COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Investigation of Multiphase Flows near Micro-textured Walls using
Lattice Boltzmann Method
Y. T. Chew*, J. J. Huang, C. Shu, H. W. Zheng
Department of Mechanical Engineering, National University of Singapore, 117576 Singapore
Email: [email protected]
Abstract During the last decade, interest in multiphase flows near textured surface has revived due to its
importance in microfluidics and the study of the self-cleaning property of some biological surfaces (often called
“Lotus Effect”). Here we investigate the three dimensional multiphase flows near solid walls with textures, e.g., array
of small pillars, using the free energy based lattice Boltzmann method (LBM). The model uses two sets of
distribution functions along some fixed directions which evolve every discrete time step. It incorporates the physics
of surface tension and wetting on a solid wall using a prescribed free energy functional which effects are reflected
through the equilibrium distribution functions and the collision step of LBM. Through the Chapman-Enskog
multiscale expansion, the Navier-Stokes equations including the surface tension effect and the Cahn-Hilliard
equation for the order parameter can be obtained at the long wavelength and long time limits. In dealing with the
wetting boundary conditions, the method proposed in [2] is adopted. Drop behaviors near a single pillar and multiple
pillars as shown in Fig. 1 have been studied. It is found that the model can simulate the static equilibrium and
dynamic evolution of drops to a reasonably satisfactory level. It is noted that similar work has been performed [1]
using the single component model for which only one set of distribution function is used. However it is limited to
very small density ratio. In the present simulations, with the newly improved model [3], the density ratio can be
much larger and represents practical fluids.
Key Words:
lattice Boltzmann method, micro-textured wall, multiphase flows, lotus effect
(a)
(b)
Figure 1: Drop on a hydrophobic wall with (a) One pillar (b) Pillar arrays
INTRODUCTION
Surface tension and wetting related phenomena are commonly seen in nature. They also play important roles in
coating industry, chemical engineering and recently in microfluidics. The interplay of them with a patterned substrate
⎯ 259 ⎯
is of special interest due to its close relation with the intriguing self-cleaning property of some plants’ leaves. The
lotus leaf is a typical example [4]. Under the microscope, tiny structures of micron size can be observed on a lotus
leaf which appears smooth macroscopically. These structures and even smaller ones are the key factors of the “Lotus
Effect” [5].
Currently there exists much related work in the physical chemistry community. Most of them are experimental and
focus on the properties of the hydrophobic surfaces with microstructures. To quantify the hydrophobicity of the
surface, it is common to measure the apparent contact angle (CA) of a static drop on such a surface and compare it
with that on a smooth surface. It has been observed that CA on textured hydrophobic surfaces is larger than that on
their smooth counterparts under the same external conditions. In other words, the microstructures increase the
hydrophobicity of a hydrophobic surface, making it “superhydrophobic”. With modern advanced fabrication and
measurement technology, it is no longer an especially difficult task to perform such an experiment.
By contrast, there is much less computational work in this field due to the interweaving of complex physics
(hydrodynamics, surface tension and wetting) and the complicated geometry. In [6], a first principle simulation was
performed to study the motion of a droplet on a pillar surface. Three stages for the spreading process were
highlighted. Constrained by the molecular dynamics simulation method, the size of the system is prohibitively small
and the duration of simulation is quite short. Recently, the lattice Boltzmann method (LBM) has been employed to
study these problems. LBM is often regarded as a mesoscopic method which has certain advantages in incorporating
complex physics and in dealing with complicated geometry. In [1] two different equilibrium states for a droplet on
textured hydrophobic surfaces were studied by a lattice Boltzmann model. The model was for single component
multiphase flows and requires only one set of distribution functions. A pseudo van der Waals fluid was assumed. The
density ratio for the simulations was restricted to be small.
In this paper, we study similar problems, namely, drop motions near walls with certain regular structures using a
newly developed model capable of simulating two phase flows with large density ratios. Two sets of distribution
functions are used, one for the hydrodynamics and the other for the interface evolution. It might be expected that
such a decoupling reduces the effects of unrelated physics (e.g., evaporation) which is not very clear in current lattice
Boltzmann models yet. Drop behaviors near textured walls with different wetting properties (i.e., hydrophilic, neutral
wetting and hydrophobic) will be investigated. The paper is organized as follows. In Section 2, the new lattice
Boltzmann model is briefly introduced. Then, the wetting boundary condition and how to incorporate them in the
present model are given in Section 3. Some numerical results are presented in Section 4 and Section 5 concludes this
paper.
FREE ENERGY BASED LBM FOR MULTIPHASE FLOWS WITH LARGE DENSITY RATIO
In the lattice Boltzmann framework, there are three types of models for multiphase and multicomponent flows,
namely, the color model, the potential model and the free energy model. The one used here adopts the free energy
concept [7]. It may be categorized into the general diffuse interface methods [8]. It has been recently developed for
multiphase flows with large density ratios. Here we just roughly recapture some essential constituents. More details
can be found in [3].
1. Free Energy and Chemical Potential We assume that the free energy functional in a control volume V with a
mixture of two fluids takes the form
[
]
F (φ , ∂ α φ , ρ ) = ∫ ψ (φ ) + 12 κ (∂ α φ )(∂ α φ ) + cs2 ρ ln ρ dV
(1)
V
where φ is the order parameter characterizing the density difference, ∂ α φ = ∂φ ∂xα represents its spatial gradient,
κ is a coefficient related to surface tension, ρ is the total density and cs is the lattice “sound speed”. Note
summation over repeated subscript for spatial coordinates (only) is assumed throughout. The first term is the bulk
free energy (per unit volume) which is chosen to be a double-well form. The second term models the interfacial
energy and the third is used to enforce incompressibility [9]. The chemical potential μ can be computed from the
variation of the free energy functional as
μ = δF δφ = ∂F ∂φ − (∂ α φ )[∂F ∂ (∂ α φ )] = ψ ′(φ ) − κ∂ αα φ
(2)
2. Lattice Boltzmann Model In LBM, fluids are described by discrete particle distribution functions along some
fixed directions. Here the D3Q15 model (3 dimensions, 15 discrete velocities) is used (see Fig. 2 for the detailed
numbering).
The weights associated with the discrete velocities, wi ( i = 0,1,L,14 ), are given by
⎯ 260 ⎯
Figure 2: D3Q15 velocity model
⎧2 9
⎪⎪
wi = ⎨ 1 9
⎪1 72
⎪⎩
(err
(re
(e
2
i
2
i
2
i
)
= c)
= 3c )
=0
(3)
The master lattice Boltzmann equations (LBEs) are as follows,
(
)
r r
r
r
r
g (x + e δ , t + δ ) − g ( x , t ) = − (g ( x , t ) − g ( x , t )) τ
r r
r
r
r
r
f i ( x + eiδ t , t + δ t ) − f i ( x , t ) = − f i ( x , t ) − f i eq ( x , t ) τ f + δ t wi eiα (μ∂ α φ ) c s2
i
i
t
t
i
eq
i
i
g
(4)
(5)
where δ t is the time step, τ f and τ g are the relaxation parameters related to the fluid viscosities and the mobility
controlling diffusion respectively. For simplicity, we only consider the cases in which the two fluids have the same
kinematic viscosity ν and the mobility θ M is constant as well. The equilibrium distribution functions, f i eq and g ieq ,
are determined from the local density, velocity, order parameter and also the chemical potential. For the specifics,
the readers are referred to [3]. These distribution functions satisfy the following relations,
∑ f =∑ f
i
i
eq
i
=ρ
(6a)
i
∑eα f = ∑eα f
i
i
i
i
eq
i
= ρuα
(6b)
i
∑eαeβ f
i
i
= ρuα u β + (ρcs2 + φμ )δ αβ
eq
i
(6c)
i
∑g = ∑g
i
i
eq
i
=φ
(7a)
i
∑e α g = ∑e α g
i
i
i
i
eq
i
= φuα
(7b)
i
∑e αe β g
i
i
eq
i
~
= Mμδαβ
(7c)
i
Given an appropriate initial condition, the computation follows the simple recurrent collision-streaming procedure as,
(1) Collision:
]
( ) [
r
r
r
g (x , t ) = [1 − (1 τ )]g ( x , t ) + (1 τ )g ( x , t ) ;
r
r
r
r
f i x , t + = 1 − (1 τ f ) f i ( x , t ) + (1 τ f ) f i eq ( x , t ) + δ t wi eiα (μ∂ α φ ) c s2 ,
+
i
g
i
g
eq
i
(2) Boundary conditions;
(3) Streaming:
⎯ 261 ⎯
r r
r
f i ( x + eiδ t , t + δ t ) = f i (x , t + ),
r r
r
g i (x + eiδ t , t + δ t ) = g i x , t + ;
(
)
(4) Calculation of macroscopic variables: ρ , uα , φ , ∂ α φ and μ ;
eq
eq
(5) Calculation of equilibrium distribution functions: f i and g i .
It is noted that on the solid wall, the conventional bounce back (by link) scheme in LBM is applied for the
distribution functions. In the above framework, even when the bounce back condition is used to enforce no-slip
condition, the fluid near the interface can move because of the diffuse interface modeling. The conventional contactline singularity problem does not bother any longer in such models.
3. Chapman-Enskog Expansion and Macroscopic Equations In applying the multiscale Chapman-Enskog expansion
on Eqs. (4) and (5) as follows,
( )
r r
r
f i ( x + eiδ t , t + δ t ) = f i ( x , t ) + ε (∂ t + eiα ∂ α ) f i + 12 ε 2 (∂ t + eiα ∂ α )(∂ t + eiβ ∂ β ) f i + O ε 3
( )
f i = f i eq + εf i (1) + O ε 2
r r
r
g i ( x + eiδ t , t + δ t ) = g i ( x , t ) + ε (∂ t + eiα ∂ α )g i + 12 ε 2 (∂ t + eiα ∂ α )(∂ t + eiβ ∂ β )g i + O (ε 3 )
( )
+ O (ε )
(8)
(9)
(10)
g i = g ieq + εg i(1) + O ε 2
(11)
∂ t = ∂ t0 + ε∂ t1
(12)
2
with ε = δ t being small compared to the macroscopic time scales, one can find that the macroscopic equations to
O(ε ) are approximately
∂ t ρ + ∂ α (ρuα ) = 0
(13)
(
)
∂ t (ρuα ) + ∂ β ρuα u β + ρc s2δ αβ = ρν∂ β (∂ β uα + ∂ α u β ) − φ∂ α μ
(14)
∂ tφ + ∂α (φuα ) = θ M ∂ αα μ
(15)
The first is the continuity equation, the second is the Navier-Stokes equation including the surface tension force and
the last is the Cahn-Hilliard equation describing the evolution of the interface.
WETTING BOUNDARY CONDITION (WBC) AND ITS IMPLEMENTATION
1. WBC in the Free Energy Framework Following [10], we add a surface energy term into the free energy functional
to model the wetting property of the wall. Then the full free energy functional becomes
[
]
F (φ , ∂ α φ , ρ , φ S ) = ∫ ψ (φ ) + 12 κ (∂ α φ )(∂ α φ ) + cs2 ρ ln ρ dV + ∫ ϕ (φ S )dS
V
(16)
S
Here S denotes the solid surface, φ S is the value of the order parameter on the wall and ϕ (φS ) is the surface energy
(per unit area). For simplicity, the following linear relation is assumed
ϕ (φS ) = −ωφS
(17)
where ω is the parameter related to the wetting property of the wall. If one minimizes the above full free energy
functional, one can obtain the following natural boundary condition for the order parameter
r
κn ⋅ (∇φ )S = −ω
(18)
where n is the local normal direction of the wall pointing into the fluid. The contact angle θ on the wall (measured
in the fluid with φ = φ * ) can be shown to satisfy the following equation
(
) (
)
3
3
cosθ = 12 ⎛⎜ 1 + ω~ − 1 − ω~ ⎞⎟
⎝
⎠
(19)
⎯ 262 ⎯
~ is given by
where the dimensionless coefficient ω
ω~ = ω
( 2κA (φ ) )
* 2
(20)
2. Implementation of WBC in LBM In implementing the WBC (specifically, Eq. (18)) on a flat wall, we follow
the way given in [2]. For boundary nodes at the intersection of two orthogonal planes we utilize simple coordinate
transformation. That is, we regard that there exists an inclined plane (the dashed line in Fig. 3) as a transition
between the horizontal and vertical planes. In three dimensions, there is a type of points resulting from the
intersection of three orthogonal planes. The derivatives and Laplacian of the order parameter on these points are
evaluated with only six nearest neighbouring points as if they are interior fluid points.
Figure 3: Transition points at the intersections of two orthogonal walls
MULTIPHASE FLOWS NEAR MICRO-TEXTURED WALLS
1. Drop near a Single Pillar First a drop near a single square pillar is studied. Initially the drop is in a nonequilibrium state. In other words, the total energy of the system is not minimized. Due to the specific geometry and
wetting property the drop will move toward an equilibrium state under the action of the surface tension force. For
~ = 0 which corresponds to θ = 90 0 . The initial drop radius is
this case, the solid wall is neutral wetting, i.e., ω
about 7.5 lattice units. Some other parameters are given in Table 1. The initial and final states are shown in Figs. 4(a)
and 4(b). The apparent contact angle is larger than 90 0 finally.
Table 1: Some parameters for simulation
Parameters
Dimensionless Value in LBM Simulation
Liquid Density ρ L
1.5
Gas Density ρ G
0.5
Surface Tension σ
0.001
Interface Width W
3
Kinematic Viscosity ν
0.005
Pillar Height
8
Pillar Side Length
10
31 × 31 × 30
Grid Size
2. Drop near Multiple Pillars Now we study a drop near a 5 × 5 pillar array. Three cases with the same initial
condition but different wetting properties for the lower wall have been simulated. The parameter controlling wetting,
ω~ , takes the value 0.477, 0 and − 0.477 which correspond to the theoretic contact angle being θ ≈ 450 , 90 0 and
⎯ 263 ⎯
(a) Initial Configuration
(b) Configuration after 10, 000 steps
Figure 4: Drop evolution near a single pillar
1350 . Some other parameters are listed in Table 2. It is noted that for these cases the fluid densities are chosen to be
ρ L = 1 and ρ G = 0.001 resulting a density ratio as large as 1000. The initial drop radius is about 12.5. The drop
configurations after 100, 000 steps are given in Figs. 5(a-c) and the interface evolutions in the middle y − z plane
are shown in Figs. 6(a-c) for every 25, 000 steps.
(a) θ ≈ 450
(b) θ = 90 0
(c) θ ≈ 1350
Figure 5: Drop configurations on a pillar array after 100, 000 time steps
⎯ 264 ⎯
Table 2 Some common parameters for simulation
Parameters
Dimensionless Value in LBM Simulation
Liquid Density ρ L
1
Gas Density ρ G
0.001
Surface Tension σ
0.0001
Interface Width W
3
Kinematic Viscosity ν
0.01
Pillar Height
7
Pillar Side Length
6
Distance between Neighbouring Pillars
4
51 × 51 × 35
Grid Size
From Figs. 5 and 6, one can find the different behaviors of the drop as described below. For the hydrophilic case the
drop tends to spread over the wall surfaces and the final drop becomes very flat, similar to that on a flat wettable
surface. Such spreading seems to be even enhanced due to the presence of the pillars as seen from the evolution of
the interface. Normally on a flat wall, the speed for the drop approaching its equilibrium configuration decreases;
however, in Fig. 6(a), significant changes still occur at consecutive sampling times even after long time evolution.
For the neutral wetting and hydrophobic cases, the change of the interface position becomes smaller and smaller
indicating that it is approaching true static equilibrium. It is also interesting to note that finally the drop is in contact
with nine nearest pillars in the neutral wetting case whereas it is only in contact with five nearest pillars in the
hydrophobic case. As the hydrophobicity increases, it might be expected that the drop would just sit on one pillar
similar to the above situation for a single pillar.
Evolution
Initial
z
10
0
Final
-10
-20
-10
0
10
20
y
(a) θ ≈ 450
Evolution
10
Evolution
Final
10
z
z
Initial
Initial
Final
0
0
-10
-10
-20
-10
0
10
20
-20
y
-10
0
10
20
y
(b) θ = 90 0
(c) θ ≈ 1350
Figure 6: Interface evolutions in the middle y − z plane every 25, 000 time steps
CONCLUSION
In summary, we have studied 3D drop behaviors near textured walls using the free-energy based lattice Boltzmann
model for large density ratios. Both statics and dynamics of the drop near either a single pillar or a regular pillar array
can be simulated to a satisfactory degree. The increase of apparent contact angle on textured hydrophobic walls has
⎯ 265 ⎯
been observed and drop motions from an initial non-equilibrium state to equilibrium state have been well captured.
This may suggest that LBM is a useful tool in studying phenomena involving surface tension and wetting, especially
under complex geometries. It is also noted that with the recently developed model the multiphase flow problems with
large density ratios can be dealt with in a simple way. Further quantitative validations for various problems are still in
progress.
Acknowledgements
The authors are grateful to the Computer Center of National University of Singapore for providing the computing
facilities and technical support.
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