dsmc (direct simulation monte carlo) analysis of microchannel flow with gas–liquid boundary
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DSMC (direct simulation Monte Carlo) analysis ofmicrochannel flow with gas–liquid boundaryDong-Hun Ryu a & Jinho Lee ba Korea Testing Laboratory , Seoul, 152-718, South Koreab School of Mechanical Engineering, Yonsei University , Seoul, 120-749, South KoreaPublished online: 23 Mar 2007.
To cite this article: Dong-Hun Ryu & Jinho Lee (2006) DSMC (direct simulation Monte Carlo) analysis of microchannelflow with gas–liquid boundary, International Journal of Computational Fluid Dynamics, 20:9, 611-620, DOI:10.1080/10618560601071166
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DSMC (direct simulation Monte Carlo) analysisof microchannel flow with gas–liquid boundary
DONG-HUN RYU† and JINHO LEE‡*
†Korea Testing Laboratory, Seoul 152-718, South Korea‡School of Mechanical Engineering, Yonsei University, Seoul 120-749 South Korea
(Received 17 October 2005; in final form 21 May 2006)
High heat capacity and constant operation temperature make a 2-phase heat remover tool promising forsolving high heat dissipation problems in MEMS devices. However, microscale analysis of the flowwith the conventional Navier–Stokes equation is inadequate, because the non-continuum effect isimportant when the characteristic dimension is comparable to the local mean free path. DSMC is adirect, particle-based numerical simulation method that uses no continuum assumption. In this paper,the gas–liquid boundary effects in microchannel flow are studied using this method. Modified DSMCcode is used to simulate low-speed flow—under which viscous heating produces no significanttemperature change—and MD results are incorporated into the DSMC boundary condition. SteadyCouette flow simulation results show that the gas–liquid boundary affects the density distribution andthe temperature dependence of the slip velocity. Unsteady simulation results show that mass transfer bydiffusion is faster than momentum transfer by collision.
Keywords: DSMC; Gas–liquid; Evaporation; Condensation; Micro fluidics
AMS Subject Classification: 65K05; 90C30
1. Introduction
Microfluidics recently attracted considerable attention as
the analysis and prediction of microscale flows become
important for precise flow control in Micro Electro-
Mechanical System (MEMS) related devices. In micro-
devices, the molecular mean free path is comparable to the
characteristic dimension of the system and the continuum
assumption does not hold. A microscopic approach that
can replace the conventional Navier–Stokes equation is,
therefore, required for analysis of heat and mass transfer in
MEMS devices. DSMC is a direct, particle-based
numerical simulation method. Bird (1994) first proposed
using a DSMC method for simulating rarefied gas
dynamics in re-entry vehicles or for supersonic motion
in low-density fluids. However, if the mean free path is
comparable with the dimensions of macroscopic gradient,
then the flow can be considered to be rarefied even for
normal- and high-density flows. The ratio of the local
mean free path to characteristic length of macroscopic
gradient, known as Knudsen number, is used to quantify
flow rarefaction. The flow is considered to be in the “slip-
flow” regime when 1023 , Kn , 1021 and in the
“transition regime” when 0.1 , Kn , 3. Because of its
extremely small size, the flow in MEMS devices is
generally in one of these two categories.
DSMC has been successful in predicting microflows with
high Kn number. Mavriplis et al. (1997), applied DSMC to
the fluid and thermal analysis of a short microchannel where
the flow is between the continuum and transition regimes.
Piekos and Breuer (1996) studied the microflow in the slip
flow and high-Kn number regimes with an unstructured grid
in the microchannel and a supersonic micro nozzle. Wu and
Tseng (2001) used DSMC to study the microscale gas flows
in various devices incorporating unique pressure boundary
condition treatment. Various microgeometries like orifice
and corner flows were recently simulated using DSMC
(Wang and Li 2004).
DSMC studies of 2-phase flows are rarely found,
however, because the DSMC method cannot be used to
simulate liquid flow. The DSMC method is based on a
molecular chaos assumption and MD or lattice Boltzmann
International Journal of Computational Fluid Dynamics
ISSN 1061-8562 print/ISSN 1029-0257 online q 2006 Taylor & Francis
http://www.tandf.co.uk/journals
DOI: 10.1080/10618560601071166
*Corresponding author. Email: [email protected]
International Journal of Computational Fluid Dynamics, Vol. 20, No. 9, October–November 2006, 611–620
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simulation is used instead of liquid simulation. But the gas
part of a 2-phase flow in which the gas and liquid flows are
in contact can be simulated by DSMC method if a proper
gas–liquid boundary condition is incorporated. This
condition is met in 2-phase annular and stratified flows
in microchannel. Two-phase flow analysis in microdevices
is of great importance when the potential advantage of
using a phase change in a heat exchanger is considered.
Also, the liquid molecules adsorbed on the channel
surface are much stronger than gas molecules, and so most
collisions at surfaces may be gas–liquid interactions,
rather than gas-wall interactions (Gad-el-Hak 1999,
Karniadakis and Beskok 2002). Recently, the heat and
fluid characteristics of two-phase microflows were
researched extensively to clarify the microfluidic
phenomena in 2-phase flow. Stanley et al. (1997)
conducted experiments using rectangular aluminum
microchannels with hydraulic diameters ranging between
56 and 256mm. Their results show that the measured
pressure drop is substantially lower than that predicted by
available semi-empirical relations for two-phase flow.
Peng et al. (1998) theoretically derived thermodynamic
aspects of phase transformations of liquids in micro-
channels, and their results show that no vapor bubble is
present in the microchannel regardless of the intensity of
the heat flux because of the small dimension of the
microchannel. They compared the result with previous
experiments that observed no boiling nucleation in a
microchannel with cross-sectional dimensions from
0.1 £ 0.3 to 0.6 £ 0.7 mm2. But Kandlikar (2002)
investigated the flow boiling characteristics in minichan-
nels and microchannels and addressed that the nucleating
bubbles of sizes 10 , 20mm and surface tension effects
causing spaced slug is quite significant in microchannels.
He mentioned the research of Peng et al. in this paper and
commented that a proper microscope and high-speed
video techniques are required to capture the boiling
nucleation in microchannels. Kandlikar and Steinke
(2002) conducted photographic studies to observe the
liquid–vapor interface and contact line movements
through a high-speed camera at a high resolution. They
concluded that the contact angle is dependent on the
surface roughness and that the equilibrium contact angle
depends on the history of the droplet on a given surface.
Steinke and Kandlikar (2004) experimentally studied the
control and effect of dissolved air in water during flow
boiling in microchannels that have a hydraulic diameter of
207mm and found that the heat transfer slightly decreases
as the bubbles begin to nucleate and form an insulating
bubble layer. This result was not reported by previous
investigators in large-diameter channels and shows the
potential importance of the microfluidic approach to
microflows. Zhang et al. (2005) experimented with silicon
microchannels with hydraulic diameters between 27 and
171mm and varying surface roughnesses. They concluded
that bubbly and slug flows are often absent in
microchannels but that annular flow patterns are quickly
established there. They also concluded that wall surface
roughness greatly affects the boiling mechanism and that
surface condition is important to maintaining steady
annular flow. Kosar et al. (2005) investigated flow boiling
of water in microchannels with a hydraulic diameter of
227mm and that have 7.5mm-wide re-entrant cavities on
the sidewalls. Their results showed large deviations from
available correlations; the effects of re-entrant cavities on
heat transfer enhancement needs more research, but
critical heat flux conditions agree with conventional
correlations.
Recent extensive progress in 2-phase flow microchannel
experiments shows that scale effect in microchannel 2-
phase flow can produce different phenomena than seen in
conventional 2-phase flow. The potential importance and
possibility of a microscopic approach to phase change
phenomena and its numerical simulation will be very
helpful in overcoming the difficulties related to micro-
channel 2-phase flow studies. Because of the high
intermolecular forces, the dilute gas assumption does not
hold and molecular chaos and binary collisions cannot be
assumed in liquid phase flow. Therefore, DSMC cannot be
applied to the analysis of liquid phase flows and molecular
dynamics simulation results should be incorporated in the
modelling of phase change boundary in DSMC. Tsuruta
et al. (1999) conducted molecular dynamics simulations
for a system of argon molecules to study the phase change
process. Their results show that surface normal com-
ponent of translational energy affects condensation and
evaporation coefficients.
In this study, the effects of the gas–liquid interface
boundary on steady and unsteady Couette microflows are
examined using DSMC. The modified DSMC algorithm is
validated and used to simulate low-speed microflows. Five
temperature cases for both the gas–liquid interface and
solid wall cases are simulated for steady flow, and one
temperature case is simulated using ensemble averaging
for unsteady flow.
2. Numerical procedure
2.1 Outline of DSMC method
As in general DSMC methods, the movement and
collision of particles are decoupled time steps much
smaller than the mean collision time. Cells are used for
macroscopic property sampling, and so the cell dimension
should be smaller than the macroscopic fluid property
gradient. The number of real molecules in the simulation
region is usually too large, so a very large number of real
molecules is replaced by a single representative molecule
for practical simulation. Representative molecules move
deterministically in the movement phase and interact with
boundaries. In current code, the cell coordinates of a
molecule are identified and updated simultaneously when
the molecule moves. The nearest cell boundary is
determined by comparing distances to each boundary
along its movement path, and the molecule moves to the
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nearest boundary if the distance per time step is longer. In
the collision phase, binary collision is simulated with two
randomly selected molecules; the variable soft sphere
(VSS) model is used to calculate the scattering parameter.
More details of general DSMC procedures can be found in
the text of Bird (1994).
2.2 Statistical scatter reduction procedure
DSMC results include statistical scatter associated with
fluctuation velocity related to thermodynamic tempera-
ture. For air under standard conditions, fluctuation
velocity is on order of 102 m/s. This statistical scatter
can be reduced if sample numbers are increased. But
increasing the numbers of sampling steps simulated
molecules increases the computing resources required,
and is ability to reduce statistical scatter is limited. This
statistical scatter is critical in simulating subsonic flow or
low-speed flow. High-speed microflows often involve
viscous dissipation that induces a temperature gradient.
Low-speed microflows are simulated to avoid the problem
of evaporation rate changes due to temperature changes,
and so statistical scatter should be reduced. Pan et al.
(2000) showed that, in contradiction to statistical theory,
statistical scatter does not decrease proportionally to the
inverse square of the number of sampling steps. They
showed that the reduction of statistical scatter by
increasing the simulated molecules count is limited and
suggested a modification of DSMC algorithm. According
to their study, large statistical scatter from thermodynamic
temperature can be avoided by splitting the thermal and
macroscopic velocities of a molecule into independent
random variables.
Details can be found in the reference; and similar
modifications are incorporated in the present work except
a few modifications. The modifications of Pan et al.
require special treatment of slip velocity, because of the
low slip velocity induced by a low wall temperature, and
in particular require a second run with modified slip
velocity. Also, the calculated shear stress value is much
lower than the theoretical value. In current code, the
movement length and collision rate are corrected as
follows.
crDt ¼ crDtl
l1
� �ð1Þ
sTcr¼pd2refc
2v21r 2kT ref=mrc
2r
� �v21=2=Gð5=22vÞ
h i
� T=T1
� �12vð2Þ
where sT is the collision cross section, cr is the relative
speed, dref is the reference diameter of molecule, v is the
viscosity index (0.81 for argon), and mr is the reduced
mass. Using these modifications, the correct collision rate
and mean free path can be computing, so that a second run
or slip velocity modification is unnecessary. Comparing
the calculated and theoretical collision rates gave a very
small average error of 0.45%. Based on experimental
results relating viscosity and temperature, density and
shear stress are calculated as follows:
r¼r1
T1
T
� �þr0
T0
T
� �ð3Þ
t¼t1
T1
T
� �1=2
: ð4Þ
2.3 Boundary conditions
Figure 1 shows a schematic of a common silicon
microchannel with trapezoidal cross-section. L is the
microchannel length, W the width, and H the height. L and
W are usually very large compared to H. For example,
microchannels used in experiments have a high aspect
ratio of L to Dh, usually 1,600 , 4,000 (Arkilic et al.
1997, Araki et al. 2002). Although Poiseulle flow is more
similar to the common experimental conditions, it is still
difficult to apply the DSMC method to simulate full-scale
realistic flow, and so only Couette flows with a gradient in
the y-direction were considered in the study. This
condition can be observed in an electrically driven
micro-Couette flow. The present simulation domain is the
shaded 2-D region with l much shorter than L, because
DSMC code developed for 2-D flow simulation is used.
One-dimensional Couette flow can be simulated by
applying periodic boundary conditions to inflow and
outflow boundaries.
The boundary conditions are shown in figure 2. Periodic
boundary conditions are applied to inflow and outflow
boundaries as described. The top surface is a moving wall
with constant velocity. In case 1, the bottom surface is a
solid wall with complete thermal diffusion, the same as the
top surface boundary condition. Molecules impinging on
the top and bottom surfaces are reflected with an
equilibrium velocity distribution corresponding to the
temperature of the wall. In case 2, the bottom surface is a
gas–liquid interface. An impinging molecule is reflected
or removed from the simulation according to whether the
molecule condenses. Microscopically, the condensation
coefficient sc is defined as the fraction of incident
molecules that condense. The surface normal velocity
W>>H
H
L>>H
H
l
x
zy
Figure 1. Schematics of a simulation domain.
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component of an impinging molecule is related to the
condensation coefficient, according to the MD study of
condensation and evaporation interface of Tsuruta et al.
(1999). They showed that the condensation coefficient is
proportional to the energy of incident molecule and
inversely proportional to the temperature of the liquid
surface.
sc ¼ a 1 2 b exp 2Ein;y
kTc
� �� �ð5Þ
where a and b are constants determined by MD
simulation, Ein,z is the normal component of translational
energy, k is the Boltzmann constant, and Tc is the
condensation temperature. In their study, the velocity
distribution functions of the molecules leaving phase
change interface are different from Maxwellian velocity
distribution and are given as follows.
f y;e ¼1 2 b exp 2mV2
y=2kTc
1 2 b=2
m
kTc
� �Vy exp 2
mV2y
2kTc
!
ð6Þ
f y;r ¼1 2 aþ ab exp 2mV2
y=2kTc
1 2 aþ ab=2
m
kTc
� �Vy
� exp 2mV2
y
2kTc
!ð7Þ
where a, b, k, and Tc are as above, m is the molecular
mass, Vy is normal component of the velocity. The
velocity distribution functions for the tangential velocity
components are consistent with the Maxwellian form.
These MD results were incorporated into current DSMC
code for simulation of the gas–liquid interface boundary.
3. Results
3.1 Statistical scatter reduction results for low-speedmicro flows
Couette flow in a microchannel with height 5mm was
simulated to validate the present statistical scatter
reduction code. The simulation condition is almost the
same as that of Pan et al. (2000), and the results are shown
in figures 3 and 4.
Figure 2. Boundary conditions of simulation cases.
Figure 3. Velocity distribution of Couette flow, a modified DSMC resultwith 2 £ 104 molecules, and original DSMC results with 2 £ 104
molecules and 2 £ 105 molecules.
Figure 4. Shear stress distribution of Couette flow, a modified DSMCresult with 2 £ 104 molecules, and original DSMC results with 2 £ 104
molecules and 2 £ 105 molecules.
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Figure 3 shows the velocity distribution results for a
modified DSMC result with 2 £ 104 molecules, an
original DSMC result with 2 £ 104 molecules, and an
original DSMC result with 2 £ 105 molecules. Results
obtained with the original DSMC method have a statistical
scatter larger than even that of the velocity of upper plate,
and it is impossible to get a slip velocity result: Increasing
the number of simulated molecules decreased the
statistical scatter, but the error remained too large to
calculate slip velocity. The slip velocity from the result
obtained by the modified DSMC method was 0.0108 m/s
near upper plate and 0.0098 m/s at the bottom plate. The
slip velocity obtained from first-order slip model is
0.0099 m/s; DSMC results were in agreement with first-
order slip velocity results.
Figure 4 gives the shear stress distribution results for the
same cases as mentioned above. The statistical scatter of
shear stress is much larger than that of velocity.
Modifications made by Pan et al. do not include the
collision rate modification, so an unrealistic shear stress
result is obtained. The present current, however, includes
corrections made to the collision rate and shear stress, and
so can predict the shear stress accurately. The modified
DSMC has a small statistical scatter, and the average shear
stress is 4.19 N/m2. This corresponds to a viscosity of
2.14 £ 1025 N s/m2, which agrees well with a nominal
value of 2.117 £ 1025 N s/m2 (at a temperature of 273 K).
3.2 Reproduction of modified velocity distributionfunction
The velocity distributions of reflected and evaporated
molecules are different from the Maxwellian distributions.
The generation of a given velocity distribution function
from random numbers is tested to incorporate the MD
results into DSMC. The procedure is given below. First,
dividing by the maximum value normalizes the distri-
bution function:
f normalizedðVyÞ ¼ f ðVyÞ=f max ðvyÞ: ð8Þ
Then, a Vy value is randomly chosen. The distribution
of Vy values is assumed to be uniform along some finite
interval, and the value of Vy is readily decided using the
random fraction Rf.
Vy ¼ Vy;lower þ ðVy;upper 2 Vy;lowerÞ £ Rf : ð9Þ
Rf is chosen again, and f(Vy) and Rf are compared to
decide whether to accept the Vy value. Because Rf is
uniformly distributed, the probability of acceptance of the
value Vy is proportional to f(Vy). Figures 5–7 show the
derived modified velocity distributions for reflected
molecules. The distribution functions for evaporated
Figure 5. Reproduction of velocity distribution, 104 samples.
Figure 6. Reproduction of velocity distribution, 105 samples.
Figure 7. Reproduction of velocity distribution, 106 samples.
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molecules and Maxwellian distribution are given to
illustrate the differences between distribution functions.
Results show that 104 samples are not enough for
reproduction of the distribution, but that a sample size of
105 is acceptable. A sample size of 106 shows better
results but requires more computation. In light of these
velocity distribution reproduction results, the number of
iteration steps and simulation molecules for macroscopic
property averaging should be at least 105.
3.3 Unsteady 1-D microchannel flow with gas–liquidinterface boundary
In case 1, the simulation region is a microchannel 22.4mm
high, and very wide and long. The simulated gas is argon,
which has a molecular diameter of 4.092 £ 10210 m and a
molecular mass of 6.64 £ 10226 kg. The representative
number of real molecules is 5 £ 1011, the number density
is 2.6875 £ 1024 m23, and the corresponding number of
simulated molecules is 36120. The Kn number is 0.0223
and so the flow is in the slip-flow regime. The simulation
domain is divided into 30 cells and 300 subcells, and the
timestep is 10210 s, less than 1/10 of the mean collision
time. The ambient temperature is held constant at 84, 90,
102, 120, or 130 K. Initially, the flow is stationary and in
equilibrium with a Maxwellian distribution. The top plate
suddenly accelerates to the constant velocity of 3 m/s
while the bottom surface is stationary. Results are sampled
for 6 £ 106 timesteps after 1.2 £ 107 timesteps elapse to
ensure that the solution is in the steady state.
In case 2, the bottom surface is a liquid–gas interface
with a condensing and evaporating liquid surface. All
other computation conditions are the same as in case 1,
and the condensation coefficient is given by equation (5).
The mass transfer rate and condensation coefficient
reproduction results are given in table 1. The results are in
units of representative molecules per timestep, and the
calculated condensation coefficients agree well with the
MD results. Figure 8 shows that the number of
impingement molecules increases with temperature,
because the average thermal speed of molecules increases
with temperature. However, the number of condensed
molecules decreases and number of reflected molecules
increases, because the condensation coefficient decreases
with temperature.
Figures 9 and 10 show the velocity distribution results
for both cases, and show that linear Couette flow is well-
established and that slips occur at the top and bottom. Slip
velocity results are tabulated in table 2. Because the
difference in slip velocities between the two cases is very
small, it is difficult to say whether slip velocity is affected
by the presence of the gas–liquid interface boundary: Any
difference may be occluded by statistical scatter, even
with the modified DSMC code. Qualitatively, results show
that slip velocity increases with temperature and that slip
in the two cases is similar. According to collision models,
the mean free path is expected to increase with
Table 1. Mass transfer rate and condensation coefficient reproduction with DSMC.
Temperature(K)
Impingement(molecule/Dt)
Reflection(molecule/Dt)
Condensation(molecule/Dt)
Condensation coefficient(Tsuruta et al.)
Condensation coefficient(DSMC)
84 8.834 0.621 8.213 0.929 0.93090 9.146 1.525 7.621 0.832 0.833102 9.741 2.077 7.664 0.785 0.787120 10.588 4.170 6.418 0.605 0.606130 11.024 5.540 5.485 0.495 0.498
Figure 8. Impingement and condensation rates as a function oftemperature. Figure 9. Velocity distribution of flowfield, case 1.
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temperature to the power of 0.19 for argon. In the present
work, the slip velocity increased with temperature to the
power of 0.16. This difference is due to the local Kn
number variation along the y-direction in the channel.
Because the density is higher near the boundary than at the
center, collision is more likely to increase with
temperature at bottom and top.
Figures 11 and 12 show the density distributions of the
flowfield for cases 1 and 2, respectively. Density is higher
near boundaries and the density near a gas–liquid
interface is a little smaller than that near a solid wall.
This is because evaporated molecules have higher mean
velocity than Maxwellian molecules. As temperature
increases, the density at top decreases and that at the
bottom increases. This is because the condensation
coefficient decreases with temperature, which causes the
number of evaporated molecules to decrease with
temperature. Therefore, density distribution in case 2
became similar to the symmetric result of case 1 as
temperature increased, but the density distribution was
still asymmetric at 130 K.
Shear stress distributions are shown in figures 13 and 14
for cases 1 and 2, respectively. The statistical scatter of the
shear stress is larger than that of primary flow properties
like velocity and density, because more variables are
multiplied and combined. When this large statistical
scatter is considered, the shear stress distributions of case
1 and case 2 seem to be the same. The slight decrease of
shear stress at the bottom and the increase at the top are
due to imperfect reproduction of the Maxwellian
distribution in boundary cells. At boundaries, reflected
or evaporated molecules have velocity distribution
functions of the wall, and this is mixed with other
molecules that have the velocity distribution of the
boundary cell; this makes it difficult to reproduce
accurately the Maxwellian distribution function numeri-
cally. Therefore, the shear stress is almost constant
throughout the flow. Shear stress increases with tempera-
ture, and so the viscosity coefficient increases with
temperature due to the increased collision rate. Because
there is no appreciable difference between cases 1 and 2,
the gas–liquid boundary appears not to affect the shear
stress.
3.4 Unsteady 1-D microchannel flow with gas–liquidinterface boundary
In the unsteady flow simulation, most of the conditions are
the same as in steady case. Only one temperature case is
simulated, 102 K, and the evaporation rate is fixed at
8.472 £ 10212 kg/m2 s (this number is taken from the
steady simulation result) and the condensation coefficient
is 0.787. During one sampling step, an average of
3.06 £ 105 molecules evaporate and 8.30 £ 104 mol-
ecules reflect from gas–liquid boundary. Therefore, the
evaporated and reflected molecules are expected to follow
the known distribution function.
Figures 15 and 16 show the evolution of the flow
velocity distribution time-averaged with 4 £ 103 samples
and ensemble-averaged with 102 samples. The number in
the box is the chronological step and each corresponds to a
time interval of 4 £ 1027 s. The simulation was executed
for 40 sampling steps, but only seven sampling steps are
plotted to illustrate the qualitative change. After the flow
reaches steady state, the flow has a slip velocity of
0.072 m/s at the bottom surface and 0.073 m/s at the top
surface. In case 2, it takes longer for the velocity to reach
the steady state, because of mass transfer at the gas–liquid
boundary. The velocity has a slip of 0.072 m/s at the top
Table 2. Slip velocity results according to temperature.
Case 1 (1-phase) Case 2 (2-phase)
Slip velocity Slip velocity Slip velocity Slip velocityTemperature (K) Bottom (m/s) Top (m/s) Bottom (m/s) Top (m/s)
84 0.0682 0.0702 0.0654 0.070190 0.0666 0.0701 0.0673 0.0710102 0.0734 0.0720 0.0715 0.0724120 0.0735 0.0736 0.0715 0.0753130 0.0746 0.0741 0.0723 0.0752
Figure 10. Velocity distribution of flowfield, case 2.
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surface and 0.074 m/s at the bottom surface. The slip
velocity difference between the solid wall case and the
gas–liquid boundary case are indistinguishable when the
macroscopic velocity is small and statistical nature of
DSMC method are considered like it was in steady flow
simulation.
In figures 17 and 18, the density distribution of the
unsteady flowfield is shown for both cases. The density
is higher at the boundary than in the middle, and it takes
less time for the density to reach the steady state than
does the velocity. This means that mass transfer by
diffusion is faster than momentum transfer by collision.
Density distribution in case 2 is somewhat different from
that in case 1. This is because the mean velocity of
evaporated molecules is higher than that of Maxwellian
molecules. At 102 K, the number of molecules that
evaporate is greater than the number of molecules that
reflect from the bottom gas–liquid interface boundary.
This results in lower density at the bottom surface than
at the top.
Figures 19 and 20 show the shear stress distribution for
the unsteady flowfield is both cases. It takes the longest
time to reach the steady state, and the statistical scatter is
large. The statistical scatter associated with shear stress is
larger than that of primary flow properties like
temperature and velocity because, as mentioned, more
variables are multiplied and combined to sample the shear
stress. The shear stress distribution is almost the same in
both cases, except that the shear stress at gas–liquid
interface seems to be a little lower than that at the solid
wall. However the difference is indistinguishable if the
high statistical scatter of shear stress is considered and the
shear stress is regarded as the same for both cases in the
present work.
Figure 11. Density distribution of flowfield, case 1.
Figure 12. Density distribution of flowfield, case 2.
Figure 13. Shear stress distribution of flowfield, case 1.
Figure 14. Shear stress distribution of flowfield, case 2.
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The temperature is held constant at 102 K in both cases,
and the constant temperature field assumption is satisfied.
In the simulation, it took about 2.8ms for Couette flow
velocity field to reach the steady state in a channel with
height 22.4mm. It took about 1.6ms for the density field to
reach the steady state, which means that the mass transfer
by diffusion is faster than the momentum transfer by
collisions. Temperature is constant and uniform for both
cases; shear stress is initially high at the top surface and
propagates slowly to the bottom surface. The effects of the
gas–liquid interface on Couette flow are very small,
because the phase change model has no tangential
momentum change while the mass transfer in the vertical
direction is affected by changes in the velocity
distribution, according to MD results. Tangential momen-
tum accommodation between gas and liquid molecules
should be studied to clarify the microfluidic phenomena
involving 2-phase boundaries.
4. Conclusion
The effects of gas–liquid interface boundary condition on
Couette flow in microchannel were examined using
DSMC. To exclude the temperature effects due to
condensing and evaporating boundary effects, low-speed
flow without viscous heating was simulated with DSMC
code modified to reduce statistical scatter. MD results are
reproduced accurately and incorporated into the DSMC
boundary model. Results show that the mass transfer at
gas–liquid boundary cause lower densities near boundary,
due to the high mean normal velocity of evaporated
Figure 15. Velocity distribution of unsteady flowfield, case 1.Figure 17. Density distribution of unsteady flowfield, case 1.
Figure 18. Density distribution of unsteady flowfield, case 2.Figure 16. Velocity distribution of unsteady flowfield, case 2.
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molecules. This behaviour weakens as temperature
increases, because the number of evaporated molecules
decreases. There are no appreciable differences between
the slip velocity and shear stress distributions between
the solid wall and gas–liquid boundary cases. As the
temperature increases, shear stress increases because the
collision rate does. Slip velocity also increases with
temperature due to the increase of gas mean free path.
Unsteady simulation results show that the time required
for the shear stress distribution to reach the steady state is
longer than that for density distribution. This means that
the mass transfer by diffusion is faster than the momentum
transfer by collision process. The simulation was
restricted to argon at 80 , 130 K. Further investigation
incorporating tangential momentum accommodation
effects between gas and liquid molecules with more
advanced models is needed to clarify microfluidic
phenomena involving 2-phase boundary.
Acknowledgements
This work was supported by Korea Research Foundation
Grant (KRF-2002-041-D00069).
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Figure 19. Shear stress distribution of unsteady flowfield, case 1.
Figure 20. Shear stress distribution of unsteady flowfield, case 2.
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