Motion Of A Spherical Aerosol Particle In A Micro-Pipe
M. Ota*, K. Kuwahara* and S. Stefanov**
*
Tokyo Metropolitan University, Tokyo, Japan
Institute of Mechanics, BASci, Sofia, Bulgaria
**
Abstract. We present the DSMC analysis of the motion of a small spherical particle in a circular micro-pipe filled with
gas molecules with given number density. We simulate a variety of steady-state flow regimes: from free molecular to
near continuum. Drag and Brownian force are computed for different particle and pipe sizes.
INTRODUCTION
Recently it becomes clear that the studies of aerosol particle motion in micro-scale systems are very important due to
their numerous industrial applications in micro- and nano-scale technologies. The mean free path of the gas
molecules in such a micro-scale gas flow can be comparable to the dimensions of the flow domain. Consequently,
the continuum description of the flow can be not valid and therefore kinetic formulation must be applied. The
DSMC method [1] seems to be the best numerical technique that is naturally suited for computation of micro-sized
molecular flow. For instance, a general equation describing the particle motion through a gas should include three
forces acting on the particle: an external force (gravitational, laser trapping, electromagnetic etc.), drag force and the
Brownian force which cannot be neglected for micro- and nano-sized particles. The DSMC method can be applied
directly for estimation of the drag and the Brownian forces [2] by using two different time-averaging procedures:
long time (for the drag force) and short time (for the Brownian force) averaging of the impulses of the molecules
striking the aerosol particle surface. The numerical results presented in the paper illustrate this approach. On other
hand, the micro-flows are difficult for a DSMC analysis considered with respect to macroscopic field calculations.
The reason is that the macroscopic gradients and bulk velocity in the micro-flows of practical interest are usually
small and the statistical error of the DSMC calculations is of order comparable to the magnitudes of the flow
macroscopic characteristics. We overcome these contradiction demands by using a simple data filter for computation
of the averaged flow fields. An alternative approach is the information preservation method proposed by Fan and
Chen [3].
In our work we simulate the translation of a small spherical particle in a micro-sized circular pipe by using DSMC
method. A simple hard sphere model of air molecules is chosen for the gas phase filling the pipe. Particularly, a
special attention is paid to the analysis of the forces acting on the particle. A comparison between free-molecular,
transition and near continuum flow regimes is presented. The treated problem is important for MEMS devices,
semiconductor manufacturing etc. The possibility for control of the particle motion by using laser trapping method is
considered.
PROBLEM FORMULATION AND COMPUTATIONAL CONSIDERATION
We consider the motion of a small spherical particle with radius RP in a circular pipe with length L and radius R (RP
< R). The pipe is filled with gas with density ρ0. In our consideration we use a hard sphere model for presenting the
air molecules. The molecule mass and diameter are 4.815×10-26 kg and 3.7×10-10 m, respectively. In general, the
equation of a spherical micro-particle motion in a gas can be written as follows:
mp
dU p
dt
= FE + FD + FB .
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
776
(1)
where mp is the aerosol particle mass, Up(t) is the instantaneous particle velocity, FE is a constant external force
acting on the particle. Here we assume an arbitrary external force FE that might be: the particle weight FE =Fg= mpg
(g is gravity acceleration), an electrostatic force acting on a charged particle, a laser trapping force acting on an
electrically neutral particle etc. In our consideration we treat only the established flow when the particle is moving
with a constant velocity UP. It means that for simplicity we have neglected the fluctuations of the aerosol particle
velocity due to the action of the random Brownian force. Under these conditions a local cylindrical coordinate
system is imposed in which the spherical particle is fixed in the center of the pipe axis and the pipe wall is moving
along the pipe axis with a constant velocity UP (see Fig. 1). In this way, the pipe wall, the ingoing and outgoing gas
fluxes are supposed to have temperature T0 and velocity UP. Temperature of the aerosol particle is also T0. Diffuse
reflection is supposed on both pipe and particle surfaces. For an established flow regime we also suppose that the
external force is equal to the summarized drag and Brownian force action and use this assumption to estimate the
magnitude of the external force needed to trap a particle moving in a micro-pipe with velocity Up.
r
UP
R
R
P
x
L
FIGURE 1. Geometry of the flow
The DSMC calculations are carried out in a two-dimensional cylindrical domain (x,r)∈ (0,L)×(0,R) by using the
standard DSMC procedures for simulation of flows with axial symmetry [1]. A uniform rectangular basic grid with
Nx×Nr cells covers the computational domain. Several runs are carried out with computational parameters one-toone equal to physical requirements. An example is a pipe with dimensions (L,R)=(16×6) µm and a radius of the
spherical particle Rp=0.5 µm. For sea-level pressure p0=105 Pa air number density is equal to n0=2.6866×1025. The
total number of the air molecules in the pipe volume is about 8×106. In this case the computational domain is
covered by grid with 1200×600 basic cells. Within every time step the basic cells near to the sphere surface are
subdivided dynamically into subcells in order to meet the spatial resolution requirements; the size of a cell is less
than the local mean free path. The flow field sampling is conducted on a coarser grid with larger cells containing
20×20 basic cells. This multilevel grid scheme [4] allows calculation of the molecular processes correctly on an
adaptive fine grid and at the same time provides a meaningful sample size for the macroscopic variables. In addition,
a simple filtering procedure [5] is applied along with the standard time averaging of the accumulated flow field data.
For all cases computed temperature is T0=288oK. The smallest time step used for computation of near continuum
cases ∆t=1.0×10-10 s. The force acting on the spherical particle is determined from the rates of delivery and removal
of momentum by incident and reflected molecules from the particle surface within time interval ∆ts
Fx =
1
∆ts
∑ m(V
(+)
x
− Vx( − ) ).
(2)
Thus the drag force is equal to FD = Fx estimated over a large ∆ts=105∆t. The Brownian force is determined as
follows
FB = < ( Fx' − FD ) 2 > =
1
1
( Fx' − FD ) 2 , Fx' = '
'
N
∆ts
∑ m(V
(+)
x
− Vx( − ) ).
(3)
In our calculations the short time interval is chosen ∆t’s=500∆t. In eq. (3) N’=200-300 is the number of short
intervals used for estimation of the Brownian force. Thus the forces acting on the sphere can be estimated directly
without additional assumptions. Recently, an interesting alternative DSMC approach using essentially the
777
assumption for a local free-molecular character of the flow near to the aerosol particle has been proposed in Gallis et
al. [6].
ANALYTICAL APPROXIMATIONS FOR THE CASE R>>RP
In the case when the pipe radius R is much larger than the particle radius Rp the influence of the pipe walls can be
neglected, and respectively, the drag force can be expressed on the base of continuum description of viscous gas by
the famous Stokes’ formula
FD = Fs = 6πηU P RP ,
(4)
which is valid for small Reynolds numbers and large particle radius Rp. In eq. (4) the dynamic viscosity is equal to
the first approximation of the Chapman-Enskog theory
η = (5 16) ρ 0λ0 2π RT0 M , λ0 = ( 2π d 2 n0 ) −1 ,
(5)
for “hard sphere” gas (here, λ0 is mean free path of the gas molecules with diameter d, M is the molecular weight of
the gas,). As the particle size considered here is comparable to mean free path of gas molecules for computation of
the drag force FD instead eq. (2) we use the approximation, suggested by Beresnev et. al [7] on the base of kinetic
theory:
FDRG =
8+π
2π RT0 
Kn
0.310 Kn


ρ 0 RP2U P
1 +
 .

2
3
M  Kn + 0.619  Kn + 1.152 Kn + 0.785  
(6)
In eq. (6) a complete accommodation is supposed on the particle surface. The Knudsen number is equal to Kn=λ0/Rp.
Figure 2 illustrates the difference between both formulations of the drag force. One can see that the downwards
velocity of the micro-sphere with radius Rp=5 µm in the sea-level atmosphere (pressure is 105 Pa) computed from
eq. (6) is larger than the computed from eq. (4).
1.4
x 10
R
-11
particle
= 5µm
1.2
1
F 0.8
[N]
0.6
F=mg
0.4
U
Stokes
0.2
U
0
0
0.002
0.004
0.006
U
p
rarefied gas
0.008
0.01
[m/s]
FIGURE 2. Dependence of the drag force on the velocity of a water drop with radius RP = 5 µm computed on the base of
the Stokes formula, eq. (4), (circles) , and the rarefied gas approximation, eq. (6) (triangles), respectively. The cross-points with
the horizontal line of the particle weight force define the corresponding velocity of sedimentation.
From Fig. 2 it is clearly seen that for micro- and nano-sized particles the velocity of sedimentation is very small.
This is sufficiently beyond the accuracy of the DSMC simulation and further we do not consider cases with gravity
action.
Other important limit is the free-molecular regime. In this case the drag force can be written in the form [1]
778
FDFM =
π
2
ρ 0 RP2U P
2 RT0
( s.CD ) .
M
(7)
where the non-dimensional drag coefficient CD for complete diffuse reflection on the sphere surface is equal to
CD =
2s 2 + 1
4s 4 + 4s 2 − 1
2π 1/ 2
2
exp(
−
s
)
+
erf
(
s
)
+
, s=
π 1/ 2 s 3
2s 4
3s
For s<<1 from (8) we come to the simple formula
CD =
UP
.
2 RT0
M
8 +π 2
.
3π 1/ 2 s
(8)
(9)
Replacing CD in eq. (7) we obtain the well-known Epstein’s formula [8]
'
FDFM
=
8+π
2π RT0
,
ρ 0 RP2U P
3
M
(10)
valid for isothermal diffuse reflection and small UP compared to thermal velocity in the gas. Further we use these
analytical results for validation of the DSMC calculations for RP.<< R.
NUMERICAL RESULTS
All DSMC calculations are conducted for hard sphere molecular model. The molecular magnitudes are based on air
with mass m=4.81×-26 kg and diameter d=4.19×10-10 of a single molecule [1]. The molecular weight of sea-level air
is M=29 kg kmol-1. It is worth noticing that all computations of free molecular regime have been done for number
density equal to n0=2.6866×1025, which is much larger than density in a real free-molecular flow. However, this
artificial collisionless regime helps us to estimate the limits of the assumption for local free-molecular regime [6]
Besides, in free-molecular regime the non-dimensional drag coefficient does not depends on gas density and can be
estimated correctly for arbitrary stream density. The drag force depends on the local Knudsen number and for a real
free molecular flow is much less than the computed for n0 in our paper. The advantage here is that we can compare
results for collisional and collisionless flow and show the influence of the collisional process on the drag and
Brownian force directly.
The results of comparison of the drag coefficient CD computed by using DSMC method with the analytical
approximation (9) for low speed free-molecular flow when Rp<<R is shown in Fig. 3.
3
10
eq. (9)
MC
2
Cd
10
1
10
0
10
1
10
2
10
U [m/s]
3
10
p
FIGURE 3. The drag coefficient CD for spherical particle versus stream velocity UP.
779
One can see that the simple analytical approximation (9) is excellent for UP < 200 m/s (s<1/2) and still good
enough for UP<300 m/s. The small systematic deviation of the DSMC results above the analytical line is apparently
due to a small influence of the pipe radius R. Here R=4.0×10-6 m, L= 16.0×10-6 m, and Rp=0.5×10-6 m. More clear
the influence of the pipe radius on the drag coefficient CD and drag force FD is shown in Figs. 4, (a) and (b),
respectively. In both figures the upper pair of lines presents the results of simulation of a free molecular flow with
the same gas density such as the one in the near-continuum flow presented with the lower lines. From Fig. 4 (b) one
can conclude that the collisional process decreases sufficiently the drag force comparing with the artificially
simulated free molecular flow with the same inflow density.
-8
300
1.6
x 10
eq.(9)
MC
1.4
250
1.2
200
1
collisionless flow
FD [N]
Cd
150
0.8
0.6
100
0.4
50
near-continuum flow
0.2
0
1
1.5
2
2.5
3
3.5
R [mm]
4
4.5
5
5.5
0
6
1
1.5
2
2.5
3
a
3.5
R [mm]
4
4.5
5
5.5
6
b
FIGURE 4. Drag coefficient (a) and drag force (b) dependence on pipe radius R. (Rp=0.5 µm , L=16.0 µm)
Apparently, the numerical results show that for R>4.0 µm (R>8RP) the influence of the pipe boundaries is weak and
can be neglected. Another important conclusion is related to the assumption for a local free molecular character of
the flow in vicinity of the submicron particles. Our computations for the collisionless flow with different L=8.0, 16.0,
32.0 showed that for R>(8-10)RP the results were stabilized. This fact can give an estimation of the maximal size of
the domain around the particle where the assumption for local free-molecular character of the flow [6] is valid. For
a near-continuum regime the boundary influence increases but the numerical results show that a condition R > (815)RP might be sufficient for a consideration of the limit R>>RP. Further we use this assumption when simulating a
near-continuum regime.
In Table 1 the magnitudes of the drag forces acting on the particle are summarized for different stream velocities
of a near continuum flow regime. The drag force obtained by DSMC method for R=4.0 µm is larger than the
computed by using eq.(6). This might be explained by the boundary influence but a further detailed calculation
should be carried out to clarify this point.
-9
11
-9
x 10
3.5
x 10
10.5
3
10
9.5
2.5
F'x(t)
F'x(t)
9
8.5
8
2
7.5
7
1.5
6.5
6
0
0.5
1
1.5
t [s]
2
2.5
1
3
-6
x 10
a
0
0.5
1
1.5
t [s]
2
2.5
3
b
FIGURE 5. Drag force oscillations for (a) collisionless and (b) collisional regimes (for parameters see Table 2).
780
-6
x 10
TABLE 1. Drag coefficients and forces for different stream velocities ( n0=2.68×1025 , T0=2880K, RP=0.5 µm, R=4.0 µm)
UP [m/s]
CD
FD [N], eq. (6)
FD [N] , DSMC
50.0
9.9
0.74×10-8
1.26×10-8
10.0
41.4
0.148×10-8
0.21×10-8
-9
1.0
583.0
0.148×10
0.29×10-9
The force oscillations (see eq. (3)) obtained within the simulation time are shown in Fig. 5. We should note here
that keeping the axial symmetry we have neglected the oscillations in radial direction. But, in our view, the obtained
results can give a magnitude of the Brownian force correct enough for our aim to consider it in the light of a laser
trapping control of the particle motion. In our paper we have presented the most problematic regime of particle
motion when the gas pressure in the pipe is atmospheric. From this viewpoint this limit case shows that the laser
trapping force should be of the same order of the summarized drag and Brownian force. In the paper we present
some preliminary results and a more detailed investigation on the Brownian force acting on submicron particles in a
gas flow is in progress.
The DSMC computations of the Brownian force for two flow regimes (collisionless and collisional) are
presented in Table 2. We should note that first paper showed that the DSMC method can be used effectively for
estimation of the Brownian force is due to Nanbu [9]. From Table 2 one can see that the Brownian force is (10-20)%
from the drag force, and consequently cannot be neglected in a submicron flow consideration. An interesting fact is
that the absolute magnitude FB is larger in the collisionless simulation than in the collisional one. But related to the
corresponding drag force it is relatively larger in the collisional regime.
TABLE 2. Brownian force (.( n0=2.68×1025 , T0=2880K, RP=0.5 µm, R=4.0 µm, UP=10.0 m/s)
Flow regime
CD
FD [N]
FB [N]
Collisionless
170.3
0.87×10-8
0.97×10-9
Collisional
41.4
0.21×10-8
0.5×10-9
FB/FD
0.115
0.23
25
x 10
2.72
290
2.712
289.7
2.704
-6
-6
289.1
temperature
x 10
2.688
3
y [m]
289.4
2.696
density
x 10
288.8
3
2
y
[m] 2
2.68
1
2
4
6
8
x [m]
10
12
288.5
1
2.672
288.2
14
-6
x 10
2
2.664
4
6
8
x [m]
10
12
14
-6
x 10
2.656
287.6
2.648
287.3
2.64
-6
longitudinal (u) velocity
x 10
53.3667
7.5533
48.03
6.3253
42.6933
5.0973
-6
3.8693
transversal(v) velocity
x 10
32.02
2.6413
3
2
y [m]
y [m]
287
37.3567
3
26.6833
1
4
6
8
x [m]
10
12
14
-6
x 10
2
1.4133
1
21.3467
2
287.9
0.1853
2
16.01
10.6733
4
6
8
x [m]
10
12
14
-6
x 10
-1.0427
-2.2707
5.3367
-3.4987
0
-4.7267
781
25
x 10
2.72
2.712
2.704
-6
velocity vector field
x 10
-6
x 10
2.688
3
2
y [m]
y [m]
3
2.696
flow streamlines and gas density
1
2
2.68
1
2
4
6
8
x [m]
10
12
14
2.672
2
16
4
6
8
x [m]
-6
x 10
10
12
14
2.664
-6
x 10
2.656
2.648
2.64
FIGURE 6. Flow fields around a particle with radius RP=0.5 µm for stream velocity UP=50 m/s (R=4.0 µm, L=16.0 µm)
25
x 10
289
2.7
288.7
2.696
288.4
2.692
-6
-6
288.1
2.688
3
3
2.684
y [m]
y [m]
temperature
x 10
density
x 10
2
2.68
1
2
4
6
8
x [m]
10
12
287.8
2
287.5
1
2.676
14
2
4
6
-6
x 10
8
x [m]
10
12
287.2
14
-6
x 10
2.672
286.6
2.664
286.3
2.66
286
15.2889
3.1044
13.76
2.505
12.2311
-6
longitudinal (u) velocity
x 10
2
7.6444
1
transversal(v) velocity
1.3061
3
9.1733
y [m]
y [m]
1.9056
-6
x 10
10.7022
3
0.7067
2
0.1072
1
2
4
6
8
x [m]
10
12
6.1156
14
2
-6
x 10
286.9
2.668
4
6
4.5867
8
x [m]
10
12
-0.4922
14
-6
x 10
3.0578
-1.0917
-1.6911
1.5289
-2.2906
0
-2.89
25
x 10
2.7
2.696
2.692
-6
velocity vector field
x 10
-6
3
2.688
3
y [m]
y [m]
flow streamlines and gas density
x 10
2
2.684
2
2.68
1
1
2
2
4
6
8
x [m]
10
12
14
-6
x 10
4
6
8
x [m]
10
12
2.676
14
-6
x 10
2.672
2.668
2.664
2.66
FIGURE 7.
Flow fields around a particle with radius RP=2.0 µm for stream velocity UP=10 m/s (R=4.0 µm, L=16.0 µm)
782
CONCLUSIONS
The DSMC method can be applied directly for estimation of the drag and the Brownian forces [2] by using two
different time-averaging procedures: long time (for the drag force) and short time (for the Brownian force)
averaging of the impulses of the molecules striking the aerosol particle surface. The numerical results presented in
the paper illustrate this approach. In our work we simulate the translation of a small spherical particle in a microsized circular pipe by using DSMC method. A simple hard sphere model of air molecules is chosen for the gas phase
filling the pipe. Particularly, a special attention is paid to the analysis of the forces acting on the particle. A
comparison between free-molecular, transition and near continuum flow regimes is presented. The treated problem
is important for MEMS devices, semiconductor manufacturing etc. The possibility for control of the particle motion
by using laser trapping method is considered.
ACKNOWLEDGEMENTS
One of the authors, M.O., would like to acknowledge the financial support provided by the Japanese Ministry of
Education,Science,Sports and Culture with Grant-in-Aid Scientific Research (B),No.13555048.The third authors, S.
S., would like to acknowledge the financial support provided by the Bulgarian Ministry of education with Grant No.
MM806/98.
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
Bird, G. A., Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Clarendon Press, Oxford, 1994.
Nanbu, K., J. Chem. Phys. 82, 3329-3334 (1985).
Fan, J., and Shen, C., J. Comp. Phys. 167, 393-412 (2001).
Stefanov, S., Boyd, I., and Cai, C., Phys. Fluids 12, 1226-1239 (2000).
Golshtein, E., and Elperin T., J.Thermophysics and Heat Transfer 12, 250-269 (1996).
Gallis, M., Torczynski, J., and Rader, D., Phys. Fluids 13, 3482-3492 (2001).
Beresnev, S., A., Chernyak, V., G., and Fomyagin, G., A., J. Fluid Mech. 219, 405-421 (1990).
Epstein, P., Phys. Rev. 23, 710-733 (1924).
Nanbu, K., J. Chem. Phys. 82, 3329-3334 (1985).
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