Using DSMC to Compute the Force on a Particle
in a Rarefied Gas Flow
J. R. Torczynski, M. A. Gallis, and D. J. Rader
Engineering Sciences Center; Sandia National Laboratories; Albuquerque, New Mexico 87185-0826 USA
Abstract. An approach is presented to compute the force on a spherical particle in a rarefied flow of a monatomic gas. This
approach relies on the development of a Green’s function that describes the force on a spherical particle in a delta-function
molecular velocity distribution function. The gas-surface interaction model in this development allows incomplete
accommodation of energy and tangential momentum. The force from an arbitrary molecular velocity distribution is
calculated by computing the moment of the force Green’s function in the same way that other macroscopic variables are
determined. Since the molecular velocity distribution function is directly determined in the DSMC method, the force
Green’s function approach can be implemented straightforwardly in DSMC codes. A similar approach yields the heat
transfer to a spherical particle in a rarefied gas flow. The force Green’s function is demonstrated by application to two
problems. First, the drag force on a spherical particle at arbitrary temperature and moving at arbitrary velocity through an
equilibrium motionless gas is found analytically and numerically. Second, the thermophoretic force on a motionless particle
in a motionless gas with a heat flux is found analytically and numerically. Good agreement is observed in both situations.
INTRODUCTION
Particle transport in rarefied gas flow is of interest in applications such as MEMS devices (small length scales) and
semiconductor processing (low pressures). Rarefied particle-transport calculations have two aspects: computing the
rarefied gas flow and computing the particle transport in this flow. The Direct Simulation Monte Carlo (DSMC)
method is a well-established method for simulating rarefied gas flow [1-2]. Particle transport calculations require
determination of the force on and heat transfer to a particle in a rarefied gas flow.
An approach is presented for calculating these quantities with an arbitrary molecular velocity distribution function.
The approach involves Green’s functions for the force on and heat transfer to a spherical particle from a delta-function
molecular velocity distribution function. The moments of these Green’s functions with respect to the molecular
velocity distribution function yield the desired force and heat transfer.
This approach is demonstrated by application to two problems. First, the drag force on a spherical particle at
arbitrary temperature moving at arbitrary velocity through an equilibrium motionless gas is found analytically and
numerically. Second, the thermophoretic force on a motionless particle in a motionless gas with a heat flux is found
analytically and numerically.
FORCE ON AND HEAT TRANSFER TO A PARTICLE
Under the following assumptions, expressions for the force on and heat transfer to a particle in a rarefied gas flow
with an arbitrary molecular velocity distribution function can be found. The particle concentration is dilute, the gas is
monatomic, and the gas flow is strictly free-molecular on the particle scale (but not necessarily on the system scale).
The particle is spherical, with uniform temperature (not necessarily the gas temperature) and uniform surface
properties. A gas-surface interaction model is used that linearly combines features of the Maxwell and Lord models to
allow incomplete accommodation of energy and tangential momentum [3-5]. Brownian motion is neglected here.
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
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The force F and heat transfer Q to a spherical particle are given by integrals of the molecular velocity distribution
function f [ u ] multiplied by the force and heat-transfer Green’s functions F δ and Q δ , which represent the force on
and heat transfer to a spherical particle from a delta-function molecular velocity distribution (see Figure 1) [5]:
F [ f ] = ∫ F δ [ u – u p ] f [ u ] du ,
(1)
Q [ f ] = ∫ Q δ [ u – u p ] f [ u ] du ;
(2)
2
F δ [ c ] = mn ( πR p )c{a 1 c + a 2 ( π
1⁄2
⁄ 3 )c p } ,
2
2
2
(3)
Q δ [ c ] = a 2 mn ( πR p ) c { ( 1 ⁄ 2 ) c – c p } ;
(4)
a1 = 1 + ( 4 ⁄ 9 ) ( 1 – ε p ) ( 1 – α p ) ,
(5)
a 2 = ( 1 – ε p )α p ,
(6)
c p = ( 2k B T p ⁄ m )
∫ f [ u ] du
1⁄2
,
= 1.
(7)
(8)
Here, u and u p are molecule and particle velocities, c = u – u p is their difference, m is the molecule mass, n is the
molecule number density, R p is the particle radius, T p is the particle temperature, 0 ≤ ε p ≤ 1 is the fraction of all
reflections that are specular, 0 ≤ α p ≤ 1 is the fraction of diffuse reflections that are isothermal, and k B is Boltzmann’s
constant. Thus, the fraction of all reflections that are diffuse and isothermal is ( 1 – ε p )α p , and the fraction of all
reflections that are diffuse and adiabatic (the molecular speed is unchanged but the direction is randomized upon
reflection) is ( 1 – ε p ) ( 1 – α p ) [5].
The above expressions can be implemented analytically and numerically. Analytical implementation requires a
closed-form representation of the molecular velocity distribution function to be provided (e.g., a Maxwellian
distribution). Numerical implementation requires a numerical representation of the molecular velocity distribution
function to be provided (e.g., from DSMC). In either situation, the provided molecular velocity distribution function
3
is used to compute the moments 〈 u – u p〉 , 〈 u – u p 〉 , 〈 ( u – u p ) u – u p 〉 , and 〈 u – u p 〉 , which are computed like
moments for other macroscopic quantities [2].
NUMERICAL IMPLEMENTATION
The above approach has been implemented in the DSMC codes DSMC1 [2] and Icarus [6] in the following manner.
The number density used in the Green’s functions corresponds to a computational molecule and is taken to be the
number of real molecules the computational molecule represents divided by the volume of the mesh cell within which
it is located. The particles are taken to be small and dilute: the volume they occupy is negligible, the particles do not
collide with each other, and the computational molecules are not affected by collisions with a particle (i.e., neither their
momenta nor their energies are altered). Since both the force and the heat transfer are exactly proportional to the crosssectional area of the particle in the locally free-molecular limit considered here, the force and heat transfer per unit
cross-sectional area are computed so that the results can be applied to particles of different size. Since three of the four
moments indicated above include odd powers of u – u p , these moments cannot be expanded as polynomials in u
and u p . Thus, these moments are determined either by computation at the prescribed value of the particle velocity u p
or by interpolation from values computed for nearby particle-velocity values.
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SPHERE-DRAG APPLICATION
An example of the Green’s function approach is the calculation of the drag force on a spherical particle with
arbitrary temperature T p and complete accommodation ( α p = 1 ) that is moving at arbitrary velocity u p through a
motionless Maxwellian distribution with number density n , temperature T , and molecular mass m :
2k B T
 1 
 u 2
2
2
- exp –  ----- , u = u ⋅ u , c 0 =  ------------- .
f [ u ] =  ---------------2
3⁄2 3

m 
 π c 0
 c 0
(9)
Application of the above approach yields the following expressions for the force and heat transfer in this case [5]:
up
up  up
Tp 1⁄2
2
2 
F = – ( mnc 0 πR p )  m 1 ------ + ( 1 – ε p )  -------
m 2 ------  ------ ,


c0
T
c 0  c 0 

(10)
up
Tp
up 

3
2
Q = ( mnc 0 πR p ) ( 1 – ε p )  k 1 ------ –  ------- k 2 ------  ,
T
c0
c0 

(11)
2
4
1⁄2
2
( – 1 + 4s + 4s )erf [ s ] + ( 1 + 2s )serf [ s ]
π
m 1 [ s ] = ------------------------------------------------------------------------------------------------------ , m 2 [ s ] = ----------- ,
3
3
4s
2
4
2
(12)
2
( 3 + 12s + 4s )erf [ s ] + ( 5 + 2s )serf [ s ]
( 4 + 8s )erf [ s ] + ( 4 )serf [ s ]
k 1 [ s ] = ---------------------------------------------------------------------------------------------------- , k 2 [ s ] = -------------------------------------------------------------------- ,
8s
8s
(13)
s
2
2 
d
2
2
- exp [ – t ] dt , serf [ s ] = s  ----- ( erf [ s ] ) =  ----------- exp [ – s ] s .
erf [ s ] =  --------- 1 ⁄ 2 ∫
 ds
 1 ⁄ 2
π
π
0
(14)
In the limit of small speed ratio s = u p ⁄ c 0 , the force simplifies to the well known expression of Epstein [7], and the
heat transfer has a corresponding expression:
T 1 ⁄ 2 π1 ⁄ 2  u p
8 
2
2 
 ------p-
 -----------  ------ ,
+
F Epstein = – ( mnc 0 πR p )   -------------(
1
–
ε
)
p  T 
 1 ⁄ 2
 3   c 0 
 3π

(15)
 2   T p  2  
3
2
- – ------- ----------- .
Q Epstein = ( mnc 0 πR p ) ( 1 – ε p )   --------- 1 ⁄ 2  T   1 ⁄ 2 
 π

π
(16)
DSMC computations with the Green’s function approach are performed using Icarus [6]. A two-dimensional
10 mm × 1 mm rectangular domain with 100 × 10 square cells is used. The 10-mm boundaries are specular, and the 1mm boundaries are diffuse and isothermal at a temperature of 273 K. Each cell has 100 computational molecules on
average. A time step of 100 ns is used, and statistics are accumulated for 0.5 million time steps. The gas is at 273 K
and 13.33 Pa (100 mtorr). The VSS collision model is used with argon molecular properties from Bird [2]:
– 26
– 10
m = 6.63 ×10
kg , d ref = 4.11 ×10
m , ω = 0.81 , and α = 1.40 . The particle has a diffuse, isothermally
reflecting surface (i.e., ε p = 0 and α p = 1 ). Two cases are considered: equal particle and gas temperatures (as in
Baines et al. [8]), and zero heat flow to the particle (physically realistic for a constant particle velocity). Figure 2 shows
comparisons of the theoretical and computed drag force ratio (the actual force divided by the above low-speed result
of Epstein [7]). Good agreement is seen for both cases at all values of the particle speed ratio.
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εp
αp
θ
m, n
φ
θ
Rp
θ
Tp
shadow region
c = u - up
FIGURE 1. Geometry to determine Green’s functions in the particle’s reference frame.
FIGURE 2. Drag force ratio for a spherical particle in a Maxwellian distribution.
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THERMOPHORESIS APPLICATION
Another example of the Green’s function approach involves the force on a motionless particle in a motionless gas
with a nonzero heat flux (see Figure 3). In this situation, the particle experiences a “thermophoretic” force in the
direction of the heat flux [9]. When the gas flow around the particle is free-molecular, the thermophoretic force F
depends on the heat flux q , the molecular speed c = 8k B T ⁄ πm , and the particle radius R p according to the relation
2
F = ξ ( qπR p ) ⁄ c , where ξ is an order-unity nondimensional quantity termed the “thermophoresis parameter” [10].
The value of ξ depends on the molecular velocity distribution function f [ u ] and the gas-surface interaction for the
particle. Here, the particle is taken to have a diffuse, isothermally reflecting surface (i.e., ε p = 0 and α p = 1 ).
The thermophoresis parameter can be determined for a Chapman-Enskog molecular velocity distribution function
corresponding to an intermolecular force inversely proportional to the intermolecular separation to a positive power.
ν
A repulsive intermolecular force proportional to 1 ⁄ r is known to produce a temperature-dependent viscosity of the
ω
form µ ⁄ µ ref = ( T ⁄ T ref ) , where ω = ( 1 ⁄ 2 ) + ( 2 ⁄ ( ν – 1 ) ) and “ref” denotes a reference value [2,11]. A hardsphere molecule has ω = 1 ⁄ 2 ( ν → ∞ ), and a Maxwell molecule has ω = 1 ( ν = 5 ). Values of the viscosity
temperature exponent ω for real molecules typically lie between these two values [2]. Table 1 gives ξ values to be
used in the force expression for intermediate values of ω in the range 1 ⁄ 2 ≤ ω ≤ 1 based on infinite-approximation
Chapman-Enskog theory [11] and the Green’s function approach [12]. Also included for reference are the ratios of the
infinite-approximation viscosity and thermal conductivity, µ and k , to the first-approximation values, µ 1 and k 1 .
DSMC computations with the numerical implementation of the Green’s function approach are performed using a
modified version of DSMC1 [2]. A one-dimensional 1-mm linear domain with 100 uniform cells is used. The lower
and upper boundaries are diffuse and isothermal at temperatures of 263 K and 283 K, respectively. Each cell has 30
computational molecules on average. A time step of 10 ns is used, and statistics are accumulated for 500 million time
steps. The accumulation method in DSMC1 is modified from move-collide-sample [2] to move-sample-collide-sample
to improve determination of cell-based values for the heat flux and the force. The gas is initially at 273 K and 133.3 Pa
(1000 mtorr). The VHS collision model (i.e., the VSS collision model but with α = 1 ) is used with the following
– 26
–5
argon-like molecular properties from Bird [2]: m = 6.63 ×10
kg , T ref = 273 K , and µ ref = 2.117 ×10 Pa ⋅ s .
Six ω values are used: 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0. In order to maintain the same value of the reference viscosity at
the reference temperature, the reference diameter for each ω value is prescribed as below [2]:
1⁄2 1⁄2
d ref
 5 ( α + 1 ) ( α + 2 ) ( mk B T ref ⁄ π ) 
=  ---------------------------------------------------------------------------------
4α ( 5 – 2ω ) ( 7 – 2ω )µ ref


1⁄2
 15 ( mk B T ref ⁄ π )

→  -------------------------------------------------------
 2 ( 5 – 2ω ) ( 7 – 2ω )µ ref
1⁄2
for α → 1 .
(17)
Thus, the Knudsen number is Kn = 0.0475 in all six simulations, where Kn = λ ⁄ L and λ = 2µ ⁄ mnc .
The results from these six simulations are shown in Figures 4-6. The temperature profiles shown in Figure 4 exhibit
only a weak dependence on ω , as expected since the Knudsen number is fixed. The profiles of the thermophoresis
parameter ξ , shown in Figure 5, exhibit well-defined (Knudsen) layers adjacent to the walls and uniform central
regions between the Knudsen layers. The theoretical Chapman-Enskog values for ξ are also shown and are seen to be
in good agreement with the ξ values in the central regions. This comparison is made quantitative in Figure 6, which
compares the theoretical Chapman-Enskog values for ξ and the numerical DSMC values for ξ (averaged over the
central 20% of the domain) as functions of ω . The uncertainty in the DSMC values is comparable to the size of the
symbols in the figure. The theoretical and numerical values agree closely. This comparison could be extended by using
the VSS model with the α value that best matches the inverse power law: α = 2 A 2 [ ν ] ⁄ ( 2 A 1 [ ν ] – A 2 [ ν ] ) based on
equating Schmidt numbers [2,11], where Chapman and Cowling [11] tabulate the functions A 1 [ ν ] and A 2 [ ν ] .
CONCLUSIONS
An approach has been presented for determining the force on and heat transfer to a spherical particle in a locally
rarefied gas flow. The approach uses Green’s functions that describe the force and heat transfer from a delta-function
molecular velocity distribution function. The force and heat transfer from an arbitrary molecular velocity distribution
function are found by taking the moments of the Green’s functions.
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T2
x=L
heat flux q
force F
x=0
T1
FIGURE 3. Thermophoresis geometry: gas and particle are motionless (particle size is greatly exaggerated).
FIGURE 4. Temperature profiles: left and right wall temperatures are 263 K and 283 K, respectively.
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FIGURE 5. Thermophoresis-parameter profiles: solid curves, DSMC simulations; dashed lines, Chapman-Enskog theory.
FIGURE 6. Thermophoresis parameters from DSMC (averaged over central 20% of domain) and Chapman-Enskog theory.
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This Green’s function approach is applied to two situations. The first involves the drag on a sphere at arbitrary
temperature and moving with arbitrary velocity through an equilibrium gas. The second involves the thermophoretic
force on a motionless sphere in a motionless gas with a nonzero heat flux. For both applications, analytical and
numerical results (from DSMC) are obtained and are found to be in good agreement.
When combined with DSMC simulations of rarefied gas flow, the Green’s function approach offers a convenient
method for determining the force on and heat transfer to particles, as needed for rarefied particle-transport simulations.
TABLE 1. Thermophoresis parameter as a function of viscosity temperature exponent from Chapman-Enskog theory.
ω
ν
ξ
µ/µ1
k/k1
Description
0.5
∞
0.697618
1.016034
1.025218
hard-sphere
0.6
21
0.693698
1.010193
1.016019
0.7
11
0.689896
1.005702
1.008959
0.8
23/3
0.686196
1.002524
1.003965
0.9
6
0.682588
1.000629
1.000988
1.0
5
0.679061
1.000000
1.000000
Maxwell
0.66
13.50
0.691404
1.007339
1.011532
helium, neon
0.81
7.452
0.685832
1.002277
1.003577
argon
0.80
7.667
0.686196
1.002524
1.003965
krypton
0.85
6.714
0.684381
1.001417
1.002227
xenon
ACKNOWLEDGMENTS
This work was performed at Sandia National Laboratories. Sandia is a multiprogram laboratory operated by Sandia
Corporation, a Lockheed Martin Company, for the United States Department of Energy under Contract DE-AC0494AL85000.
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
Bird, G. A., Molecular Gas Dynamics, Clarendon Press, Oxford, 1976.
Bird, G. A., Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Clarendon Press, Oxford, 1998.
Cercignani, C., The Boltzmann Equation and Its Applications, Springer-Verlag, New York, 1988, Chapter III, Section 5.
Lord, R. G., “Some extensions to the Cercignani-Lampis gas-surface scattering kernel,” Physics of Fluids A 3, 1244-710 (1991).
Gallis, M. A., Torczynski, J. R., and Rader, D. J., “An approach for simulating the transport of spherical particles in a rarefied gas flow via the
direct simulation Monte Carlo method,” Physics of Fluids 13, 3482-3492 (2001).
Bartel, T. J., Plimpton, S. J., and Gallis, M. A., Icarus: A 2-D Direct Simulation Monte Carlo (DSMC) Code for Multi-Processor Computers,
User’s Manual - v 10.0, Sandia Report SAND2001-2901, Sandia National Laboratories, Albuquerque, NM, 2001.
Epstein, P. S., “On the resistance experienced by spheres in their motion through gases,” Physical Review 24, 710-733 (1924).
Baines, M. J., Williams, I. P., and Asebiomo, A. S., “Resistance to the motion of a small sphere moving through a gas,” Monthly Notices of the
Royal Astronomical Society 130, 63-74 (1965).
Waldmann, L., “Über die Kraft eines inhomogenen Gases auf kleine suspendierte Kugeln,” Zeitschrift für Naturforschung A14, 589-599
(1959).
Gallis, M. A., Rader, D. J., and Torczynski, J. R., “Thermophoresis in rarefied gas flows,” Aerosol Science and Technology, accepted (2002).
Chapman, S., and Cowling, T. G., The Mathematical Theory of Non-Uniform Gases, Cambridge University Press, Cambridge, 1990.
Gallis, M. A., Rader, D. J., and Torczynski, J. R., “Calculations of the near-wall thermophoretic force in a rarefied gas flow,” Physics of Fluids,
submitted (2002).
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