy scattering of molecular beams in the presence of high-temperature Viscosity cross sections.
Abstract: Collisional Processes.- Analytical Formulae for Cross Sections and Rate Constants of Elementary Processes in Gases.- Relaxation of Velocity Distribution of Electrons Cooled (Heated) By Rotational Excitation (De-Excitation) Of N2.- Effects of the Initial Molecular States in a High-Energy Scattering of Molecular Beams.- Differential Cross Sections for Ion-Pair Formation with Selection of the Exit Channel.- Low-temperature Viscosity Cross Sections Measured in a Supersonic Argon Beam II.- Excited Oxygen Iodine Kinetic Studies.- Determination of Antisymmetric Mode Energy of CO2 Injected into a Supersonic Nitrogen Flow.- Molecular Beams.- Where are we going with molecular beams?.- Cesium Vapor Jettarget Produced With a Supersonic Nozzle.- Basic Features of the Generation and Diagnostics of Atomic Hydrogen Beams in the Ground and Metastable 22S1/2-States to Determine the Fundamental Physical Constants.- Optical Pumping Of Metastable Neon Atoms in A Weak Magnetic Field.- CO2-Laser Excitation of a Molecular Beam Monitored By Spontaneous Raman Effect.- Time-of-Flight and Electron Beam Fluorescence Diagnostics: Optimal Experimental Designs.- Molecular Beam Time-of-Flight Measurements in A Nearly Freejet Expansion of High Temperature Gas Produced By a Shock Tube.- Electron Beam Diagnostics.- Electron-Beam Diagnostics of High Temperature Rarefied Gas Flows.- Excitation Models Used in the Electron Beam Fluorescence Technique.- Electron - Beam Diagnostics in Nitrogen Multiquantum Rotational Transitions.- Free Jets, Nonequilibrium Expansions.- Free Jet as an Object of Nonequilibrium Processes Investigation.- State Dependent Angular Distributions of Na2 Molecules in a Na/Na2 Free Jet Expansion.- Molecular Beam Time-of-Flight Measurements and Moment Method Calculations of Translational Relaxation in Highly Heated Free Jets of Monatomic Gas Mixtures.- Rovibrational State Population Distributions of CO (v ? 4, J ? 10) In Highly Heated Supersonic Free Jets of CO-N2 Mixtures.- Free Jet Expansion with A Strong Condensation Effect.- Measured Densities in UF6 Free Jets.- Rotational Relaxation of NO in Seeded, Pulsed Nozzle Beams.- The Free-Jet Expansion from a Capillary Source.- Rotational Relaxation in High Temperature Jets of Nitrogen.- Translational Nonequilibrium in a Free Jet Expansion of a Binary Gas Mixture.- Laser Induced Fluorescence Study of Free Jet Expansions.- Jet-Surface Interactions.- Experimental Study of Plume Impingement and Heating Effect on Ariane's Payload.- The Interaction of a Jet Exhausting from a Body with a Supersonic Free Flow of a Rarefied Gas.- Modelling Control Thruster Plume Flow and Impingement.- Impingement of a Supersonic, Underexpanded Rarefied Jet upon a Flat Plate.- Some Peculiarities of Power and Heat Interaction of a Low Density Highly Underexpanded Jet with a Flat Plate.- Condensation in Flows.- Nonequilibrium Condensation in Free Jets.- Condensation and Vapour-Liquid Interaction in a Reflected Shock Region.- Homogeneous and Heterogeneous Condensation of Nitrogen in Transonic Flow.- Investigation of Nonequilibrium Homogeneous Gas Condensation.- The Peculiarities of Condensation Process in Conical Nozzle and in Free Jet Behind it.- Investigation of Nonequilibrium Argon Condensation In Supersonic Jet By Mass-Spectrometry, Electron Diffraction and VUV Emission Spectroscopy.- Clusters and Nucleation Kinetics.- The Microscopic Theory of Clustering and Nucleation.- Kinetics of Cluster Formation and Growth in the Process of Isothermal Condensation.- Relaxation Processes in a Molecular Dynamic Model of Cluster from the Lennard-Jones Particles.- Quantum-Chemical Study Of Processes With Cluster Isomerism.- The Homogeneous Nucleation at the Continuously Changing Temperature and Vapour Concentration.- Molecular Clusters as Heterogeneous Condensation Nuclei.- Experiments with Clusters.- The Photochemistry of Small van der Waals Molecules as Studied by Laser Spectroscopy in Supersonic Free Jets.- Diagnostics of Clusters in Molecular Beams.- Experimental Studies of Water-Aerosol Explosive Vaporization.- Laser Probing of Cluster Formation and Dissociation in Molecular Beams.- Free Molecule Drag on Helium Clusters.- Vibrational Relaxation Kinetics in a Two-Phase Gas-Cluster System.- Gas-Particle Flows.- Long-Range Attraction in the Collisions of Free-Molecular and Transition Regime Aerosol Particles.- Nonequilibrium Statistical Theory of Dispersed Systems.- The Mechanism of Strong Electric Field Effect on the Dispersed Media in the Rarefied Gas.- Generation of High-Speed Aerosol Beams By Laval Nozzles.- Kinetic Model of a Gas Suspension.- Gas Mixtures.- Kinetic Phenomena in the Rarefied Gas Mixtures Flowing Through Channels.- On the Discrete Boltzmann Equation for Binary Gas Mixtures.- Peculiarities and Applicability Conditions of Macroscopic Description of Disparate Molecular Masses Mixture Motion.- Numerical Solution of the Boltzmann Kinetic Equation for the Binary Gas Mixture.- Species Isotope Separation.- Gas or Isotope Separation by Injection into Light Gas Flow.- Molecular Diffusion Through a Fine-Pored Filter Versus Resonante IR-Radiation Intensity.- On Limiting Situations of Gas Dynamic Separation.- A Study of Reverse Leaks.- Investigation of Nonequilibrium Effects in Separation Nozzles by Monte-Carlo Simulation.- Separation of Binary Gas Mixtures at their Effusion through a Capillary and a Nuclear Filler into Vacuum.- Ionized Gases.- Effects of Nonideality in Quantum Kinetic Theory.- Molecular Mass and Heat Transfer of Chemical Equilibrium Multicomponent Partially Ionized Gases in Electromagnetic Field.- Spectroscopic Study of a Plasma Flow along the Stagnation Streamline of a Blunt Body.- On Model Kinetic Operators and Corresponding Langevin Sources for a Non-Equilibrium Plasma.- Related Fields.- Rarefied Gas Dynamics as Related to Controlled Thermonuclear Fusion.- Vacuum Ejectors with Appreciably Uneven Flows in Channels at Low Reynolds Numbers.- Simulation of the Process of the Cosmic Body Formation.

2,747 citations


"A phenomenological and extended con..." refers methods in this paper

  • ...Extended gas dynamic continuum models can be derived by either perturbation methods, commonly known as the Chapman–Enskog expansion [4], or moment methods [6,9]....

    [...]

Frequently Asked Questions (2)
Q1. What contributions have the authors mentioned in the paper "A phenomenological and extended continuum approach for modelling non-equilibrium flows" ?

This paper presents a new technique that combines Grad ’ s 13-moment equations ( G13 ) with a phenomenological approach to rarefied gas flows. In this study, the proposed numerical procedure is tested on a planar Couette flow case, and the results are compared to predictions obtained from the direct simulation Monte Carlo method. 

As shown in Fig. 3f, this is partly due to the extended constitutive terms in the G13 equations and further improved by the wall scaling function. In particular, it can be shown that the proposed solution method for the G13 equations is incapable of capturing non-linearities of normal stress and tangential heat flux occurring in the vicinity of the wall.