Journal of Template - i-repository.net · classical continuum-dynamics approach using a sharp-interface model (5). Therefore, developing CFD methods to simulate these fluid flows

Journal of Template

Vol. 00, No. 0, 0000

1

Phase-field Model-based Simulation of Motions of a Two-phase Fluid on Solid Surface*

Naoki TAKADA**, Junichi MATSUMOTO** and Sohei MATSUMOTO** ** Research Center for Ubiquitous MEMS and Micro Engineering,

National Institute of Advanced Industrial Science and Technology (AIST),

1-2-1 Namiki, Tsukuba-shi, Ibaraki 305-8564, Japan

E-mail: [email protected]

Abstract A preliminary numerical simulation of the microscopic two-phase fluid motion on a solid surface was conducted using an interface-tracking method based on the phase-field model (PFM). Two variations of the lattice Boltzmann method (LBM) based on fictitious particle kinematics are proposed for solving diffuse-interface advection equations which were revised to improve volume-of-fluid conservation in the PFM simulations. The major findings are as follows: (1) the interface-tracking method accurately predicted the capillary force effect on dynamic two-phase fluid systems with a high density ratio between parallel plates; (2) the initial shape and volume of the two-phase fluid were retained adequately in linear translation with the use of the LBMs. These results proved that the PFM-based method and the LBM-based advection schemes can be used for simulating two-phase fluid motions in various macro- and microfluidics problems for devices, machineries and higher-throughput microdevice fabrication processes.

Key words: Computational Fluid Dynamics, Microfluidics, Multiphase Flow, Contact Line, Capillarity, Diffuse-interface Model, Allen–Cahn Equation, Cahn–Hilliard Equation

1. Introduction

Gas–liquid and liquid–liquid flows with a fluid–fluid interface moving on solid surfaces are widely encountered in various science and engineering fields (1). Computational fluid dynamics (CFD) simulations facilitate the understanding and prediction of such two-phase flows for flexible and accurate fluid motion and position control; the result can be used to optimally design microfluidic devices, machineries, and fabrication processes (2)–(4). It is often difficult to experimentally observe such flows, measure the flow variables (velocity, pressure, etc.) simultaneously in three dimensions, or analyze them theoretically through the classical continuum-dynamics approach using a sharp-interface model (5). Therefore, developing CFD methods to simulate these fluid flows is important.

The phase-field model (PFM) is based on the van der Waals and Cahn–Hilliard free-energy theory (6) and regards an interface as a finite volumetric zone across which physical properties vary steeply but continuously (5). Such interfacial zones are autonomously formed to retain interfacial width by a diffusion current induced by the gradient of chemical potential; this minimizes the free energy of the fluid system and leads it toward an equilibrium state through phase separation or phase change. The contact angle of a fluid phase on a solid surface is obtained from the wetting potential of the surface through a simple boundary condition of the gradient of the order parameter on the surface. The PFM simplifies the interface-tracking calculation by adopting standard finite difference

*Received XX Xxx, 200X (No. XX-XXXX) [DOI: 00.0000/ABCDE.2006.000000]

Journal of Template

2

Vol. 00, No. 0, 0000

techniques only, and without any elaborating algorithms for the advection and reconstruction of interfaces and the evaluation of interfacial force such as in other methods. Therefore, PFM-based CFD has attracted considerable attention from many researchers as a promising simulation tool for multiphase flows in the last two decades. In this study, we examined the applicability of a PFM-CFD method (7)–(10) to microscopic two-phase fluid motions induced by the wettability of a solid surface and propose numerical schemes for solving diffuse-interface advection equations revised to improve volume conservation.

This paper is organized as follows. In the next section, we outline the PFM-CFD method (7)–(10) for a two-phase flow with a high density ratio on a solid surface. The CFD simulation results of a capillarity-driven gas–liquid flow are presented in §3. Two variations of the lattice Boltzmann method (LBM) (11)–(20) for calculating the diffuse-interface advection are proposed and evaluated through a benchmark test of linear translation of a circular interface in §4. The conclusions are presented in the last section.

2. Numerical Method Based on the Phase-field Model

2.1 Basic Equations The PFM-based CFD method used in this study, which is based on the LBM (17)(19), was

proposed for incompressible, isothermal two-phase fluid flows with a high density ratio (3)(7)–(9). The method solves a set of mass and momentum conservation equations of the fluid and the advection–diffusion equation of the diffuse-interface profile (7)–(10):

0 u (1)

21 TSp

t

uu u u u (2)

( )CHMt

u (3)

where u denotes the velocity; t, the time; , the density; p, the pressure; , the viscosity; , the order parameter (5); , the chemical potential; and MCH, the mobility.

The scalar variable distinguishes between two phases with different values of and indicates an interface as a finite volumetric zone between phases across which varies continuously (5). Inside the interface, is assumed to be a sine-curve function of that varies between given constants for gas and liquid phases G and L; varies between the given constants G and L as a linear function of (17). The parameter S is set to be a constant in the entire field; this is based on the following definition of for a flat interface normal to the direction of axis (5):

12

S d

(4)

The chemical potential is derived from the bulk free energy and excessive energy

caused by the squared local gradient of (5):

2

(5)

where represents the double-well form of and is a parameter for controlling the interfacial thickness . In this PFM-CFD method, the van der Waals free energy (17)(18) is represented by (). To simplify computation, the mobility MCH is defined as MCH,0 (7)(17), where MCH,0 is a given positive constant (7). The right-hand side of Eq. (3) influences

Journal of Template

3

Vol. 00, No. 0, 0000

the formation of interfaces with a given constant thickness, and this factor disappears in the equilibrium state.

2.2 Boundary Condition on Solid Surface In a flow, a solid surface with velocity uW is regarded as a no-flux and no-slip boundary

which imposes the following constraints on the fluid motion (9):

0S p n (6)

0S n (7)

Wu u (8)

where nS is the unit vector normal to the surface boundary. In addition, for G L, the following constraint of wettability is taken into account in a manner similar to that in other PFMs (5)(19):

S S n (9)

The parameter S is called the wetting potential (19). A value of S = 0 leads to a static contact angle W = 90°(3)(9).

2.3 Numerical Scheme and Algorithm Equations (1)–(3) are solved under the boundary condition given in Eqs. (6)–(9) through

the use of the following conventional techniques (3)(7)–(9). The three-dimensional space is discretized uniformly as unit cubic cells on a fixed structured grid with a cell width of x = y = z = 1 in the Cartesian coordinate system (x, y, z), where the scalar and vector variables are located in a staggered arrangement. The gradients of p, , , and are evaluated with a fourth-order central difference scheme (CDS). The advancement in time t is based on the second-order Runge–Kutta’s scheme for increasing t. The variables p and u are obtained by using the projection method. In Eq. (2), advection is calculated with a third-order upwind finite difference scheme, whereas the viscous stress is evaluated with a second-order CDS. In Eq. (3), both of the advection and diffusion terms of are calculated using a fourth-order CDS within the framework of the finite volume method (FVM). On the solid boundary, is extrapolated by using the known data within the domain according to Eq. (9). For example, on a solid surface normal to the x direction, the values of at the cell centers outside the flow domain are set by the following extrapolation (9):

1

1 0 S x (10)

12 1 0 127 24 S x

(11)

where the subscript of denotes the cell number in the x direction and the number = 0 indicates that the cell neighbors the surface inside the flow domain.

3. Applicability Test Simulation of Capillarity-driven Two-phase Fluid Motion

The basic applicability of the PFM-CFD method presented above (3)(7)–(9) to the dynamic contact-line problem was examined through a simple test simulation of the capillarity-driven motion of a microscopic two-phase fluid with a high density ratio between parallel plates with a hydrophilic solid surface under no gravity. The fluid was assumed to be an air–water system with density and viscosity ratios of L/G = 801.7 and L/G = 73.76, respectively, in a gap space with a height h = 0.1–5 mm or an air–ethanol

Journal of Template

4

Vol. 00, No. 0, 0000

system with L/G = 636.4 and L/G = 82.39 in a gap of h = 1 mm. The static contact angle W was set to a uniform 61° or 56° (9) on the surfaces in each system. The computational domain of 32 5 128 unit cubic cells in three dimensions (x, y, z) was surrounded by the solid plates with a hydrophilic flat surface, periodic boundaries, and free inflow and outflow boundaries in the x, y, and z directions, respectively. Both the actual ends of the plates in the z direction were assumed to be located away from the computational domain boundaries by 1024z, where the pressure was fixed to be constant.

Figure 1 shows time-series snapshots of the liquid phase profile between the plates for Ohnesorge number Oh = L/(Lh)1/2 = 2.15 10-3 in the air–water system with h = 5 mm. The liquid penetrates faster at a smaller W. The numerical results of the dimensionless interfacial moving distance s* and moving velocity V* = ds*/dt* at dimensionless time t* agreed well with the one-dimensional theoretical predictions (1) for each value of W until t* = 0.4, which corresponded to t = 0.64 s in the actual system (see Fig. 2). As shown in Fig. 3, the other conditions of L/G, L/G and Oh also showed good agreement between the numerical results and theoretical predictions of s* for both values of W.

(b) W =55.9˚t* = 0 t* = 0.288

128z

h = 32x

t* = 0.375

(a) W =61.4˚t* = 0 t* = 0.288 t* = 0.375

5y

Gas

Liquid

Fig. 1 Liquid penetrating gap between parallel plates with hydrophilic contact angle W

0 0.1 0.2 0.3 0.40

0.01

0.02

0.03

0.04

0.05

Dim

ensi

onle

ss i

nter

face

m

ovin

gdi

stan

ce

s*

Dimensionless time t*

14

4*

0

2 116 n

Oh ss n

h

14

4*2

0

2 18

L

nL

t t nh

0 0.1 0.2 0.3 0.40

0.05

0.1

0.15

0.2

Dim

ensi

onle

ss i

nter

face

ve

loci

ty V

* =ds

* /dt

*


(a) The dimensionless moving distance s*.

(b) The dimensionless moving velocity V*.

Numerical results

Theoretical predictions

61.4˚ 55.9˚W

Numerical results


61.4˚ 55.9˚W

Fig. 2 Time histories of interfacial moving distance and velocity between parallel plates for Oh =

L/(Lh)0.5 = 2.15 10-3

Journal of Template

5

Vol. 00, No. 0, 0000

0 0.1 0.2 0.3 0.4 0.5 0.60

0.01

0.02

0.03

0.04

0.05

Dim

ensi

onle

ss i

nter

face

m

ovin

gdi

stan

ce

s*


Numerical results for


W = 61.4˚

W = 55.9˚

W = 61.4˚

W = 55.9˚

L/G = 801.7, L/G = 73.76, Oh = 4.8010-3 (air–water, h = 1 mm)

(a)

0 0.5 1 1.50

0.02

0.04

0.06

0.08

0.1

0.12

0.14

s*

t*

Numerical results


W = 61.4˚

W = 55.9˚

W = 61.4˚

W = 55.9˚

(b) L/G = 801.7, L/G = 73.76, Oh = 6.78 10-3 (air–water, h = 0.5 mm)

0 1 2 3 40

0.05

0.1

0.15

0.2

0.25s

*

t*

Numerical results


W = 61.4˚

W = 55.9˚

W = 61.4˚

W = 55.9˚

(c) L/G = 801.7, L/G = 73.76, Oh = 1.52 10-2 (air–water, h = 0.1 mm)

0 0.5 1 1.5 20

0.05

0.1

0.15

s*

t*

Numerical results


W = 61.4˚

W = 55.9˚

W = 61.4˚

W = 55.9˚

(d) L/G = 636.4, L/G = 82.39, Oh = 1.07 10-2 (air–ethanol, h = 1 mm)

Fig. 3 Time histories of interfacial moving distance

Our previous study (9) confirmed that the capillary force is accurately calculated using the PFM-CFD method in static contact-line problems on a hydrophilic solid surface under gravity, a liquid column between vertical parallel plates and a drop on a horizontal plate. The results of the present study confirmed that this CFD method can successfully predict capillarity-driven fluid flow as well.

4. Improvement of Numerical Method for Diffuse-interface Advection

4.1 Allen–Cahn Advection Equation The numerical results shown in §3 demonstrated that the PFM-based CFD method

(3)(7)–(9) is useful for interface-tracking simulation of capillarity-driven two-phase fluid motion on a solid surface. The CH advection Eq. (3) includes the fourth-order differential term on the order parameter for forming diffusive interfaces (5)–(7)(17). To improve the computational efficiency in the PFM-based simulations, the diffusion term should be simplified. A PFM-CFD method for two-phase flow was recently proposed in which an alternative advection equation with a conservation form has a second-order differential term on for interface formation (21). We propose a numerical method for solving the diffuse-interface advection equation instead of Eq. (3).

In the PFM-CFD method proposed by Chiu and Lin (21), the diffuse-interface advection equation is based on the following Allen–Cahn (AC) equation with an advection term for the time evolution of the spatial distribution of the non-conservative order parameter :

ACMt

u (12)

where MAC represents the mobility. The chemical potential is defined by Eq. (5), in which the following double-well potential () is adopted:

Journal of Template

6

Vol. 00, No. 0, 0000

2211

2 (13)

The profile across a flat interface in an equilibrium state is expressed as (5)(21)–(23)

1( ) 1 tanh

2 2

(14)

where denotes the signed distance in the direction normal to the interface from the central position and = 1/2 for = 0. By substituting Eq. (5) with Eq. (13) into Eq. (12), the following equation is obtained:

2 1 1 2ACM

t

u (15)

The above equation can be transformed into

AC

AC

AC

Mt

M

M

u n

n n

n

(16)

where n = /|| is the unit vector normal to the interface and the second term on the right-hand side, n||, is derived by using the first and second derivatives of on .

1

(17)

1 1 2

n (18)

Equation (16) shows that the interface in Eq. (12) moves not only with fluid velocity u but also with diffusion proportional to the interfacial curvature n. Subtracting the right-hand side term of Eq. (16) from that of Eq. (12) leads to the following equation:

ACMt

u n (19)

where the interfacial curvature effect on the interfacial motion is eliminated from the original Eq. (12). The CFD method adopting Eq. (19) is called the advected phase-field (APF) approach (22)(23). After some algebraic operation for MAC = constant MAC,0, Eq. (19) can be transformed into the following equation in conservation form (21):

0Dt

u j (20)

where D0 = MAC,0 and j is the local anti-diffusion flux vector given by

Journal of Template

7

Vol. 00, No. 0, 0000

,0 1ACM j n (21)

Equation (20) is equivalent to a one-step conservative level-set (CLS) equation (24)–(27) with which both interfacial re-initialization and advection calculations are conducted at the same time. The revised AC advection Eq. (20) has a second-order differential diffusion term that reduces the computational cost in PFM-based interface-tracking simulations compared to the CH equation with a fourth-order differential term.

4.2 Lattice Boltzmann Method for Solving Revised AC Advection Equation We propose a numerical scheme based on an LBM (11)–(20) for solving Eq. (20). The LBM assumes that a fluid flow consists of fictitious mesoscopic particles repeatedly colliding with each other and translations with a set of isotropic discrete velocities (14)(16). The main variables in the LBM are the distribution functions for the number densities of particles grouped with their velocities. The time evolution of the particle distributions at position x and time t is expressed by

1, ,eqa

a a a ag

gg g t g t

t

e x x (22)

where ga is the distribution function with particle velocity ea, g is the single relaxation time in the so-called BGK collision step, and the superscript eq of ga denotes a local equilibrium state. In this study, Eq. (22) was discretized into the following equation in semi-Lagrangian form with second-order accuracy in space and time (14)(16) :

00 0, , , ,eq

a a a a ag

tg t t t g t g t g t

x e x x x (23)

where t0 represents the constant increase in time and x + eat0 denotes the position neighboring x in the direction of vector ea. In the two-dimensional Cartesian coordinate system (x, y), the space is divided uniformly into square cells with widths x and y. All the scalar and vector variables are located at each cell center. The particle velocity set ea = (ea,x, ea,y) is given as follows (14)(16):

0 1 2 3 4 5 6 7 8, , , , , , , ,

0 1 1 0 1 1 1 0 1

0 0 1 1 1 0 1 1 1c

e e e e e e e e e (24)

where c is the lattice constant that is related with x, y and t0 as follows:

0c t x y (25)

The variables of two-phase fluid, and u, are defined with ga and ga

eq by

eqa aa a

g g (26)

eqa aa

g u j e (27)

The equilibrium distributions ga

eq are defined by the following equations:

Journal of Template

8

Vol. 00, No. 0, 0000

0 0 0 2, 1 3 1

2eq

S

g t w wc

u ux (28)

2

2 4 2

( ), 3 ( 0)

2 2aeq a

a aS S S

g t w ac c c

e ue u j u ux (29)

where cS = c/30.5 is the speed of sound of the LBM, is the parameter that controls the number density of moving particles in an equilibrium state with u = j = 0, and the weight parameter wa takes 4/9 for a = 0, 1/9 for a = 1, 3, 5, 7, and 1/36 for a = 2, 4, 6, 8. The revised AC Eq. (20) can be derived from the LBM Eq. (23) through the Chapman–Enskog multiscale expansion procedure (14)(16). The constant part of the diffusion coefficient in Eq. (20), D0 = MAC,0, is given as follows:

2 00 2g

tD c

(30)

To reduce the numerical diffusion error of the semi-Lagrangian LBM, the value of g is set as follows (11)(12)(15):

0

1 11

2 3g t

(31)

The advantage of the semi-Lagrangian LBM over other numerical methods for directly solving the differential equations is the simple particle–kinematics operation in the discrete conservation form on an isotropic spatial lattice (14)(16), which is useful for high-performance computing of mass, momentum and energy conservation equations in multiphase fluid dynamics.

4.3 LBM for Revised CH Advection Equation We propose that the same idea for the AC equation be applied to the CH Eq. (3) as

follows:

,0CHMt

u n (32)

where the mobility MCH,0 is set as constant and the interfacial curvature dependency (Gibbs–Thomson effect) is removed from the diffusion flux (7) by addition of the second term on the right-hand side. The surface tension effect of on is thus removed from the advection equation, and the chemical potential balance across a curved interface is improved. By adopting Eq. (32) instead of Eq. (3) in the PFM-CFD method, not only should the surface tension force alone be considered in the Navier–Stokes equations of fluid motion, but also the volume of each phase fluid should be better conserved in the flow domain. To solve the revised CH Eq. (32), we propose another LBM based on the previous ones (13). It adopts the following equilibrium distribution functions ga

eq with the correction term ||n to Eq. (23):

0 0 02, 1 3 1

2eq

S

g t w wc

u ux n (33)

Journal of Template

9

Vol. 00, No. 0, 0000

2

2 4 2, 3 ( 0)

2 2aeq a

a aS S S

g t w ac c c

e ue u u ux n (34)

The chemical potential of Eq. (32) is the same as that of the AC equation. The mobility MCH,0 is defined (13) in the same form as the conventional diffusion coefficient D0 in Eq. (30):

2 0,0 2CH g

tM c

(35)

where g takes the same value of Eq. (31) as that for the AC equation in this study.

4.4 Benchmark Test on Advection of Interface A benchmark test problem of linear translation of a single circular interface in two dimensions (x, y) (7) was solved for evaluating the proposed LBMs. As shown in Fig. 4(a), the computational domain was uniformly divided into square cells of 100x 100y in a periodic uniform flow with velocity u = (u, v). In another case of higher spatial resolution, cells of 200x 200y were arranged. The numerical results were obtained under the conditions of x = y = 1, t0 = 1, u = v = 10-2, MAC,0 = 10-2, and = 1; this resulted in a Courant number C = ut0/x = 10-2 and theoretical interfacial thickness 41/2 = 4. The value of MAC,0 was selected to be on the same order as that of C for the above x, t0 and , to balance the diffusion with the speed of MAC,01/2 and the advection with |u| (21). To set the initial distribution of (x,0), Eq. (23) was iterated under the tentative condition of = 1 and 0 inside and outside the fluid region with diameters d = 32 or 64, respectively, to obtain the solution at a stationary equilibrium state. For calculating the interfacial curvature, the gradient of at each cell center was calculated using a second- or fourth-order CDS.

Table 1 Finite volume method (FVM) specifications

Schemetype #

Gradient of Interpolation of at cell surface

Advectionterm

Diffusionterm

Timemarching

1 2nd 2nd 2nd 2nd 2nd2 2nd 4th 2nd 2nd 2nd3 2nd 2nd 4th 4th 2nd4 2nd 4th 4th 4th 2nd5 4th 4th 4th 4th 2nd

u＝(u, v)u=v

x

y

OA contour line of =0.5

(x,t)

Time t* = 0 Periodicboundaries

100 x

100 y

1x y

(a) The computational domain andthe initial condition.

32d x x

y

O

Time t* = tu/d = 50

10 x 10 x

(x,t)

(b) The spatial distribution of afterthe linear translation .

Fig. 4 Benchmark test on linear translation of circular interface using LBM with fourth-order CDS for

revised AC advection equation

Journal of Template

10

Vol. 00, No. 0, 0000

For comparison, several FVMs for Eq. (20) were also applied to the test problem under the same conditions of C, d, MAC,0, u, (x, 0), , x, y and t as those in the LBMs. Table 1 shows the specifications of the FVMs. The variables of and u were located in a staggered arrangement on a structured grid with square cells. The time marching was based on the same second-order scheme as that used for solving the CH Eq. (3) in the two-phase flow simulations. The gradient of for the interfacial curvature, interpolation of at each cell surface, advection and diffusion were evaluated using a second- or fourth-order CDS.

As shown in Fig. 4(b), the interfacial profile in the LBM result was adequately retained without oscillation until the dimensionless time t* = ut/d = 50. Figure 6 shows that the area A, which is represented by the number of spatial square cells with > 1/2, was conserved within a -1% error to the initial value A0. The numerical result obtained with the LBM was better than those obtained with the FVMs adopting second-order CDSs (see Fig. 5).

(a) Type-1 (b) Type-2 (c) Type-3 (d) Type-4 (e) Type-5x

y

O

( , )x t( , )x t( , )x t( , )x t( , )x t

Fig. 5 Interface-profile solutions of revised AC advection equation solved with FMVs after linear

translation at dimensionless time t* = 50 at C = 0.01

0 10 20 30 40 50-0.02

-0.01

0

0.01

Dimensionless time t* = tu/d

Cir

cula

r-sh

aped

Are

aC

onse

rvat

ion

Err

or(A

/A0-

1)

LBM:

Type-1

Type-2

Type-3

Type-4

Type-5

FVMs：

Fig. 6 Area conservation error in linear translation using LBM and FVMs at C =0.01 for d=32

0 10 20 30 40 50-1

-0.5

0

0 10 20 30 40 50-1

-0.5

0

Dimensionless time t* = tu/(32x)

Are

aC

onse

rvat

ion

Err

or(A

/A0-

1)


Err

or(A

/A0-

1)

Revised AC Eq.(20)

FVM Type-5

Eq.(19)

LBM (4th-order for )

Original AC Eq.(12)

(a) Low resolution (d=32) (b) High resolution (d=64)

Fig. 7 Area conservation error in linear translation at C=0.01 using LBM and FVM Type-5 for original

and revised AC equations and for APF equation

Journal of Template

11

Vol. 00, No. 0, 0000

The area conservation errors for the revised AC Eq. (20) were compared with those for the original AC and APF Eqs. (12) and (19), both of which were solved by the use of FVM Type-5. As shown in Fig. 7, the area advected by Eq. (20) was better conserved than those in Eqs. (12) and (19) until t* = 50 for large and small interfacial curvatures of d = 32 and 64, respectively. The benchmark tests demonstrated that the anti-diffusion term added to the original AC equation effectively eliminates the curvature-dependent local diffusion flux of from the equation, as well as that the revised AC equation in conservation form is more useful than the APF equation in non-conservation form for interface-tracking CFD simulation of multiphase flows.

0 10 20 30 40 50-0.02

-0.01

0

0.01


Are

aC

onse

rvat

ion

Err

or(A

/A0-

1)

MAC,0 = 10-2

MAC,0 = 210-2

MAC,0 = 510-3


Are

aC

onse

rvat

ion

Err

or(A

/A0-

1)

0 10 20 30 40 50-0.01

0

0.01

MAC,0 = 10-2

MAC,0 = 210-2

MAC,0 = 510-3

MAC,0 = 10-2 MAC,0 = 210-2MAC,0 = 510-3

yx

O

yx

O MAC,0 = 10-2 MAC,0 = 210-2MAC,0 = 510-3

4x 4y

=1/2

8x 8y

=1/2

( , )t x ( , )t x

(a) Low resolution (d=32) (b) High resolution (d=64)

(1) LBM with second-order CDS for gradient of

0 10 20 30 40 50-0.02

-0.01

0

0.01

Are

aC

onse

rvat

ion

Err

or(A

/A0-

1)

MAC,0 = 10-2

MAC,0 = 210-2

MAC,0 = 510-3

Are

aC

onse

rvat

ion

Err

or(A

/A0-

1)

0 10 20 30 40 50-0.01

0

0.01MAC,0 = 10-2

MAC,0 = 210-2

MAC,0 = 510-3

MAC,0 = 10-2 MAC,0 = 210-2MAC,0 = 510-3

yx

O

yx

O MAC,0 = 10-2 MAC,0 = 210-2MAC,0 = 510-3

4x 4y

=1/2

8x 8y

=1/2

( , )t x ( , )t x

Dimensionless time t* = tu/d Dimensionless time t* = tu/d (a) Low resolution (d=32) (b) High resolution (d=64)

(2) LBM with fourth-order CDS for gradient of

Fig. 8 Influence of mobility MAC,0 on interface-profile solutions at t* = 50 and area conservation of linear

translation at C = 0.01 solved with LBM for revised AC equation

The influence of the mobility MAC,0 on the solution of the revised AC equation was

investigated at C = 10-2 for MAC,0 = 5 10-3, 10-2 and 2 10-2 at each of the high and low spatial resolutions. As shown in Fig. 8, the area conservation error in the results from the LBM with the second-order CDS for the gradient of varied with MAC,0 more than that from the LBM with the fourth-order CDS for . In both LBMs, all the results for MAC,0 = 10-2 were within a conservation error of 1% at each resolution. There were no major differences shown in the interfacial profile among the results for different MAC,0 at both resolutions. On the other hand, the FVMs were more sensitive to MAC,0 than the LBMs, as shown in Fig. 9. For instance, in the results obtained with FVM Type-4 and Type-5 at low

Journal of Template

12

Vol. 00, No. 0, 0000

resolution, the circular area was well conserved for MAC,0 = 10-2, whereas the region with > 1/2 disappeared for a larger MAC,0 = 2 10-2. With the other FVMs, the region either deformed to not be circular or disappeared until t* = 50 for each MAC,0 at both resolutions. These results confirmed that the LBMs give a better numerical solution to the revised AC equation for diffusive-interface advection that is more stable and more flexible with regard to the mobility than the FVMs.

0 10 20 30 40 50-1

-0.5

0

Are

aC

onse

rvat

ion

Err

or(A

/A0-

1) MAC,0 = 10-2

MAC,0 = 210-2

MAC,0 = 510-3

Are

aC

onse

rvat

ion

Err

or(A

/A0-

1)

0 10 20 30 40 50-1

-0.5

0MAC,0 = 10-2

MAC,0 = 210-2

MAC,0 = 510-3

MAC,0 = 10-2 MAC,0 = 210-2MAC,0 = 510-3

yx

O

yx

O MAC,0 = 10-2 MAC,0 = 210-2MAC,0 = 510-3

4x 4y

=1/2

8x 8y

=1/2

( , )t x ( , )t x


(1) FVM Type-1

Are

aC

onse

rvat

ion

Err

or(A

/A0-

1)

0 10 20 30 40 50-1

-0.5

0MAC,0 = 10-2

MAC,0 = 210-2

MAC,0 = 510-3

0 10 20 30 40 50-1

-0.5

0

Are

aC

onse

rvat

ion

Err

or(A

/A0-

1)

MAC,0 = 10-2

MAC,0 = 210-2MAC,0 = 510-3

MAC,0 = 10-2 MAC,0 = 210-2MAC,0 = 510-3

yx

O

yx

O MAC,0 = 10-2 MAC,0 = 210-2MAC,0 = 510-3

4x 4y

=1/2

8x 8y

=1/2

( , )t x ( , )t x


(2) FVM Type-2

0 10 20 30 40 50-1

-0.5

0

Are

aC

onse

rvat

ion

Err

or(A

/A0-

1) MAC,0 = 10-2

MAC,0 = 210-2

MAC,0 = 510-3

Are

aC

onse

rvat

ion

Err

or(A

/A0-

1)

0 10 20 30 40 50-1

-0.5

0MAC,0 = 10-2

MAC,0 = 210-2

MAC,0 = 510-3

MAC,0 = 10-2 MAC,0 = 210-2MAC,0 = 510-3

yx

O

yx

O MAC,0 = 10-2 MAC,0 = 210-2MAC,0 = 510-3

4x 4y

=1/2

8x 8y

=1/2

( , )t x ( , )t x


(3) FVM Type-3

Journal of Template

13

Vol. 00, No. 0, 0000

0 10 20 30 40 50-1

-0.5

0

Are

aC

onse

rvat

ion

Err

or(A

/A0-

1) MAC,0 = 10-2

MAC,0 = 210-2

MAC,0 = 510-3

Are

aC

onse

rvat

ion

Err

or(A

/A0-

1)

0 10 20 30 40 50-1

-0.5

0MAC,0 = 10-2

MAC,0 = 210-2

MAC,0 = 510-3

MAC,0 = 10-2 MAC,0 = 210-2MAC,0 = 510-3

yx

O

yx

O MAC,0 = 10-2 MAC,0 = 210-2MAC,0 = 510-3

4x 4y

=1/2

8x 8y

=1/2

( , )t x ( , )t x


(4) FVM Type-4

MAC,0 = 10-2 MAC,0 = 210-2MAC,0 = 510-3

yx

O

yx

O MAC,0 = 10-2 MAC,0 = 210-2MAC,0 = 510-3

0 10 20 30 40 50-1

-0.5

0

Are

aC

onse

rvat

ion

Err

or(A

/A0-

1) MAC,0 = 10-2

MAC,0 = 210-2MAC,0 = 510-3

Are

aC

onse

rvat

ion

Err

or(A

/A0-

1)

0 10 20 30 40 50-1

-0.5

0MAC,0 = 10-2

MAC,0 = 210-2MAC,0 = 510-3

4x 4y

=1/2

8x 8y

=1/2

( , )t x ( , )t x


(5) FVM Type-5

Fig. 9 Influence of mobility MAC,0 on interface-profile solutions at t* = 50 and area conservation of linear

translation at C = 0.01 solved with FVMs for revised AC equation

Ox

y

O

4x

4y

=1/24x

4y

=1/2

(a) LBM for AC eq. with2nd-order CDS in grad.

yx

4x

4y

=1/2 xy

O

4y

=1/2xy

O

4x

4x

4y

=1/2y

xO

xO

4x

4y

=1/2y

(b) LBM for AC eq. with4th-order CDS in grad.

(c) LBM for CH eq. with2nd-order CDS in grad.

(d) LBM for CH eq. with4th-order CDS in grad.

(e) FVM Type-2 for AC eq. (f) FVM Type-5 for AC eq.

0 1 (x, t)

Fig. 10 Benchmark test on linear translation of interface using LBMs and FVMs at C = 0.02 for d = 32

Journal of Template

14

Vol. 00, No. 0, 0000

In the benchmark tests at higher C = 0.02, the mobility MAC,0 = 0.02 was adopted in the FVMs for ensuring their numerical stability, whereas MAC,0 and MCH,0 in the LBMs were set at the same value of 0.01 when C = 0.01. The numerical results obtained with the LBM for the revised CH equation for d = 32 are shown in Figs. 10 and 11 for comparison with the results from the revised AC equation. The fluid region obtained with each LBM retained its initial area and interfacial shape better than that obtained with FVM Type-2 until t* = 50.

0 10 20 30 40 50

-1

-0.5

0

0.5[10-2]

Dimensionless time t* = tU/d

Cir

cula

r-sh

aped

Are

aC

onse

rvat

ion

Err

or(A

/A0-

1)

FVMs for AC eq.Type-2

Type-5AC eq.CH eq.

2nd order 4th orderLBM

Fig. 11 Area conservation error in linear translation using LBMs and FVMs at C = 0.02 for d = 32

0 10 20 30 40 50-0.1

-0.05

0

0.05


Are

aC

onse

rvat

ion

Err

or(A

/A0-

1)

Revised CHEq.(32)

2nd-order

4th-order

Original CHEq.(3)

2nd-order

4th-order

Centraldifference scheme (CDS) for

Fig. 12 Area conservation error in linear translation of interface using LBMs for original and revised

Cahn–Hilliard equations at C = 0.02 for d = 32

To investigate the area conservation of the revised CH Eq. (32) for comparison with the original Eq. (3), both equations were solved under the LBM framework with a second-order or fourth-order CDS applied to the gradient of . As shown in Fig. 12, the circular area in Eq. (32) was better conserved than that in Eq. (3) until t* = 50. These results demonstrated that the anti-diffusion term successfully eliminates the interfacial curvature-dependent local diffusion flux of from the original CH equation (7) in the same way as the AC equation.

5. Conclusions

In this study, we examined the applicability of a computational fluid dynamics (CFD) method based on a phase-field model (PFM) to microscopic two-phase fluid motions induced by solid-surface wettability. Two variations of the lattice Boltzmann method (LBM) were proposed for solving the Allen–Cahn (AC) and Cahn–Hilliard (CH) diffuse-interface advection equations, which were revised to improve the volume conservation in the PFM simulations. The PFM-CFD method was applied to the capillarity-driven motion of two-phase fluids with a high density ratio between parallel plates with a uniformly

Journal of Template

15

Vol. 00, No. 0, 0000

hydrophilic surface. The numerical results were compared with the theoretical predictions. The LBMs were evaluated using the benchmark test problem of linear translation of a single circular interface for comparison with several variations of the finite volume method (FVM).

The following major findings were obtained: (1) the PFM-CFD method accurately predicted the gas–liquid motion driven by the capillary force within the narrow gap spacing; (2) the use of the LBMs allows the initial interfacial shape and volume of a fluid to be adequately retained during translation on a spatial structured grid; (3) the LBMs in a semi-Lagrangian form solved the advection equations more accurately than the FVMs adopting second-order central difference schemes.

The above results prove that the PFM-CFD method and LBMs can be used for analyzing two-phase fluid motions on solid surfaces in various microfluidic devices and high-throughput MEMS device fabrication processes. In future work, the PFM-CFD method will be improved by adoption of either the revised AC or CH equation. New PFM-CFD methods will also be constructed under a LBM-based framework for solving the Navier–Stokes equation with the AC or CH equation. They will be evaluated through simple benchmark tests on contact-line motion and practical applications in microengineering by comparison with theoretical, experimental, and/or other numerical simulation data.

Acknowledgments

This research was supported through a grant from the Japan Society for the Promotion of Science (JSPS) through the Funding Program for World-Leading Innovative R&D on Science and Technology (FIRST Program) initiated by the Council for Science and Technology Policy (CSTP).

References

(1) Ichikawa, N., Hosokawa, K. and Maeda, R., Interface Motion of Capillary-driven Flow in

Rectilinear Microchannel, Journal of Colloid and Interface Science, Vol. 280, No. 1

(2004), pp. 155–164.

(2) Matsumoto, S., Takada, N. and Matsumoto, J., Microfabrication Process of Cellular

Structures in Hollow Fiber-shaped Substrates, Transactions of the Japan Society of

Mechanical Engineers Series C, Vol. 76, No. 768 (2010), pp. 1911–1913.

(3) Inoue, Y., Matsumoto, M., Hojo, M., Takada, N., Adachi, T. and Ishida, K., Simulation

Study on Dynamics of Resin–Air Interface during Resin–Air Flows between Filaments

Using Phase-field Navier–Stokes Model, Journal of the Japan Society for Composite

Materials, Vol. 36, No. 3 (2010), pp. 94–103.

(4) Lu, J., Takagi, H. and Maeda, R., Chip to Wafer Temporary Bonding with Self-alignment

by Patterned FDTS Layer for Size-free MEMS Integration, Proceedings of IEEE Sensors

2011 conference (2011), pp. 1121–1124.

(5) Anderson, D. M., McFadden, G. B. and Wheeler, A. A., Diffuse-interface Methods in Fluid

Mechanics, Annual Review of Fluid Mechanics, Vol. 30 (1998), pp. 139–165.

(6) Cahn, J. W. and Hilliard, J. E., Free Energy of a Nonuniform System. I. Interfacial Free

Energy, Journal of Chemical Physics, Vol. 28, No. 2 (1958), pp. 258–267.

(7) Takada, N. and Tomiyama, A., A Numerical Method for Two-phase Flow Based on a

Phase-field Model, JSME International Journal Series B Fluids and Thermal Engineering,

Vol. 49, No. 3 (2006), pp. 636–644.

(8) Takada, N. and Tomiyama, A., Application of Interface-tracking Method Based on

Phase-field Model to Numerical Analysis of Free Surface Flow, Theoretical and Applied

Mechanics Japan, Vol. 55 (2006), pp. 149–156.

Journal of Template

16

Vol. 00, No. 0, 0000

(9) Takada, N., Matsumoto, J., Matsumoto, S. and Ichikawa, N., Application of a Phase-field

Method to the Numerical Analysis of Motions of a Two-phase Fluid with High Density

Ratio on a Solid Surface, Journal of Computational Science and Technology, Vol. 2, No. 2

(2008), pp. 318–329.

(10) Matsumoto, J. and Takada, N., Two-phase Flow Analysis Based on a Phase-field Model

Using Orthogonal Basis Bubble Function Finite Element Method, International Journal of

Computational Fluid Dynamics, Vol. 22, No. 8 (2008), pp. 555–568.

(11) Orlandini, E., Swift, M. R. and Yeomans, J. M., A Lattice Boltzmann Model of

Binary-fluid Mixtures, Europhysics Letters, Vol. 32, No. 6 (1995), pp. 463–468.

(12) Qian, Y. H. and Orszag, S. A., Scalings in Reaction A+B→C: Numerical Simulation by

Lattice Boltzmann Models, Journal of Statistical Physics, Vol. 81, Nos. 1&2 (1995), pp.

237–253.

(13) Swift, M. R., Orlandini, E., Osborn W. R. and Yeomans, J. M., Lattice Boltzmann

Simulations of Liquid–Gas and Binary Fluid Systems, Physical Review E, Vol. 54, No. 5

(1996), pp. 5041–5052.

(14) Chen, S. and Doolen, G. D., Lattice Boltzmann Method for Fluid Flows, Annual Review of

Fluid Mechanics, Vol. 30 (1998), pp. 329–364.

(15) Hirabayashi, M., Chen, Y. and Ohashi, H., The Lattice BGK Model for the Poisson

Equation, JSME International Journal Series B Fluids and Thermal Engineering, Vol. 44,

No. 1 (2001), pp. 45–52.

(16) Succi, S., The Lattice Boltzmann Equation for Fluid Dynamics and Beyond, (2001),

Oxford at the Clarendon Press, UK.

(17) Inamuro, T., Ogata, T., Tajima, S. and Konishi, N., A Lattice Boltzmann Method for

Incompressible Two-phase Flows with Large Density Differences, Journal of

Computational Physics, Vol. 198, No. 2 (2004), pp. 628–644.

(18) Seta, T. and Okui, K., The Single Component Thermal Lattice Boltzmann Simulation of

Pool Boiling in Two Dimensions, Journal of Thermal Science and Technology, Vol. 1, No.

2 (2006), pp. 125–137.

(19) Yoshino, M. and Mizutani, Y., Lattice Boltzmann Simulation of Liquid–Gas Flows through

Solid Bodies in a Square Duct, Mathematics and Computers in Simulation, Vol. 72, Nos.

2-6 (2006), pp. 264–269.

(20) Yan, Y. Y. and Zu, Y. Q., A Lattice Boltzmann Method for Incompressible Two-phase

Flows on Partial Wetting Surface with Large Density Ratio, Journal of Computational

Physics, Vol. 227, No. 1 (2007), pp. 763–775.

(21) Chiu, P.-H. and Lin, Y.-T., A Conservative Phase Field Method for Solving Incompressible

Two-phase Flows, Journal of Computational Physics, Vol. 230, No. 1 (2011), pp. 185–204.

(22) Biben, T. and Misbah, C., Tumbling of Vesicles under Shear Flow within an

Advected-field Approach, Physical Review E, Vol. 67, No. 3 (2003), 031908.

(23) Beaucourt, J., Biben, T., Leyat, A. and Verdier, C., Modeling Breakup and Relaxation of

Newtonian Droplets Using the Advected Phase-field Approach, Physical Review E, Vol. 75

No. 2 (2007), 021405.

(24) Olsson, E. and Kreiss, G., A Conservative Level Set Method for Two Phase Flow, Journal

of Computational Physics, Vol. 210, No. 1 (2005), pp. 225–246.

(25) Oshima, N. and Gong, J., Level-set and Phase Field Approaches for Interface of

Conservative System, Proceedings of the 24th JSME Computational Mechanics

Conference, No.11-03 (2011.10), pp. 313–314.

(26) Tan, N., Aoki, T., Inoue, K. and Yoshitani, K., Numerical Simulation of Two-phase Flow

Driven by Rotating Object, Transactions of the Japan Society of Mechanical Engineers

Series B, Vol. 77, No. 781 (2011), pp. 1699–1714.

(27) Sato, Y. and Niceno, B., A New Contact Line Treatment for a Conservative Level Set

Method, Journal of Computational Physics, Vol. 231, No. 10 (2012), pp. 3887–3895.

Documents

Journal of Template - i-repository.net · classical continuum-dynamics approach using a sharp-interface model (5). Therefore, developing CFD methods to simulate these fluid flows