Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 10.4.4 - Modeling Interfacial Viscous Flows and Colloid Transport in a 2D Microchannel
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 Material Information
Title: 10.4.4 - Modeling Interfacial Viscous Flows and Colloid Transport in a 2D Microchannel Computational Techniques for Multiphase Flows
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Shi, X.
Lazouskaya, V.
Jin, Y.
Wang, L.-P.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: multiphase flow
contact line dynamics
lattice Boltzmann equation
colloid transport
 Notes
Abstract: To better understand colloid and contaminant transport and retention in the vadose zone where pore-scale multiphase (i.e., air-water) ows through a soil porous medium play a role, we apply the mesoscopic LBM approach to simulate an interfacial ow in a micro-channel. A Lagrangian colloid tracking is then performed to study the trajectories of colloids near the moving uid- uid (air-water) interface. The effects of density ratio and viscosity ratio on the detailed ow near the interface are examined. It was found that the ow on the water side is not sensitive to the density ratio. The viscosity ratio is important in determining the stability of the interface. For the air front case when the less viscous uid invades the more viscous uid, the ngering instability was realized. The normalized nger widths at different Capillary numbers were found to be in good agreement with previous experimental and numerical studies. For both the air-front and the water-front cases, we found that colloids follow closely the ow streamlines. Even though the air-water interface is unfavorable collector by design, the colloids can still be brought close to the interface and may appear to be temporally retained due to the ow stagnation. Colloids may slide along the interface primarily due to hydrodynamic force. These observations agree with our parallel micro-model experimental observations.
General Note: The International Conference on Multiphase Flow (ICMF) first was held in Tsukuba, Japan in 1991 and the second ICMF took place in Kyoto, Japan in 1995. During this conference, it was decided to establish an International Governing Board which oversees the major aspects of the conference and makes decisions about future conference locations. Due to the great importance of the field, it was furthermore decided to hold the conference every three years successively in Asia including Australia, Europe including Africa, Russia and the Near East and America. Hence, ICMF 1998 was held in Lyon, France, ICMF 2001 in New Orleans, USA, ICMF 2004 in Yokohama, Japan, and ICMF 2007 in Leipzig, Germany. ICMF-2010 is devoted to all aspects of Multiphase Flow. Researchers from all over the world gathered in order to introduce their recent advances in the field and thereby promote the exchange of new ideas, results and techniques. The conference is a key event in Multiphase Flow and supports the advancement of science in this very important field. The major research topics relevant for the conference are as follows: Bio-Fluid Dynamics; Boiling; Bubbly Flows; Cavitation; Colloidal and Suspension Dynamics; Collision, Agglomeration and Breakup; Computational Techniques for Multiphase Flows; Droplet Flows; Environmental and Geophysical Flows; Experimental Methods for Multiphase Flows; Fluidized and Circulating Fluidized Beds; Fluid Structure Interactions; Granular Media; Industrial Applications; Instabilities; Interfacial Flows; Micro and Nano-Scale Multiphase Flows; Microgravity in Two-Phase Flow; Multiphase Flows with Heat and Mass Transfer; Non-Newtonian Multiphase Flows; Particle-Laden Flows; Particle, Bubble and Drop Dynamics; Reactive Multiphase Flows
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Resource Identifier: 1044-Shi-ICMF2010.pdf

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7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


Modeling Interfacial Viscous Flows and Colloid Transport in a 2D Microchannel


X. Shi* V. Lazouskayat Y. Jint and L.-P. Wang*

Department of Mechanical Engineering, University of Delaware, Newark, DE 19716, USA
t Department of Plant and Soil Science, University of Delaware, Newark, DE 19716, USA
graceshi@udel.edu and lwang@udel.edu
Keywords: multiphase flow, contact line dynamics, lattice Boltzmann equation, colloid transport




Abstract

To better understand colloid and contaminant transport and retention in the vadose zone where pore-scale multiphase
(i.e., air-water) flows through a soil porous medium play a role, we apply the mesoscopic LBM approach to simulate
an interfacial flow in a micro-channel. A Lagrangian colloid tracking is then performed to study the trajectories of
colloids near the moving fluid-fluid (air-water) interface. The effects of density ratio and viscosity ratio on the detailed
flow near the interface are examined. It was found that the flow on the water side is not sensitive to the density ratio.
The viscosity ratio is important in determining the stability of the interface. For the air front case when the less
viscous fluid invades the more viscous fluid, the fingering instability was realized. The normalized finger widths at
different Capillary numbers were found to be in good agreement with previous experimental and numerical studies.
For both the air-front and the water-front cases, we found that colloids follow closely the flow streamlines. Even
though the air-water interface is unfavorable collector by design, the colloids can still be brought close to the interface
and may appear to be temporally retained due to the flow stagnation. Colloids may slide along the interface pri-
marily due to hydrodynamic force. These observations agree with our parallel micro-model experimental observations.


Introduction

Colloidal particles are present in large quantities in wa-
ter systems that enter the subsurface. Colloids, due
to their large available surface area, may help strongly
sorbing contaminants penetrate deeper into the under-
ground environment. The role of mobile colloids in the
subsurface transport of contaminants has been recog-
nized for some time (McCarthy & Zachara 1989), and
colloid-facilitated transport of contaminants and micro-
organisms has been an important research topic in soil
and environmental sciences (Saiers & Ryan 2006; Sen
& Khilar 2006). The widespread use of nanoparti-
cles in industrial and commercial products in recent
years also makes the topic of nanoparticle transport rel-
evant as the environmental impact of nanoparticles has
been recognized (Fortner et al. 2005; Wiesner et al.
2006). Nanoparticles and natural colloids share simi-


lar characteristics in the subsurface environment. There-
fore, understanding the fate and transport of colloids and
nanoparticles is of great importance to pollution con-
trol and natural resource preservation. A quantitative
description of these subsurface transport phenomena is
required before we can effectively monitor and manage
our environmental quality.
Compared to the saturated system, where the soil ma-
trix is filled with water, the unsaturated system is com-
plicated by the presence of air-water interface and mov-
ing contact line. Wan & Wilson (1993, 1994) performed
the pioneering mechanistic study on the interaction of
colloids with air-water interface. They found that col-
loids could irreversibly attach to stationary and moving
air bubbles. However, studies conducted by Chen &
Flury (2005) found that isolated bubbles and air-water
interfaces were unfavorable site to retention of colloids.
The similar conclusion was reached in Crist et al (2004).









Much remains to be understood regarding the mechanis-
tic and quantitative description of colloid transport be-
havior in the vicinity of air-water interface.
Several colloid retention mechanisms in unsaturated
porous media have been previously reported in the liter-
ature: retention at solid-water interface (SWI), retention
at air-water interface (AWI) and retention at the contact
line. For example, Zhuang et al (2007) found that
the lower water saturation results in less colloid con-
centration in the bulk, and the retained colloids could
re-gain mobility when the system is flushed with water.
The reason for a higher retention rate in an unsatura-
tion system is likely the presence of air-water interface
and contact line. There have also been studies to isolate
interfacial forces from hydrodynamic effects so their in-
dividual roles can be better investigated. For example,
to separate the effect of solid grain surface from that of
air-water interface, isolated bubbles are used by Chen
& Flury (2005) to avoid further complications of con-
tact line where additional mechanisms such as the capil-
lary force resulting from air-water interface could have a
component tangential to the solid wall (Gao et al 2008).
This current computational study focuses on colloids
transport near AWI and contact lines. Our study is moti-
vated by the recent experimental studies of Lazouskaya
et al (2006) and Lazouskaya (2008). They utilized mi-
cro channels containing fixed or moving air-water inter-
faces to model different aspects of microscale flows rel-
ative to AWI in unsaturated porous media. They found
that colloid retention at AWI was assisted by the stag-
nant flow conditions. They also found that the hydrody-
namic effects dominate transport of colloids away from
interfaces. Near SWI and AWI interfaces, the hydrody-
namic effect is coupled to physicochemical processes.
At AWI or near a contact line region, the local flows of-
ten contain stagnation points. This increases the interac-
tion time between a colloid and the interface, making it
more likely for other interface-related interaction forces
to alter the motion of colloids.
There are two components to our computational
model. The first is the simulation of microscale flow
near an air-water interface and the second is the mod-
eling of colloid transport. We consider air-water inter-
facial flow in a 2D model microfluidic channel. Both
the capillary number and Reynolds number are small,
namely, the fluid inertia may be neglected and the flow is
mainly governed by viscous force and surface tension ef-
fects. The mesoscopic lattice Boltzmann method (LBM)


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


is used to simulate the flow.
As an alternative CFD approach for multiphase
flows, Lattice Boltzmann method is still undergoing
rapid development. For problems involving large den-
sity/viscosity ratios, LBM suffers from challenges such
as numerical instability and spurious currents. Several
attempts have recently been made to identify the origin
of spurious currents in order to reduce such currents.
Lee & Fischer (2006) showed that the use of potential
form ofintermolecular force can greatly reduce the mag-
nitude of spurious currents in free-energy based LBM
two-phase model (He & Doolen 2002). Shan (2006)
argued that the anisotropic form of pressure in Shan-
Chen model causes undesirable spurious velocity. He
proposed to use more non-local lattice nodes to bet-
ter restore isotropy and thus suppress spurious currents.
However, the downside of this approach is that the in-
terface becomes more diffusive when two or more belts
of neighboring lattice points are employed to represent
nonlocal interactions. Particular caution is required to
make sure that the thickness of interface is much less
than the characteristic length of flow.
In this work, we employ the LBM multiphase model
of Shan & Chen (1993) to simulate flow field near the
interfacial region, but will incorporate a different equa-
tion of state (Yuan & Schaefer 2006) to enhance the
model capability in dealing with large density contrast
while maintaining a sharp interface. When the flow
field is established, we then simulate the motion of col-
loids in the interfacial region and compare the simu-
lated colloid trajectories with our parallel experimental
observations reported in Lazouskaya (2008); Shi et al
(2010). To study colloid transport behavior and reten-
tion mechanism, we track colloids by numerically inte-
grating the colloid's equation of motion with physico-
chemical, hydrodynamic, Brownian and body forces. It
will be demonstrated that the simulated trajectories are
in reasonable agreement with our experimental observa-
tions.


Multicomponent two-phase flow
LB model

In the pore-scale experimental study of Lazouskaya
(2008), a trapezoidal channel with characteristic width
of 40 pm was used. The typical mean flow speed was
around 10-5 m/s. The resulting Capillary number Ca is









then of the order of 10-7, namely, surface tension dom-
inates the viscous effects. It is challenging to match ex-
actly such a small Capillary number in our simulation
as a very small velocity scale would be needed and as
such spurious currents can easily contaminate the phys-
ical flow. The Capillary number in our simulation is
made to be much smaller than one. We assume the flow
pattern near the moving air-water interface is not sensi-
tive to Ca as long as Ca is reasonably small.
Due to its physical simplicity and relative ease of im-
plementation, the multiphase LBM model of Shan &
Chen (1993) has been used to study a variety of mul-
tiphase flow problems such as droplet displacement in
a channel (Kang et al 2002), fingering flow of two im-
miscible fluids (Kang et al 2004), and coalesce and
breakup (Shan & Chen 1993, 1994). The Shan-Chen
model is based on the concept of non-local fluid-fluid
force interaction and can treat fluid-fluid and fluid-solid
interaction using a uniform formulation of interaction
potentials.
Consider a complex fluid containing multiple compo-
nents. The distribution function for the k-th component
is governed by


ff (x, t) f() (x, t)


where Tk is the relaxation time of the k-th component,
fk(eq) (x,t) is the corresponding equilibrium distribu-
tion function. We use the standard D2Q9 lattice so i =
0, 1, 2,..., 8 and the particle velocities are: eo = (0, 0),
ei = (1,0),e2 = (0,1), e3 = (-1,0), e4 = (0, -1),
e5 = (1,1),e6 = (-1,1), e7 = (-1, -1), and e8 =
(1, -1). The density and velocity of the k-th fluid com-
ponent are defined as

Pk =EfI' A = fiei. (2)
i i

The equilibrium velocity ueq is modified to take into ac-
count the presence of multicomponent fluids and their
interactions, as

eq Ek t7k TFk, (3)
Uk k Pk/Tk + Fk, (3)P

where Fk = Flk + F2k represents the net force due to
mesoscopic fluid-fluid interaction Flk and mesoscopic
fluid-solid interaction F2k,.


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


Shan and Chen (1993, 1994) represented the fluid-
fluid interaction by the following non-local interaction
force

Flk = --Ek(x) Gkk(x, X ',, (x')(x'-x), (4)
x' k

where the interaction coefficients Gkk(x, x') will be dis-
cussed below.
The scalar function .'. in Flk is known as the ef-
fective mass density and is given as a function of par-
ticle local number density, which results in a non-ideal
equation of state that permits co-existence of two phases
for each component if the effective temperature is be-
low a critical temperature. The exact form of .',. deter-
mines the density ratio of two phases. More will be dis-
cussed when specific forms of .',. are introduced. In the
D2Q9 lattice structure, each lattice node is surrounded
by 4 nearest neighbor nodes with an interaction distance
of 1 lattice unit and the next 4 nearest neighbor nodes
with an interaction distance of /2 lattice units. The
weights GkT must be carefully designed to satisfy sym-
metry and isotropy conditions associated with gradient
operators (Shan 2006). When only the 4 nearest neigh-
bors and the 4 next nearest neighbors are used, the opti-
mal setting is


x x' =
x X' =
otherwise;


where the sign of gkk determines the nature (attractive or
repulsive) of the interaction between component k and
component k.
Similarly, the fluid-solid interaction is modeled as

F2k = -Pk(x) Y kwP,(x')(x' x), (6)
x'
where the summation is over lattice nodes inside the wall
only. The relative magnitude and sign of gkw determines
the wetting property of a solid wall. We follow the im-
plementation of Kang et al (2002) and consider only the
4 nearest neighbor lattice nodes in the above summation
for solid wall interaction. If we assume that the channel
wall is located half link away from the fluid lattice node
closest to the wall and the wall is parallel to x-axis,
then only one wall link is considered in the above sum-
mation. The fluid-wall interactions amounts to an ad-
ditional wall-normal force applied to those fluid lattice


fik (Xt-ei6t, tt-6t) fik (X, t)


Gkk (XI X.') nken i









nodes closest to the wall before the local collision op-
eration. The usual bounce-back conditions still apply to
those same lattice nodes to yield the approximate no-slip
boundary condition.
It is important to note that, in the Shan and Chen
scheme, the effect of interaction forces is integrated into
the particle distribution function to make the scheme
formally explicit, while in reality the interaction forces
are added using the implicit trapezoidal time integration.
Namely, the distribution functions being solved is re-
ally the sum of the original particle distribution func-
tions after a net momentum shift resulting from the net
interaction force. As a consequence, the macroscopic
fluid velocity should be computed as pu = k I p, I, +
Ei Fk, where the factor reflects the merging of the
trapezoidal time integration of the force into the distri-
bution function.
Finally, the specific detail on the implementation of
boundary conditions at the inlet and outlet is given in Shi
et al (2010).



Single-component two-phase
flow LBM simulations

Besides the difficulties of matching exactly the Reynolds
and Capillary numbers in our simulations to the exper-
iment of Lazouskaya (2008), another challenge is the
high density ratio (around 1000) and high viscosity ratio
(around 50) for the air-water flow system due to the nu-
merical instability and spurious currents. Our approach
here is to use a different equation of state in LBM, as
suggested by Yuan & Schaefer (2006), to handle high
density and viscosity ratios. Still we are currently un-
able to treat the true density and viscosity ratio in the
air-water system. Instead, we shall investigate the sen-
sitivity of the resulting flow on the assumed density and
viscosity ratios in order to understand if lower density
and viscosity ratios can be used to effectively simulate
the microscale flows in the air-water system for our pur-
pose of studying the transport of colloidal particles.
For simplicity, here we shall consider the application
of Shan-Chen model at the single-component two-phase
flow setting; this is a special case (k = 1) of the general
multiphase LB model discussed in the last section. In
this case, the equation of state can be derived in terms of


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


the effective mass density function as
1 3
P = 3P+ 2ip ))2. (7)

While in their original model, Shan and Chen consid-
ered only a specific form of t(p), it was later recog-
nized by Yuan & Schaefer (2006) that other equations
of state (EOS) can be incorporated to extend the ca-
pability of the model. Yuan & Schaefer (2006) con-
sidered Peng-Robinson EOS, modified Redlich-Kwong
EOS, and a non-cubic form of Carnahan-Starling EOS
in the extended Shan-Chen model and showed that these
EOS can greatly enhance the algorithm stability for large
density ratios. By choosing a proper reduced tempera-
ture on the coexistence curves, different density ratios
can be modeled. For example, by assuming the follow-
ing form of t(p),

) 2pRT 2a2p 2p
3g 3g(1 bp) 3 Tgn (1 + bp) 9911'
(8)
we can model the Redlich-Kwong equation of state
given as
pRT ap2
S1- bp i(1 + bp)'
where a = 0.4274'-' ', b = 0.08664RTc/pc. By
taking the first and second derivatives of pressure with
respect to the density, we can obtain the critical den-
sity, pressure and temperature: pc = 2.7292, pc =
0.1784, Tc = 0.1961. The unique feature of this ap-
proach is that the reduced temperature is explicitly built
into the model.
We adopted the above Redlich-Kwong EOS to sim-
ulate a two-phase interfacial flow in the micro-channel
with a density ratio of the order of 100. The walls of the
2D micro-channel is parallel to the x-axis, with y being
the cross channel direction. A parabolic velocity pro-
file with a maximum velocity of 0.025 (in lattice units)
is imposed at both the inlet and outlet. For the single-
component two-phase flow setting (i.e., Case 2), the less
dense side (representing the gas side) is advancing while
the more dense side (the liquid side) is receding. For
comparison, a two-phase flow with the same density on
two sides of the interface (i.e., Case 1) is also simulated.
The parameters used in the simulations are listed in
Table 1. The same maximum velocity is assumed for
two cases because we intend to compare the detail of the
simulated flow fields at the interfacial region.






7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010



Table 1: Setup of parameters for interfacial flow simulation of different density.


Symbol


Channel width
Channel length
Fluid densities
Viscosity
Maximum inlet velocity
Average front velocity
Reynolds number
Channel length
Initial transition steps
Total time steps
Surface tension
Capillary number
Fluid-fluid interaction strength
Fluid-solid interaction strength
Contact angle


H
L
pl /p2

Vo
Vf
Re = HVf/v
L/H

Nsteps
a
Ca = Vfup/a
9
g
glw
0


Single-
component(Case
1)
64
800
1.0/1.0
1 /1
0.025
0.0167
6.41
12.5
1000
20,000
0.0794
0.035
912 = 0.2
0.06/-0.06
700


We make sure that the steady flow is achieved when
both the interface shape and the velocity field relative
to the interface are independent of time. The interface
locations for Case 1 and Case 2 are found to be only 5
grid spacings apart after 20,000 lattice time steps, cor-
responding to a dimensionless time of tVo/H = 5.21,
where Vo is the mean flow speed and H is the channel
width. We found that the interface shape in Case 2 over-
laps with that of Case 1, so we simply shift the interface
in order to compare the velocity distributions of the two
cases.
In Figure 1, we compare the distribution of the
streamwise (x-component) velocity along the x direc-
tion, at two distances from the wall. At both distances
from the walls, the velocity far away from the interface
are identical. In the interfacial region, the velocity dis-
tributions on the gas side are different, due to different
local flow patterns for different density ratios near the
interface on the gas side. Interestingly, however, the ve-
locity distributions on the liquid side are very similar.
Therefore, the density difference alters the flow on the
low density side but has a negligible effect on the flow
of the high density side. For our application, the flow
on the liquid side is responsible for the transport of col-
loids, therefore, a lower than actual density ratio used in
the simulation could be allowed as long as the flow on


the liquid side is of primary concern.
The y-component velocity dominates the relative
transport of colloids towards the wall region when col-
loids are close to the interface. It determines whether
colloids would slide along the interface or simply slow
down at the stagnation region. In Figure 2, we compare
the distributions of the y-component velocity at a dis-
tance 1/8 channel width from the channel wall. Again
it appears that the velocity field on the liquid side is not
sensitive to the density ratio, while flow on the gas side
can be different from different density ratios. This same
observation holds for the water front case when the liq-
uid is advancing and the gas is receding.


LBM simulations with different

viscosity ratios

In order to further verify our 2D two-phase flow simu-
lation in a micro-channel, in this section we will con-
sider finger instability and compare our simulation re-
sults with results from previous experimental and nu-
merical studies.
It is well known that the interface tends to be un-
stable when a less viscous fluid is moving into a more
viscous fluid, and a fingering instability develops. The


R-K(Case 2)


64
800
7.53/0.079
1/1
66
0.025
0.0167
6.41
12.5
1000
20,000
0.5079
0.041
gii=-0.2
0.06/-0.06
700






















1.0


I , I , I I


' I I I I '
0.0 0.2 0.4 0.6 0.8 1.0
x/L


rnterfa.
u@18WCsl.
u@1/8W Case.2



Gas Liquid




Locate at x=0.54





0.00 0.20 0.40 0.60 0.80 1.00


Figure 1: Comparison of x-component velocity near
the interface: (a) at the center of channel, (b) at a dis-
tance of one eighth channel width. Case 1 has a density
ratio of 1 and Case 2 a density ratio of 94. The location
of interface is marked by a vertical line.



most studied topic of this instability is the the relation-
ship between the normalized width Wf /W of finger and
the Capillary number, where Wf is the finger width and
W is typically the spanwise with of the Hele-Shaw cell.
Reinelt & Saffman (1985) simulated the finger penetra-
tion of in a 2D channel by solving the Stokes equation


Interface
-u@center Case.l
u@center ofCas2


S= 1- 0.417 (1- e-1.69Cao 25)


(10)


Experimental studies on finger instability went back
to the classical work of Saffman & Taylor (1958) who
showed, both theoretically and experimentally, that the
interface is unstable when a less viscous fluid is invad-
ing a more viscous fluid. This was followed by several
other studies including the work by Tabeling et al (1987)
who, based on their experimental data, proposed to use
one single dimensionless number = 12Ca(b)2 to
better fit all data for different ratios of spanwise chan-
nel width W of their Hele-Shaw cell to the distance b
between the two plates. The interface between air and
oil (viscous fluid used in the experiments) is a meniscus
which lives in a three-dimensional space. When the in-
terface is advancing, the meniscus could leave a thin film
behind it at the cross section of two closely spaced chan-


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


2.0 I I
Interface
--v@lW Case.l
1.6 -v@ Case.2


1.2
Gas Liquid
0.8


0.4
Locate at x=0.54
0.0


-0.4

0.00 0.20 0.40 0.60 0.80 1.00
x/L

Figure 2: The distributions of the y-component velocity
at the 1/8 channel width away from the channel wall.


with a normal-stress boundary condition at the inter-
face through a least squares iteration method. Halpem
& Gaver (1994) used a different method known as the
boundary element analysis to study this problem. The
governing equation can be simplified to Stokes equation
when Ca Re << 1. Halpem & Gaver (1994) obtained
a relationship between 3 =- Wf/W and the Capillary
number Ca for Ca = 0.35 1.25. They further pro-
posed the following regression formula


Liquid





f-


Gas


Locate at x=0.543










nel walls. The finger interface has a concave shape at the
center where the maximum film thickness is measured.
Tabeling et al (1987) fitted their experimental results of
film thickness tmax by a stretched exponential as

tmax = 0.119b (l1 e-0038(W/b) (l -8.58 ) ,
(11)
which can be transformed to the ratio of finger width to
the width b as


b tmax


1 0.119 (1 e-0.038(W/b))


x(i


e-8.58C ) (12)


Their prediction is good for Ca < 0.2, which is the Ca
range simulated on our study.
It must be stressed that, when comparing experimen-
tal data with our 2D simulations, we must take into ac-
count of the effect of the aspect ratio W/b on the liquid
film thickness left by advancing interface. Our 2D simu-
lation could be viewed as the special case of W/b -+ oo,
the limiting case of very large aspect ratio of the Hele-
Shaw cell studied in the experiments. The 2D simulation
data of Reinelt & Saffman (1985) would fall on experi-
mentally predicted curve in the limit of W/b -+ oo.
Here we simulate two general situations of two-phase
displacement in a 2D channel. The first is the air-front
case when a less viscous liquid is advancing into a more
viscous fluid, while the second is the water front case
when the flow direction is reversed. In the simulations,
the basic parameter settings for the these are the same,
except that the flow direction is reversed. We consider
a slow flow with the Reynolds number changing from
1.06 to 3.47 on the dense liquid (water) side. Two vis-
cosity ratios are considered and their values are shown
in Table 2, along with the flow speed used to define the
velocity profile at the inlet and outlet. Case 3 is for the
viscosity ratio equal to 8.6, and Case 4 for 13.3. Other
parameters are the same as Case 1 shown in Table 1. The
resulting ratio of the finger width to the channel width is
also listed in Table 2.
The typical time evolution of the fingering interface
for the air front case is shown in Figure 3, while the sta-
ble interface for the water front case is shown in Figure
4. The detail velocity fields and streamlines for these
cases are shown in Figures 5 and 6, respectively. For the
air-front case, the mean velocity is calculated by defini-
tion: the average value of the stream velocity cross the


60.


40.0
0.0
0.0
0.0


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010

-- I I-i-I-


200.0 400.0 I600.0 80.
CONTOUR FROM 0 TO 0 BY 0


Figure 3: The time evolution of the interface shape for
Case 3 at the air-front setting, the steady interface shape
develops into finger shape due to fingering instability.
Umax = 0.025 and the mean velocity of interface is
0.021. The flow goes from left to right.



CONTOUR FROM TO.2 BY.1

Figure 4: Interface evolution for the water-front set-
ting in Case 3, the steady interface is a stable interface.
Same parabolic inlet profile with Umax = 0.025, the
mean velocity is 0.017 which is consistent with the given
parabolic profile. The flow is moving from right to left.


channel. It is also used as a monitoring parameter to
determine whether the steady state is reached. Take the
case of Umax = 0.015 in Case 4 as example, the mean
velocity falls into 0.2%-0.8% percent from time steps
10,000 and 20,000 (total time steps used as Case 3 and
Case 4). Therefore, we believe that the steady-state fin-
ger shape and propagation of interface are established.
Capillary number is calculated with the mean velocity
and viscosity from the liquid side.
From the vector plots (Figure 5 and Figure 6), we
find that the interface shape does have an effect on flow
field near the interface and contact line and stagnation
regions. A pair of eddies appear on the air side at the tip
of the air-front finger interface (a little bit away from the
interface). This phenomenon has been visualized in the




--------


---------=-



Figure 5: The resulting flow field relative to the interface
and streamlines in the air-front case shown in Figure 3.






7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010



Table 2: Setup of parameters of two-phase flow simulations of different viscosity.


Viscosity
trighht=0.43,ie/tl =0.05
Viscosity
/tright=0.66,/tIet =0.05


Surface Tension
0.0794
Surface Tension
0.0794


Umax=0.015
Ca=0.081, 0= 0.833
Umax=0.015
Ca=0.099, O= 0.80


Umax=0.025
Ca=0.119,0=0.815
Umax=0.025
Ca=0.153,0=0.76


Umax=0.035
Ca=0.159,0=0.78
Umax=0.035
Ca=0.322,0=0.732


- - - - - - -
-- - -


Figure 6: The resulting flow field relative to the interface
and streamlines in the water-front case shown in Figure
4.


experimental work of Tabeling et al (1987) by releasing
small fish-skin particles into the interfacial flow. It is
also reported in numerical simulations: for example, in
the 2D simulation by Reinelt & Saffman (1985) and the
3D simulation by Wong et al (1995a,b). For the water-
front case, the flow relative to the interface exhibits a
smaller vortex pattern near contact line at the interfacial
region.
In Figure 7, we show the normalized finger width
for the air front case from our simulations as a func-
tion of the Capillary number. The fitting, Eq. 12,
from the experimental work of Tabeling et al (1987)
with W/b -+ oo as well as the regression formula, Eq.
(10), from the numerical simulation by Halpern & Gaver
(1994) are plotted in Figure 7 for comparison. We chose
the viscosity ratio so that our Capillary numbers can ex-
tend below the minimum of 0.2 simulated in Kang et
al (2004). There is a reasonable level of agreement be-


1.10


1.00


0.90


0.80


0.70


I I I I


0.00 0.10 0.20
Ca


Figure 7: The normalized finger width as a function of
Capillary number.



tween our simulation data and these previous results.
This implies that our simulated interface shape are re-
alistic for the parameter range we considered here.

We should point out that our Capillary number and
Reynolds number are much larger than those realized
in the experiment of Lazouskaya (2008). At this point,
it is not completely clear whether the fingering for the
air front case would remain in the limit of very small
Capillary number. While in Lazouskaya (2008), a liquid
film was found in the air front case, it is much thinner
in scale. Whether this is a result of smaller Ca number
with the same physical origin macroscopicc fingering in-
stability) or it is of a different origin such as molecular-
scale adhesion remains to be studied.


Case 3
rightlyl Itleft
Case 4
/right I / left


Targeted area when Ca goes to zero

Targeted area when Ca goes to zero









Transport of Colloids near the
Fluid-Fluid Interface

The simulated flow field near the fluid-fluid interface can
be used to study the motion of colloids near the interface
and contact line. This is relevant to colloid transport and
retention in unsaturated soil. We integrate the equation
of motion for colloids under the influence of drag force,
Brown fluctuations, and physicochemical forces. A de-
tailed description of the equation of motion and relevant
parameters used for the colloids and solution conditions
can be found in Shi et al (2010). Keeping the same
parameters and treatment for colloid tracking as in Shi
et al (2010) but using the new flow fields discussed here,
we shall briefly study the trajectories of colloids near the
fluid-fluid interface.
In the air-front case, the trajectories are plotted in Fig-
ure 8 for several colloids released on the water side near
the walls away from the interface. This is because the
relative motion is towards the interface near the wall re-
gion but away from the interface in the center region of
the channel (Shi et al 2010). We carefully chose the re-
lease locations so that no colloids-wall physicochemical
interaction occurred. The air-water interface is unfavor-
able retention site since a repulsive electrostatic force is
applied. We set a cut-off distance for colloids-AWI in-
teraction to 10-6 m, as stated in Shi et al (2010). For
the colloids shown in Figure 8, none of them approaches
close enough to the interface to experience this repul-
sive interaction force. Thus the motion of colloids is
governed by hydrodynamic force and Brownian motion,
with the hydrodynamic force dominating the motion.
Therefore, colloids appear to follow the flow streamlines
relative to the interface.
When colloids travel into the interfacial region, we
observed that some of them are sliding along the inter-
face towards the center of the channel without contacting
the interface. They move very slowly towards the stag-
nation point located at the tip of the interface and may
appear to be temporally retained. The curved stream-
lines therefore play a major role in determining the tra-
jectories relative to the interface. There is also a pos-
sibility for a colloid very close to the channel wall to
continue its path into the liquid films between the finger
and the channel walls. These observations are similar to
what were observed in Lazouskaya (2008).
The dependence of colloid transport on the detail flow


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


pattern implies that it only makes sense to compare dif-
ferent micro-model experiments with similar geomet-
rical and interfacial configurations. Since the contact
line, the interface shape, and corresponding interfacial
flow field are critical to colloids transport, care should
be taken, for example, when generalizing the conclusion
obtained from single-bubble experiments (Chen & Flury
2005) to air-water interface contacting grain surface in
pore-scale experiments (Johnson & Elimelech 1995).
Our simple 2D simulations so far provide a reasonable
agreement with our parallel experiments with regard to
the hydrodynamic effects on colloids transport. There
are, however, other complications that are not yet con-
sidered, including colloid-colloid interactions when col-
loids clustered at the AWI and additional Capillary force
when colloids are trapped at interface or near the con-
tact line. The minimum velocity in our simulation is also
limited as we wanted to make sure that the flow velocity
is larger than spurious flow for most flow regions, which
prevents us from simulating a much smaller Capillary
number and associated micron-size thin liquid films at-
tached to the channel walls. In real soil matrix, the inter-
connected pores and irregular channel formed by soil
particles would make flow field much more complicated
than in our simple micro-channel.
Colloids trajectories in the water-front case are shown
in Figure 9. Here colloids released near the center fol-
low the streamlines as shown in Figure 6, approaching
the stagnation region at the tip of the interface. Again,
since the flow is so slow there, colloids may appear to
move with the interface visually in the micromodel ex-
periment (Lazouskaya et al 2006; Lazouskaya 2008).
The colloids then follow the curved interface and slowly
migrate towards the channel walls. They may reside
close to the contact line before returning to the bulk liq-
uid side along the walls. These are similar to the visual-
izations from the micro-model experiment, as shown in
Figure 15 of Shi et al (2010).


Conclusions and Summary

In this paper, we presented results on microscale inter-
facial flow simulation and transport of colloids near the
fluid-fluid interface, in order to better understand colloid
transport behavior in unsaturated soil porous media. At
this stage, the lattice Boltzmann method for two-phase
flow is under rapid development. For the purpose of our





7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


480 520 560 600 640 680


Figure 8: Colloid trajectories in the air-front setting from Case 3 with Umax = 0.025.


60.0 5 4B B 1 a
r) ,1 I ,1 11
68.0
-- M#3 colloAld
.a-a1 #2 cilod
##1 collkid
20.0


0.0

360 400 440 480 520

Figure 9 Colloids trajectory in the water-front setting from Case 3 with U 0.025.
Figure 9: Colloids trajectory in the water-front setting from Case 3 with Umax = 0.025.


application, the ability to treat large density and viscos-
ity ratio is crucial. We have demonstrated that the use
of the Redlich-KWong EOS in LBM can handle a high
density ratio. An important observation is that the den-
sity difference has a small effect on flow field at dense
fluid (water) side, although the flow field on the low den-
sity (gas) side is greatly affected with larger spurious
currents. On the water side, not only do the interface
shapes from different density ratio simulations overlap
with each other, the velocity profiles are also insensitive
to the density ratio. As far as the water-side flow field
in the vicinity of the interface is concerned, we may ne-
glect the influence of density difference and just focus
on the effect of the viscosity ratio.

We then studied the effect of viscosity ratio. First, the
interface shape evolutions for the air-front and the water-


front cases are simulated. Consistent with previous ex-
perimental observations (Saffman & Taylor 1958), we
found that the finger instability is associated with the
case of a less viscous fluid driving a more viscous fluid
(the air front case), while a stable interface is maintained
for the water front case. Quantitatively, we compared
our simulated finger width and its dependence on the
Capillary number to previous experimental and numeri-
cal results (Tabeling et al 1987; Halpem & Gaver 1994)
and concluded that a good agreement was achieved. The
flow pattern near the interface is also found to be quali-
tatively similar to previous results (Shikhmurzaev 1997,
2008).

Colloid trajectories relative to the interface were ob-
tained by integrating the equation of motion. For both
the air-front and the water-front cases, we found that col-


60.0

40.0

20.0

0.0










loids follow closely the flow streamlines. Even though
the air-water interface is unfavorable collector by design,
the colloids can still be brought close to the interface and
may appear to be temporally retained due to flow stag-
nation. They could slide along the interface primarily
due to the hydrodynamic forces. These observations are
similar to experimental observations in Lazouskaya et
al (2006); Lazouskaya (2008). These results must be
viewed as preliminary as the parameter conditions are
not exactly the same as the experimental conditions and
there are a number of simplifications in our current nu-
merical models.


Acknowledgements

This study is supported by the US Department of Agri-
culture (NRI-2006-02551, NRI-2008-02803), US Na-
tional Science Foundation (NSF CBET-0932686), and
National Natural Science Foundation of China (Project
No. 10628206). We thank Dr. Qinjun Kang for provid-
ing his LBM codes and for his comments on LB multi-
phase flow models.


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