NUMERICAL SIMULATION OF CONTACT LINE PROBLEMS USING PHASE FIELD MODEL

With no-slip boundary condition on the top and thesubstrate and the periodic condition on the lateral are imposed, the initial staticdroplet on the inclined substrate will finally slide a

NUMERICAL SIMULATION OF CONTACT LINE PROBLEMS USING PHASE FIELD

MODEL

HAN JUN

NATIONAL UNIVERSITY OF SINGAPORE

2015

NUMERICAL SIMULATION OF CONTACT LINE PROBLEMS USING PHASE FIELD

MODEL

HAN JUN

(B.Sc., Beijing Normal University)

A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SCIENCE

DEPARTMENT OF MATHEMATICS

NATIONAL UNIVERSITY OF SINGAPORE

2015

I hereby declare that the thesis is my original work and it has been written by me in its entirety I have duly acknowledged all the sources

of information which have been used in the thesis.

This thesis has also not been submitted for any degree in any university

previously.

HAN JUN June 18, 2015

It is my great honor to take this opportunity to thank those who made this thesispossible First and foremost, I owe my deepest gratitude to my supervisor, Prof RenWeiqing, whose generous support, patient guidance, constructive suggestion, invaluablehelp and encouragement enabled me to conduct such an interesting research project Hissupport and advice have been invaluable, in terms of both personal interaction and profes-sionalism I have benefited from his broad range of knowledge, deep insight and thoroughtechnical guidance in each and every step of my research I am particularly grateful forhis emphasis on simplicity and profoundness in research, an approach that has immenselyaffected my development as an academic Without his inspiration and supervision, thisthesis would never have happened

I would like to express my appreciation to senior fellow researchers, Yao Wenqi, ZhangZhen, Xu Shixin, and my friends, Li Yunzhi and Guo Jiancang, for their patient guid-ance, illuminating suggestions and inspiring discussions I take this opportunity to thankNational University of Singapore for offering me NUS Research Scholarship

Last but not the least, thanks to my parents for their love and support, and to mywife, Huang Shan, who is a PhD in Department of Mathematics, National University ofSingapore, for everything

HAN JUN

The phase field model is a continuum thermodynamical model and has beenwidely applied to deal with multi-phase systems with complicated and time-dependentinterfaces In this thesis, efficient numerical methods are developed to simulate thevapor-liquid system in a rectangle or a cube using the phase field model The fi-nite volume method constructed to simulate the continuity equation is proved toconserve mass for two different kinds of boundary conditions imposed in the liquid-vapor system When no gravity is considered and the walls are hydrophobic withno-slip boundary condition imposed on all walls, the initial random noise vapor-liquid system will evolve to a stable droplet at the center and the velocity field ofthe droplet will tend to zero at equilibrium With the same boundary conditions, theinitial system with water on the one side and vapor on the other side will also evolve

to a stable droplet at the center While all walls are hydrophilic, a stable bubblewill be formulated at the center and the water will be attracted to the walls whenno-slip boundary condition is imposed on all walls The final stable state admitsone of local minimums of Helmoltz free-energy functional and thus is a physicallystable state A lot of numerical experiments are done to verify the relation betweenthe static contact angle and the wettability of the flat solid substrate derived fromthe Young’s relation by Borcia During the simulation, we find that this analytical

Abstract Abstract

relation between the static contact angle and the wettability of the substrate can

be approximated by a simple linear function with very small error Numerical periments are carried out to simulate the droplet sliding on the inclined substratewhen microgravity is imposed With no-slip boundary condition on the top and thesubstrate and the periodic condition on the lateral are imposed, the initial staticdroplet on the inclined substrate will finally slide along the inclined substrate at aconstant velocity due to the balance between the tangential component of the grav-ity force and the friction against the motion of the droplet The numerical resultsreveal that the final stable velocity of the droplet sliding on the inclined substratelinearly depends on the microgravity imposed when other parameters which affectthe final stable velocity of the droplet sliding on the sloped substrate are fixed Thenthe linear relation between the microgravity imposed in the liquid-vapor system andthe final stable velocity of the droplet obtained from the numerical simulation isanalyzed from the theoretical perspective

1.1 The phase field model 1

1.2 The contact line and contact angle 4

1.3 Outline of the thesis 6

2 Numerical Methods 8 2.1 Introduction 9

2.2 The dimensionless equations of the phase field model 10

2.3 Numerical methods for the phase field model 11

2.3.1 Finite volume method for the continuity equation 11

2.3.2 The explicit difference method for the momentum equation 14

List of Tables

List of Figures

noise liquid-vapor system will evolve to a stable droplet at the center,

at equilibrium state No-slip boundary condition is imposed on all

side without gravity No-slip boundary condition is imposed on all

No-slip boundary condition is imposed on all walls and the wettability

evolve to a bubble at the center and water will be attracted to the

LIST OF FIGURES LIST OF FIGURES

No-slip boundary condition is imposed on all walls and the wettability

0, 0.1, 0.3, 0.5, 0.7, 0.9, respectively No-slip boundary condition is

im-posed on the top and substrate and the periodic boundary condition

angles under two different kinds of criterions The range of the

the periodic boundary condition is imposed on the lateral of (c) The

is Mueller potential Two endpoints of the initial string are chosen as

LIST OF FIGURES LIST OF FIGURES

(0.6235, 0.0280) to a saddle point (0.2125, 0.2930), obtained by

climb-ing strclimb-ing method, and the exact MEP The red is the exact MEP

climbing string method, and the exact MEP The red is the exact

climb-ing strclimb-ing method, and the exact MEP The red is the exact MEP

4.10 Comparison between the numerical MEP from one local minimum

climbing string method, and the exact MEP The red is the exact

LIST OF FIGURES LIST OF FIGURES

4.11 The minima and saddle point of the Lennard-Jones potential for

seven-atom cluster model The saddle point is found by the climbing

4.12 Comparison between the energy path of the numerical MEP obtained

by the climbing string method The red one is the exact MEP of the

Chapter 1

Introduction

Since its proposition, it has attracted much interest and then generalized to manyother areas, such as multi-phase systems The phase field model is a continuumthermodynamical model The key point in the interfacial dynamical system is how

to characterize the boundary conditions at the interface The phase field modelsubstitutes boundary conditions at the interface by differential equations for theevolution of the auxiliary field which acts as the role of an order parameter, thusavoiding the explicit treatment of the boundary conditions at the interface Thephase field model usually takes two distinct values 0 and 1 for two phases with asmooth change between both values in the zone around the interface For example,for the liquid-vapor system, the density function can be used as the phase field

The phase field model has been successfully and widely applied to deal with modynamically multi-phase systems with complicate and time-dependent interfaces

ther-By introducing the phase field, the multi-phase systems can be treated continuously

Chapter 1 Introduction 1.1 The phase field model

system with a deformable interface to understand the Marangoni convection and

system parameters such as thermal conductivity and viscosities can be expressedcontinuously from the liquid to gas system Therefore, the problem can be treatedsimilar to an entire one phase problem and the interface conditions are substituted

by some nonclassical phase field terms in the Navier-Stokes equation The phasefield variable is governed by a partial differential equation over the entire domainand is coupled with velocity, temperature, and concentration fields Therefore, extraterms are included into the model equation by minimizing the free-energy functional.The interface of two fluids is captured implicitly by gradient terms of the densityand the phase field in the Helmoltz free-energy functional of the system They havebeen extended to simulate the Marangoni migration, where a droplet placed in atemperature gradient tends to move towards the hotter wall, i.e., attracted by hot

A number of formulations of the phase field model are based on a free energy tional depending on the phase field and a diffusive field (variational formulations).Equations of the model are then obtained by using general relations of StatisticalPhysics Such a functional is constructed from physical considerations, but contains

func-a pfunc-arfunc-ameter or combinfunc-ation of pfunc-arfunc-ameters relfunc-ated to the interffunc-ace width Pfunc-arfunc-ame-ters of the model are then chosen by studying the limit of the model with this widthgoing to zero, in such a way that one can identify this limit with the intended sharpinterface model

where the nonclassical phase field term is included into the classical Navier-Stokes

Chapter 1 Introduction 1.1 The phase field model

shear stress balance at the droplet interface The Helmoltz free-energy functional

Navier-Stokes equation Therefore, the basic equations of the phase field model forthe liquid with its own vapor can be written as

assuring the shear stress balance at the droplet interface between the liquid phaseand the vapor phase

Compared with the phase field method, another similar method, the so-called,ghost fluid method, was developed by Osher and his collaborators for describing

Chapter 1 Introduction 1.2 The contact line and contact angle

and his collaborators use the level set function to deal with the interface in phase systems The zero level usually labels the location of the interface, the positivevalues correspond to one fluid, and the negative ones correspond to the other fluid

multi-A ghost fluid can be defined through the level set function and it has the samepressure and velocity of the real fluid but the entropy of the other one for each ofthe two fluids Therefore, the interface conditions can be captured by the ghost fluidjust defined Because the ghost fluids have the same entropy as the real fluid which

is not replaced The one-phase problem can be solved exactly in the same way as inthe phase filed formulation One difference from the phase field model in which theinterface is diffuse thus allowing the diffusive transport between two phases around

When two immiscible fluids are placed on a substrate or one kind of liquid withits own vapor is put in a container, the line where the interface of the two fluidphases or the interface of liquid and its own vapor intersects the substrate is named

as the contact line The contact angle is the tangent angle along the contact lineintersecting the substrate Figure 1.1 presents an example of droplet-vapor systemand the definitions of the contact line and the contact angle are clearly illustrated

in this situation When no-slip boundary condition is imposed and the velocity oftwo fluids or the liquid with its own vapor is zero, we refer to the static contact lineand static contact angle While slip boundary conditions induce the definition ofthe moving contact line and moving contact angle

Contact line problems have wide applications, e.g., industrial emulsification, uid/liquid extraction and hydrodesulfurization of crude oil, polymer blending and

contact line problems are interesting since they have distinct mathematical and

Chapter 1 Introduction 1.2 The contact line and contact angle

denote the fluid-vapor, fluid-solid, and solid-vapor interfaces, respectively

physical features, such as singularities, hysteresis, instabilities, and competing

attracted a lot of interests and much effort has been made to research on addressthese difficulties The equilibrium configuration of the static contact line was re-searched by Laplace and Young And the important Young’s relation is proposed[12]and the analytic relation between the static contact angle and the solid substrate

for the moving contact line problem based on thermodynamic principles are derived

is derived for the moving contact line problem through a combination of

also systematically investigated the physical processes near a moving contact line

phys-ical models for moving contact line problems, a number of numerphys-ical methods havebeen put forward to verify the accuracy of these models An efficient numericalscheme for the two phase moving contact line problem with variable density, viscos-

applied for solving a coupled system of the Cahn-Hilliard and Navier-Stokes

Chapter 1 Introduction 1.3 Outline of the thesis

two-phase flows with moving contact line and insoluble surfactant is proposed by

flow field, a convection-diffusion equation for the surfactant concentration, togetherwith the Navier boundary condition and a condition for the dynamic contact an-

Motivated by Borcia’s work, the purposes of this paper are twofold First, merical simulation of static contact angle problems in three-dimension liquid-vaporsystem is presented To ensure the conservation of mass, finite volume method isproposed to solve the continuity equation Explicit finite difference method andSemi-implicit finite difference method are put forward to simulate the momentumequation Rather than simulating the contact line problem of two dimensional case

nu-in Borcia’s paper, the static contact angle calculated nu-in the numerical experiments

is consistent with the analytic static contact angle obtained from the following tion,

s − 4ρ3

which will be derived from the Young’s relation in the following chapter according

droplet sliding on the inclined substrate When microgravity is imposed, the initialstatic droplet on the inclined substrate will finally evolve to a stable droplet sliding

on the inclined substrate at a constant velocity As the microgravity increases, thefinal stable velocity of the droplet will also increase We investigate the relationbetween the microgravity imposed and the final constant velocity of the droplet

Chapter 1 Introduction 1.3 Outline of the thesis

when other parameter affecting the final stable velocity is fixed

This thesis is organized as follows The phase field model for the liquid-vaporsystem and the introduction to the contact line and contact angle are presented

in chapter 1 In chapter 2, we develop both explicit difference method and implicit difference method for the momentum equation, and finite volume methodfor the continuity equation for the conservation of mass The linear system derivedfrom the semi-implicit difference method can be solved by the conjugate gradientmethod Numerical experiments are carried out to simulate the liquid-vapor systemfor different initial conditions and boundary conditions in chapter 3 These numer-ical results demonstrate that the numerical schemes are very efficient in simulatingthe liquid-vapor system The theoretical relation between the contact angle and thewettability of the substrate has been presented In chapter 3, a lot of numericalexperiments are carried out to verify the accuracy of the relation the static contact

sliding on the inclined substrate with microgravity imposed are also simulated inChapter 3 After a long-time simulation, the initial static droplet will side along theinclined substrate at a constant velocity When other parameters affecting the finalstable velocity of the droplet are fixed, the relation between the final stable velocityand the microgravity is numerically investigated The implementations of the stringmethod, the simplified and improved string method and the climbing string method

to the Muller potential and for the Lennard-Jones potential for the seven-atom ter model are carried out in chapter 4 The conclusions are drawn and the furtherresearch plan is discussed in chapter 5

clus-Chapter 2

Numerical Methods

In this chapter, we will focus on developing efficient numerical methods to ulate the phase field model for the liquid-vapor system First, we should rewritethe equations for the phase field model in the dimensionless form before develop-ing numerical methods Since the conservation of mass is very important in thisliquid-vapor system, finite volume method are put forward to simulate the continu-ity equation Explicit and semi-implicit difference methods are proposed to simulatethe momentum equation As discussed in the numerical results, the reason why thesemi-implicit difference method is constructed is that the explicit difference methoddemonstrates obvious oscillation and the semi-implicit difference method can reducethe degree of oscillation Another advantage of the semi-implicit difference method

sim-is that the matrices of the linear systems derived from the semi-implicit differencemethod are sparse, positive definite and symmetric Therefore, these linear systemscan be solved efficiently and fast by the conjugate gradient method The conjugategradient method will be presented in detail at the end of this chapter

Chapter 2 Numerical Methods 2.1 Introduction

Numerical methods play a very important role in scientific research Since thesolutions of partial differential equations are not available in most cases, the nu-merical methods which simulate the partial differential equations can provide us astrong support to understand the solutions of the partial differential equations weare interested in When appropriate methods are applied, the numerical solutionsobtained from numerical experiments are significantly approximate to the analyticsolutions of the partial differential equations The most widely used numerical meth-ods are finite difference method, finite volume method, finite element method andspectral method Usually, it is much easier to construct the finite difference schemesboth explicit and implicit for partial differential equations we encounter Therefore,the finite difference method is widely used for its simplicity To improve the accu-racy and stability of the finite difference schemes, higher-order schemes and implicitschemes are often constructed to satisfy the requirement When the conservationlaw is required in the physical system, finite volume method is usually constructed

In our thesis, finite volume method are constructed to simulate the continuity tion for the conservation of mass Both explicit and semi-implicit finite differencemethods are constructed to simulate the momentum equation For the same initialvalue, the semi-implicit difference method converges faster to the stable droplet onthe center than the explicit finite difference method and demonstrates less oscillationthan the explicit difference method Significantly fewer iterative steps are neededfor the evolution of final steady state using implicit difference methods when thesame stopping criterion is used

equa-Chapter 2 Numerical Methods 2.2 The dimensionless equations of the phase field model

will occur Therefore

Chapter 2 Numerical Methods 2.3 Numerical methods for the phase field model

Before presenting a detailed description of numerical methods, we should duce mesh into the domain we are interested in The computational domain is acube in three dimensional case or a rectangular in two dimensional case The cube

intro-or rectangle has been equally divided in each dimension Take the cube as an

the computational domain We denote the length of the unit cell along x direction,

y direction, z direction as ∆x, ∆y and ∆z, respectively.

2.3.1 Finite volume method for the continuity equation

time step, then we have

Chapter 2 Numerical Methods 2.3 Numerical methods for the phase field model

function to approximate ρv along the normal direction n, then we will have

∫

∂A ijk

2 − ρ i,j,k W i,j,k + ρ i,j,k −1 W i,j,k −1

+ (ρ i,j+1,k V i,j+1,k + ρ i,j,k V i,j,k

2 − ρ i,j,k V i,j,k + ρ i,j −1,k V i,j −1,k

n i,j,k W n i,j,k + ρ n

conditions for all walls of the area is considered, i.e., v = 0 on the boundary, we

introduce the following uniform notation for the flux

(2.7)

Chapter 2 Numerical Methods 2.3 Numerical methods for the phase field model

When no-slip boundary conditions are considered on the top and on the bottom andperiodic conditions are considered on the lateral, we introduce the ghost mesh sothat the notation can be simplified in the simulation,

ρ n −1,j,k = ρ n N x −1,j,k , ρ n N x+1,j,k = ρ n 1,j,k , ρ n N x+2,j,k = ρ n 2,j,k;

ρ n i, −1,k = ρ n i,N y −1,k , ρ n i,N y+1,k = ρ n i,1,k , ρ n i,N y+2,k = ρ n i,2,k;

U N x+1,j,k n = U 1,j,k n , V N x+1,j,k n = V 1,j,k n , W N x+1,j,k n = W 1,j,k n ;

U i,N y+1,k n = U i,1,k n , V i,N y+1,k n = V i,1,k n , W i,N y+1,k n = W i,1,k n ;

for i = 0, , N x, j = 0, , N y, k = 0, , N z Therefore, no matter which boundary

for all the mesh points can be uniformly written as

n i,j,k+1 −F z n

i,j,k)− ∆t

n i,j+1,k −F y n

i,j,k)− ∆t

n i+1,j,k −F x n

i,j,k ).

(2.8)

Theorem 1: The finite volume scheme (2.8) for either no-slip boundary conditionfor all walls or periodic boundary condition on the lateral with no-slip boundary onthe top and bottom conserves the mass, i.e,

Chapter 2 Numerical Methods 2.3 Numerical methods for the phase field model

For the periodic boundary condition on the lateral and no-slip condition on the topand bottom, by introducing ghost mesh points, using almost the same argument as

in both cases is conserved

2.3.2 The explicit difference method for the momentum

Chapter 2 Numerical Methods 2.3 Numerical methods for the phase field model

each dimension Let G = (0, 0, G) In x-direction, the momentum equation can be

∂U

13

∂V

13

∂x∂z

)(2.13)

The momentum equation for y-component can be written as

∂V

13

∂U

13

∂W

13

∂U

13

Chapter 2 Numerical Methods 2.3 Numerical methods for the phase field model

constructed as follows

U i,j,k n+1 =U i,j,k n − c x

n i,j,k (U i+1,j,k n − U n

1,i −1,j,k)− c y

n i,j,k (U i,j+1,k n − U n

i,j−1,k)

n i,j,k (U i,j,k+1 n − U n

i,j,k−1)− C a

n i+1,j,k − µ n

i−1,j,k) +

n i,j+1,k − V n

i,j−1,k)

n i,j,k+1 − W n

Chapter 2 Numerical Methods 2.3 Numerical methods for the phase field model

V i,j,k n+1 = V i,j,k n − c x

n i,j,k (V i+1,j,k n − V n

i −1,jk)− c y

n i,j,k (V i,j+1,k n − V n

i,j −1,k)

n i,j,k (V i,j,k+1 n − V n

2,i,j,k −1)− C a

n i,j+1,k − µ n

i,j −1,k) +

n i+1,j,k − U n

i −1,j,k)

n i,j,k+1 − W n

i,j,k + V i,j−1,k n ) + c xx (V i+1,j,k n − 2V n

i,j,k + V i−1,j,k n )

+ c zz (V i,j,k+1 n − 2V n

i,j,k + V i,j,k n −1 )].

(2.17)

Chapter 2 Numerical Methods 2.3 Numerical methods for the phase field model

W i,j,k n+1 = W i,j,k n − c x

n ijk (W i+1,j,k n − W n

i −1,j,k)− c y

n i,j,k (W i,j+1,k n − W n

i,j −1,k)

n i,j,k (W i,j,k+1 n − W n

i,j,k−1)− C a

n i,j,k+1 − µ n

i,j,k −1) +

n i+1,j,k − U n

i −1,j,k)

n i,j+1,k − V n

i,j,k + W i,j,k−1 n ) + c xx (W i+1,j,k n − 2W n

i,j,k + ρ n i,j −1,k)− 1

n i,j,k+1 − 2ρ n

i,j,k + ρ n i,j,k −1)

(2.19)

Chapter 2 Numerical Methods 2.3 Numerical methods for the phase field model

The CFL condition is given as

n i,j,k

In the simulation, the explicit difference method demonstrates some oscillation

To avoid this inaccuracy, the semi-implicit method is constructed Since the cretized matrices are symmetric and positive definite, these systems of linear equa-tions can be efficiently solved by conjugate gradient method The semi-implicit

n i,j+1,k + U i,j n+1 −1,k) + 1

n+1 i,j,k+1 + U i,j,k n+1 −1)

)

= U i,j,k n − c x

n ijk (U i+1,j,k n − U n

i −1,j,k)− c y

n i,j,k (U i,j+1,k n − U n

i −1,j,k) +

n i,j+1,k − V n

Chapter 2 Numerical Methods 2.3 Numerical methods for the phase field model

n i+1,j,k + V 2,i n+1 −1,j,k) + 1

2c zz (V

n+1 i,j,k+1 + V i,j,k n+1 −1)

)

= V i,j,k n − c x

n i,j,k (V i+1,j,k n − V n

i −1,j,k)− c y

n i,j,k (V i,j+1,k n − V n

i,j −1,k) +

n i+1,j,k − U n

Chapter 2 Numerical Methods 2.3 Numerical methods for the phase field model

n i+1,j,k + W i n+1 −1,j,k) + 1

n+1 i,j+1,k + W i,j n+1 −1,k)

)

= W i,j,k n − c x

n i,j,k (W i+1,j,k n − W n

i −1,j,k)− c y

n i,j,k (W i,j+1,k n − W n

i,j,k −1) +

n i+1,j,k − U n

derive the numerical scheme of mesh points on the boundary

matri-ces When no-slip boundary conditions are imposed on all walls, we just need to

Chapter 2 Numerical Methods 2.3 Numerical methods for the phase field model

the three linear systems above It’s easy to derive another nice property of matrices,

2.3.4 Standard conjugate gradient method for the semi-implicit

difference equations

The conjugate gradient method is a very efficient algorithm for the numerical

solution of particular systems of linear equations Ax = b, namely those whose

matrix is symmetric and positive definite The conjugate gradient method is oftenimplemented as an iterative algorithm, applicable to sparse systems that are toolarge to be handled by a direct implementation or other direct methods such asthe Cholesky decomposition In our linear systems from the semi-implicit schemes,

definite Thus in the practical simulation, we can apply the standard conjugate

The conjugate gradient method for solving Ax = b can be written succinctly in

the following way:

Chapter 2 Numerical Methods 2.3 Numerical methods for the phase field model

Algorithm: Guess the initial ”solution” x0,

Chapter 3

Contact lines and contact angles

In this chapter, numerical experiments are carried out on the droplet-vapor tem for two different kinds of the boundary condition The first kind of the boundary

sys-condition is that the no-slip sys-condition is imposed on all walls(i.e., v = 0) and the

density ρ is given accordingly on different walls The second kind of the boundary

condition is that the no-slip condition is imposed only on the top and bottom of the

rectangular And the periodic boundary condition for both the velocity v = 0 and

wet-tability ρ on the top and bottom of the rectangule is defined according to different

examples All numerical experiments are carried out without gravity in this first

all walls After a long-time simulation, the initial random vapor system will finallyevolve to a stable droplet at the center The vapor will surround the droplet and thevelocity field of the system will tend to zero in the final equilibrium state A differ-ent initial condition is given and numerical experiments are carried out to simulatethe evolution of the liquid-vapor system One side of the rectangle is filled withwater and the other side is full with the vapor Also, the first kind of the boundarycondition is imposed in this case When all walls of the rectangle are hydrophobic,the water will move towards the center of the rectangle as the time goes on And

Chapter 3 Contact lines and contact angles

at equilibrium, a stable droplet will be formulated at the center of the rectangle.The velocity of the field will also go to zero after a sufficiently long-time evolution.While all walls of the rectangle are hydrophilic and the first kind of the boundarycondition is imposed, the almost same initial random noise liquid-vapor system willevolve to a bubble at the center and the water will be attracted to the walls It’seasy to interpret why we have a different numerical result Since the walls of therectangle are hydrophobic, the water will move towards to the walls and therefore

a bubble will be formulated at the center

In the following, we will focus on the three dimensional simulation For thesecond kind of the boundary condition mentioned above, when all walls of the cubeare hydrophobic, the initial random noise liquid-vapor system will evolve to a stabledroplet at the center and the droplet is a round ball in the three dimensional case.The theoretical relation between the static contact angle and the wettability of the

lot of numerical experiments are performed to verify this theoretical relation In thesimulation, we find that the complicate relation of the static contact angle and the

Numerical experiments have been done to simulate the dynamic contact linewhen the microgravity is introduced Borcia has devoted a lot of effort to simu-lating the droplet sliding on the inclined substrate when the microgravity is intro-

when the droplet finally slides along the inclined substrate, driven by the tial force of the gravity We don’t know whether the deterministic relation of thetwo kinds of dynamic contact line angles with respect to the final stable velocityexists The streamlines inside and around the droplet are presented by Borcia whenthe droplet evolves along the inclined substrate Affected by the lateral periodicboundary conditions, the streamlines of the system indicates the self-interaction ofthe droplet Borcia has found that the constant body force per unit volume, i.e., thetangential component of the gravity force parallel to the solid surface together with

tangen-Chapter 3 Contact lines and contact angles 3.1 Introduction

the boundary conditions, imposes a Poiseuille-type flow But Borcia didn’t analyzethe relation between the final stable velocity of the droplet and the microgravityimposed It’s obvious that as the microgravity increases, the final stable constantvelocity of the droplet will increase When the inclined angle is fixed, how to findthe explicit expression of the final velocity of the droplet sliding along the slopedsubstrate in terms of the microgravity is an interesting topic We will focus onfinding this relation in this chapter when the inclined angle is fixed

Numerical methods have been developed in chapter 2 Then to test the accuracyand stability of the proposed numerical methods, a lot of numerical experimentsare to be done in chapter 3 The purpose of this chapter is to verify the analytic

ways have been proposed to interpret the contact lines, particularly moving contactline problems Much effort has been devoted to studying the contact line problems

characterize the contact line problems The numerical simulations for the contactline phenomena are usually based on the sharp interfacial models The interface

of the two fluids along the substrate is tracked by a local moving mesh and haszero thickness in the sharp interface method The interface boundary conditionsare included into the discrete system of equations at the interface adjacent grid

an alternative perspective to deal with the interface between two fluids on the contactline problems These methods assume the interface between two fluids has a nonzerothickness and has been coupled with physical properties such as surface tension.The lattice Boltzmann methods are kinetic methods and simulate the evolution

of multi-fluid system based on a given inter-particle potential The distributionalfunction is proposed to interpret the microscopic properties of the system And

Chapter 3 Contact lines and contact angles 3.1 Introduction

the distributional function satisfies the Boltzmann equation with different collisionterms To ensure the time averaged motion of the particles is consistent with theNavier-Stokes equation, the rules governing the collisions between the particles havebeen designed The lattice Boltzmann methods have been widely researched And

a large amount of theoretical and numerical results have demonstrate the validation

As mentioned above, another widely used method is the phase field modelmethod Borcia has applied the phase field model to do a large amount of re-search and published more than ten papers about the phase field model, such

by Borcia’s work In the following, we will focus on performing a lot of numericalexperiments to test the accuracy and stability of the numerical methods proposed

in chapter 2

Numerical experiments have been done to simulate the static contact line lems and verify the relation between the static contact angle and the wettabilities ofthe substrate in the liquid-vapor system without the gravity The numerical resultsare quite consistent with the theoretical results obtained through the thermody-namical principle These numerical experiments demonstrate that the numericalmethods, finite volume methods for the continuity equations, explicit and implicitfinite difference methods, works well on the static contact line problems for thedroplet-bubble system In the following, we focus on generalizing the numericalmethods proposed to dynamic contact line problems

prob-When the microgravity is introduced into the liquid-vapor system, the initialstatic droplet sitting on the inclined substrate will move along the inclined substrate,driven by the tangential component of the gravity force The no-slip boundarycondition for the velocity is introduced on the inclined substrate of the cube and

the lateral On the top of the cube, the no-slip boundary condition for the velocity