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 Declaration 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 i i Acknowledgements It is my great honor to take this opportunity to thank those who made this thesis possible First and foremost, I owe my deepest gratitude to my supervisor, Prof Ren Weiqing, whose generous support, patient guidance, constructive suggestion, invaluable help and encouragement enabled me to conduct such an interesting research project His support and advice have been invaluable, in terms of both personal interaction and professionalism I have benefited from his broad range of knowledge, deep insight and thorough technical guidance in each and every step of my research I am particularly grateful for his emphasis on simplicity and profoundness in research, an approach that has immensely affected my development as an academic Without his inspiration and supervision, this thesis would never have happened I would like to express my appreciation to senior fellow researchers, Yao Wenqi, Zhang Zhen, Xu Shixin, and my friends, Li Yunzhi and Guo Jiancang, for their patient guidance, illuminating suggestions and inspiring discussions I take this opportunity to thank National 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 my wife, Huang Shan, who is a PhD in Department of Mathematics, National University of Singapore, for everything HAN JUN June 2015 ii ii Abstract The phase field model is a continuum thermodynamical model and has been widely applied to deal with multi-phase systems with complicated and time-dependent interfaces In this thesis, efficient numerical methods are developed to simulate the vapor-liquid system in a rectangle or a cube using the phase field model The finite volume method constructed to simulate the continuity equation is proved to conserve mass for two different kinds of boundary conditions imposed in the liquidvapor system When no gravity is considered and the walls are hydrophobic with no-slip boundary condition imposed on all walls, the initial random noise vaporliquid system will evolve to a stable droplet at the center and the velocity field of the droplet will tend to zero at equilibrium With the same boundary conditions, the initial 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 bubble will be formulated at the center and the water will be attracted to the walls when no-slip boundary condition is imposed on all walls The final stable state admits one of local minimums of Helmoltz free-energy functional and thus is a physically stable state A lot of numerical experiments are done to verify the relation between the static contact angle and the wettability of the flat solid substrate derived from the Young’s relation by Borcia During the simulation, we find that this analytical iii iii 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 experiments are carried out to simulate the droplet sliding on the inclined substrate when microgravity is imposed With no-slip boundary condition on the top and the substrate and the periodic condition on the lateral are imposed, the initial static droplet on the inclined substrate will finally slide along the inclined substrate at a constant velocity due to the balance between the tangential component of the gravity force and the friction against the motion of the droplet The numerical results reveal that the final stable velocity of the droplet sliding on the inclined substrate linearly depends on the microgravity imposed when other parameters which affect the final stable velocity of the droplet sliding on the sloped substrate are fixed Then the linear relation between the microgravity imposed in the liquid-vapor system and the final stable velocity of the droplet obtained from the numerical simulation is analyzed from the theoretical perspective iv Contents Acknowledgements ii Abstract iii List of Tables vii List of Figures viii Introduction 1.1 The phase field model 1.2 The contact line and contact angle 1.3 Outline of the thesis Numerical Methods 2.1 Introduction 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 2.3.3 The semi-implicit difference method for the momentum equation 19 v v CONTENTS 2.3.4 CONTENTS Standard conjugate gradient method for the semi-implicit difference equations 22 Contact lines and contact angles 24 3.1 Introduction 26 3.2 Numerical verification of the static contact angle 28 3.3 Simulation of droplets sliding on the inclined substrate 34 The string method 41 4.1 Introduction 41 4.2 The string method 42 4.3 Application of the string method 45 4.3.1 Implementation of the simplified and improved string method to Muller potential 45 4.3.2 Implementation of the climbing string method to LennardJones potential for seven-atom cluster model 47 Conclusions and further research 55 5.1 Conclusions 55 5.2 Further research plan on moving contact line problems 56 5.2.1 The diffuse interface model for one-component fluid 59 Bibliography 62 vi List of Tables vii vii List of Figures 1.1 A droplet on a substrate xA and xB are the contact lines; the tangent along contact line xA is the contact angle at the wall, denoted as Θm ; Γ, Γ1 , and Γ2 denote the fluid-vapor, fluid-solid, and solid-vapor interfaces, respectively 3.1 Droplet formation in a vapor atmosphere without gravity The initial noise liquid-vapor system will evolve to a stable droplet at the center, at equilibrium state No-slip boundary condition is imposed on all walls and the wettability ρs is chosen as ρs = 0.001 29 3.2 Droplet formation with water on the one side and vapor on the other side without gravity No-slip boundary condition is imposed on all walls and the wettability ρs is chosen as ρs = 0.001 37 3.3 Bubble formation in a liquid-vapor atmosphere without gravity Noslip boundary condition is imposed on all walls and the wettability ρs is chosen as ρs = 0.8 The initial noise vapor-liquid system will evolve to a bubble at the center and water will be attracted to the walls at equilibrium state 37 viii viii Chapter The string method 4.3 Application of the string method Final string 1.5 1.5 1 y y Initial string 0.5 0.5 0 −1.5 −1 −0.5 0.5 −1.5 −1 −0.5 x 0.5 x Figure 4.9: The MEP obtained by the climbing string method Two endpoints of the initial string are X0 = (−0.5594, 1.4405) and XN = (−1.5, −1.5) The saddle point found in this numerical experiment is (−0.8220, 0.6243) Energy along MEP 120 100 80 60 40 20 0 Thermodynamic integration along string Exact 0.2 0.4 0.6 0.8 Figure 4.10: Comparison between the numerical MEP from one local minimum (−0.5594, 1.4405) to a saddle point (−0.8220, 0.6243), obtained by climbing string method, and the exact MEP The red is the exact MEP and the blue one is the numerical MEP 53 Chapter The string method 4.3 Application of the string method saddle minima 3 2 1 −1 −2 −1 −3 −2 −4 −5 −3 −6 −5 −6 −4 −2 (a) One of the minimas of the the (b) One of the saddle points of the the Lennard-Jones potential Lennard-Jones potential Figure 4.11: The minima and saddle point of the Lennard-Jones potential for sevenatom cluster model The saddle point is found by the climbing string method Energy path of the Lennard−Jones potential −7.9 Exact Thermodynamic integration along string −8 −8.1 −8.2 −8.3 −8.4 −8.5 −8.6 −8.7 −8.8 −8.9 0.2 0.4 0.6 0.8 Figure 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 Lennard-Jones potential and the blue one is the numerical energy path 54 Chapter Conclusions and further research 5.1 Conclusions In this thesis, the finite volume method is constructed to solve the continuity equation of the phase field model in droplet-vapor system for the conservation of mass The conservative property of this scheme is proved in two different kinds of boundary conditions Then the explicit and semi-implicit finite difference methods are developed to simulate the momentum equation The initial random noise vapor-liquid atmosphere will finally evolve to a stable droplet at the center when the wettability of the walls is close to zero The initial vapor-liquid system with the vapor on the one side and the water on the other side will also formulate a stable droplet at the center when all the walls are hydrophobic, i.e., ρs ≈ However, when the walls favor water, for the same initial random noise vapor-liquid atmosphere, a bubble will finally be formulated at the center and the water will be attracted to the walls of the rectangle Numerical experiments are carried out to verify the relation between the static contact angle and the wettability of the flat solid substrate which is derived from Young-Laplace equation by Borcia through applying the thermodynamical principle In the experiments of simulation, we find that this analytic relation can be well approximated by a linear function with very small error 55 55 Chapter Conclusions and further research 5.2 Further research plan on moving contact line problems Then we simulate the droplets sliding on the inclined substrate when the microgravity is introduced and no-slip boundary condition on the top and the substrate of the rectangle or cube and the periodic boundary condition on the lateral are imposed The numerical experiments show that the final constant velocity linearly depends on the microgravity imposed when other parameters which affect the final stable velocity of the droplet sliding on the inclined substrate are fixed in the system Then the reason why the final stable velocity of the droplet obtained from the numerical simulation linearly depends on the microgravity has been theoretically analyzed in the thesis 5.2 Further research plan on moving contact line problems The numerical results on the verification of the static contact angles indicate that the numerical methods constructed work well for the static contact line problems Based on the numerical methods constructed for the static contact line problems, we hope to develop efficient numerical methods for the diffuse interface model for one-component fluids derived by Ren and his collaborators in [5] to simulate the moving contact line problems Despite a large amount of effort has been devoted to researching the boundary condition at the moving contact line problems This area still faces much controversy One of main difficulties is that the notorious contact line singularity will occur when the classical Navier-Stokes equation with the no-slip boundary condition is applied to model the contact line problem What is worse is that the contact line singularity is a rather non-physical singularity and it results in infinite energy dissipation rate The contact line singularity is not only a mathematical complication but also representing the complex physical processes encountered near the contact line region To remove the singularity, much effort has been done to modify the hydrodynamic model Most of these modified models postulate slip to occur between 56 Chapter Conclusions and further research 5.2 Further research plan on moving contact line problems the fluid-solid surface near the contact line By assuming that the tangential stress vanished near the contact line or by Navier boundary condition in which the shear stress is proportional to the slip velocity Another problem needed to deal with is that the dynamic contact angle which is assumed to be fixed and equal to its static value Although these modified models remove the singularity, their applications are restrictive and not shed much light on capturing the phenomena near the contact line [16] Another type of model for the contact line dynamics is the molecular kinetic model proposed by Blake and Haynes [29] The molecular kinetic model is based on the rate theory of Eyring There are two primary components about the contact line motion in the molecular kinetic model The first is an activated process between the adsorption sites on the solid surface; the second is that the contact line motion is driven by the unbalanced Young stress A fundamental difference between the hydrodynamic model and the molecular kinetic model is the way to interpret the energy dissipation of the system The molecular kinetic model focuses on the non-hydrodynamic dissipation near the contact line while the hydrodynamic model emphasize the energy loss in the bulk because of the viscous flow [16] This difference induces different relations between the dynamic contact angle and the contact line speed, which is not anticipated To address this issue, combinations of the hydrodynamic and the molecular kinetic models have been proposed by Charlot in 2000 [33] This proposed model considers the viscous dissipation in the bulk and the dissipation caused by the friction at the contact line and has been successfully revealed much about the dynamic near the contact line The disadvantage of the model proposed by Charlot is that it is too complicated and thus limited to the real world application To overcome these difficulties above, Ren and his collaborators have proposed continuum models based on the thermodynamic principles, both macroscopic and microscopic [5,16] Even though the idea, based on the thermodynamical principles, used by Ren looks simple, it is a very powerful approach and can be used for many other related problems A new boundary condition at the contact line for the case 57 Chapter Conclusions and further research 5.2 Further research plan on moving contact line problems of partial wetting and the case of complete wetting has been proposed by Ren The simplified model for thin liquid films of the contact line dynamics is also proposed Then Ren and his collaborators have derived the sharp interface models and diffuse interface models for the moving contact line A lot of effort has been devoted to developing numerical methods to simulate the moving contact line problems The combination of a splitting method based on a pressure Poisson equation and a convex splitting method is proposed by X Wang to simulate the moving contact line problem with variable density and viscosity The physical model is based on a phase field approach consisting of a coupled system of the Cahn-Hilliard and Navier-Stokes equations with the generalized Navier boundary condition For the moving contact line with variable density and viscosity, the main difficulty is how to deal with the situation with large density and viscosity ratio To overcome this difficulty, the splitting method based on a pressure Poisson equation is proposed to solve the Navier-Stokes equations and a convex splitting method is proposed to simulate the Cahn-Hilliard equation [18] Another widely used method is the level-set method The basic idea of the level-set method is described as follows A level-set method for two-phase flows with moving contact line and insoluble surfactant is proposed by W Q Ren and J.J Xu The physical model consists of the Navier-Stokes equation for the flow field, a convection-diffusion equation for the surfactant concentration, together with the Navier boundary condition and a condition for the dynamic contact angle derived by Ren The numerical method is based on the level-set continuum surface force method for two-phase flows with surfactant developed by Xu et al with some cautious treatment for the boundary conditions [52] Three major components are contained in this numerical method The first one is a flow solver for the velocity field The second one is a solver for the surfactant concentration The last one is a solver for the level-set function In the flow solver, the surface force is dealt with by applying the continuum surface force model The Navier boundary condition incorporates the unbalanced Young stress at the moving contact line [20] Recenlty, A finite element method has been developed 58 Chapter Conclusions and further research 5.2 Further research plan on moving contact line problems for the continuum model derived for the dynamics of two immiscible fluids with moving contact lines and insoluble surfactants based on thermodynamic principles by Z Zhang, S Xu and W Ren The continuum model consists of the Navier-Stokes equations for the dynamics of the two fluids and a convection-diffusion equation for the evolution of the surfactant on the fluid interface The interface condition, the boundary condition for the slip velocity, and the condition for the dynamic contact angle are imposed The Stokes equation is solved using the finite element method with unstructured elements in two different domains The convection-diffusion equation for the surfactant is solved by using the finite element method, on the mesh formed by the markers on the interface [53] By applying the diffuse interface model and the phase field model for the liquidvapor system, we want to further numerically investigate the diffuse interface model for one-component fluid in the following research To my best knowledge, no numerical experiment has been carried out before to verify the accuracy of the diffuse interface model for liquid-vapor system In the further research, we hope to verify that the diffuse interface model for the liquid-vapor system through numerical experiments Some preliminary work has been done so far 5.2.1 The diffuse interface model for one-component fluid The diffuse interface models have been applied to the computation of flows related to complex interface morphologies and topological changes, and have also been used to model moving contact lines [6–9] The boundary conditions are theoretically derived for the phase-field models in the cases of one-component fluid and two-component fluids The one-component fluid means a liquid with its own vapor system and the two-component fluids are two immiscible fluids The boundary conditions for moving contact line models in the cases of one-component fluids and two-component fluids are fully investigated in [5] In this thesis, we mainly focus on the simulation of moving contact line model of a liquid with its own vapor system on solid surfaces Before presenting the boundary condition for this system, we first 59 Chapter Conclusions and further research 5.2 Further research plan on moving contact line problems introduce some useful notations The viscous stress, assumed to be linear, is τ = η(∇v + ∇v T ) + λ(∇ · v)I, where η and λ are consistent with the definitions in section 2, and I is an identity matrix Define the operator B[ρ] = K ∂ρ ′ + γwf (ρ), ∂n (5.1) ′ (ρ) = − 12 Γ cos θssurf sin( π2 ρ) given in [16] and θssurf is the contact angle on where γwf √ √ ξ , where ξ = Kr is the the surface The surface tension Γ is defined as Γ = 2r2 3u interface profile thickness and u is a parameter related to ξ The viscous dissipation Jb is given by Jb = −Mb (ρ)B[ρ] ∂ρ + (v s · ∇s )ρ = Jb , ∂t (5.2) where v s is the slip velocity on the boundary The slip velocity v s on the boundary satisfies the following equation −β(ρ)v s = τ n − B[ρ]∇s ρ, (5.3) where β(ρ) is the slip coefficient This model together with boundary conditions physically means that the initial droplet on the substrate with boundary conditions will finally evolve to a stable droplet where the contact angle intersecting the substrate is equal to the given parameter θssurf Using the same scaled variables as in [16], the dimensionless forms of (5.2.1), (5.2.1) can be written as follows ∂ρ + (v s · ∇s )ρ = L(ϕ); ∂t (5.4) 60 Chapter Conclusions and further research 5.2 Further research plan on moving contact line problems v s · tx = −L∫ tx · τ · n + L(ρ)∇s ρ · tx ; (5.5) v s · ty = −L∫ ty · τ · n + L(ρ)∇s ρ · ty ; (5.6) where tx , ty are tangent vectors on the substrate plane along x, y direction, respec√ ∂ρ tively, L(ϕ) = Vs (ε − 62π cos θssurf cos( π2 ρ)) The dimensionless parameters will ∂n occur, L∫ = βηlld and Vs = ξdνl It’s easy to construct the difference scheme for these equations (5.4), (5.5), (5.6) We just present difference scheme for ∇s ρ at mesh point (i, j, 0) on the substrate, ∇s ρni,j,0 = ( ρni+1,j,0 − ρni−1,j,0 ρni,j+1,0 − ρni,j−1,0 T , , 0) 2∆x 2∆y The schemes for the model equations are (2.8), (2.16), (2.17), (2.18) We impose periodic boundary conditions on the lateral and no-slip boundary conditions with ρs = 0.001 on the top of the cube In the simulation, Vs = 5.0, ε = 0.01, and Ls = 0.0038 So the system is solvable and can be implemented on computers Several numerical experiments are carried out to verify the diffuse interface model in the vapor-liquid system by varying θssurf But I haven’t got satisfactory and reasonable results at the moment I am checking the reasons why these numerical experiments are not satisfactory and reasonable One main reason might be that the boundary is changing for this moving contact line model This means further techniques and appropriate methods are needed to deal with these boundary conditions Much further efforts are being tried to overcome the difficulty I encounter at the moment 61 Bibliography [1] G.J Fix, Free boundary problems: theory and applications, Ed A Fasano and M Primicerio, p 580, Pitman (Boston, 1983) [2] J S Langer, Models of pattern formation in first-order phase transitions, Series on directions in condensed matter physics 1, p 165(1986) [3] L.K Antanovskii, A phase field model of capillary, Phys Fluids 7(4)(1995)747 [4] S De Martino, M Falanga, and S I Tzenov, Kinetic derivation of the hydrodynamic equations for capillary fluids, Phys Rev E.70.067301, 2004 [5] W Ren and W E, Derivation of continuum models for the moving contact line problem based on thermodynaic principles,2011 [6] D Jacqmin, Contact-line dynamics of a diffuse fluid interface, J Fluid Mech., 402, 57-88, 2000 [7] T Qian, X Wang and P Sheng, Molecular scale contact line hydrodymaics of immiscible flows, Phys Rev E., 68, 016306, 2003 [8] T Qian, X Wang and P Sheng, A variational approach to moving contact line hydrodynamics, J Fluid Mech., 564, 333-360, 2006 62 62 BIBLIOGRAPHY BIBLIOGRAPHY [9] R Borcia, I.D Borcia and M Bestehorn, Static and dynamic contact angles a phase field modeling, Eur Phys, J Special Topics, 166, 127-131, 2009 [10] R Defay and I Priogine, Surface tension and adsorption, Wiley, New York, 1966 [11] D I Collias and R K Prudhomme, Diagnostic techniques of mixing effectiveness: the effect of shear and elongation in drop production in mixing tanks, Chem Eng Sci 47, 1401C1410 (1992) [12] P Somasundaran and R Ramachandran, Surfactants in flotation, in Surfactants in Chemical/Process Engineering, (Marcel Dekker, New York, 1988), Vol 28, pp 195C235 [13] A Branger and D Eckmann,Accelerated arteriolar gas embolism reabsorption by an exogenous surfactant, Anesthesiology 96, 971C979 (2002) [14] T Young, ”An essay on the cohesion of fluids,” Philos Trans R Soc London 95, 65(1805) [15] R Borcia, I Borcia, and M Bestehorn, Drops on an arbitrarily wetting substrate: A phase field description, Phys Rev E., 78, 066307(2008) [16] W Ren, D Hu, and W E, Continuum models for the contact line problem, Phys Fluids 22, 102103 (2010) [17] W Ren, and W E, Boundary conditions for the moving contact line problem, Phys Fluids, 19, 022101 (2007) [18] M Gao, and X Wang, An efficient scheme for a phase field model for the moving contact line problem with variable density and viscosity, Journal of Computational Physics 272(2014)704-718 [19] T Qian, X Wang, and P.Sheng, Molecular hydrodynamics of the moving contact line in two-phase immiscible flows, Commun.Comput.Phys.1(2006)1C52 63 BIBLIOGRAPHY BIBLIOGRAPHY [20] J Xu and W Ren, A level-set method for two-phase flows with moving contact line and insoluble surfactant, J Comput Phys 263, 71C90 (2014) [21] R Borcia and M Bestehorn, Phase field models for Marangoni convection in planar layers, Journal of optoelectronics and advnaced materials, Vol 8, No 3, June 2006, p 1037-1039 [22] R.J Braun and B.T Murray, Adaptive phase-field computations of dendritic crystal growth, Journal of Crystal Growth 174 (1), 41-53(1997) [23] N O Young, J S Goldstein, and M J Block, The motion of bubbles in a vertical temperature gradient, Journal of Fluid Mechanics 350(1959) [24] R Borcia, and M Bestehorn, Phase-field simulations for drops and bubbles, Phys Rev E., 75, 056309(2007) [25] J Glimm, J Grove, X L Li, K.-M.Shyue, Y.Zeng, Q.Zhang, Three dimensional front tracing, SIAM J Sci Comput 19 703C727(1998) [26] C Peskin, The immersed boundary methods, Acta Numer.11 479C517(2002) [27] D.M Anderson, G.B McFadden, A A Wheeler, Diffusive-interface methods in fluid dynamics, Annu Rev Fluid Mech.30 139C165(1998) [28] C Pozrikidis, Interfacial dynamics for Stokes flow, J Comput Phys.169(2001)250C301 [29] T D Blake and J M Haynes, Kinetics of liquid/liquid displacement, J Colloid Interface Sci 30, 421(1969) [30] Z Li, K Ito, The Immersed Interface MethodNumerical Solutions of PDEs Involving Interfaces and Irregular Domains, Frontiers Appl.Math., SIAM, 2006 [31] C.W Hirt, B.D Nichols, Volume of fluid(VOF) method for the dynamics of free boundaries, J Comput Phys.39201C225(1981) 64 BIBLIOGRAPHY BIBLIOGRAPHY [32] R Mittal, G Iaccarino, Immersed boundary methods, Annu Rev Fluid Mech.37(2005)239C361 [33] M J de Ruijter, M Charlot, M Voue, and J De Coninck, Experimental evidence of several time scales in drop spreading, Langmuir 16, 2363(2000) [34] T Hou, J Lowengrub, M Shelley, Boundary integral methods for multi-component fluids and multi-phase materials, J Comput Phys 169(2001)302C362 [35] G Tryggvason, B Bunner, A Esmaeeli, D Juric, N Al-Rawahi, W Tauber, J Han, S Nas, Y.-J.Jan, Afront-tracking method for the computations of multiphase flow, J Comput Phys.169 708C759(2001) [36] D Enright, R Fedkiw, J Ferziger, and I Mitchell, A hybrid particle level set method for improved interface capturing, Journal of Computational physics 183 (1), 83-116(2002) [37] R LeVeque, Z Li, Immersed interface methods for Stokes flow with elastic boundaries or surface tension, SIAM J Sci Comput 18(1997)709 [38] E.B Dussan V., On the Spreading of Liquids on Solid Surfaces: Static and Dynamic Contact Lines, Annual Review of Fluid Mechanics, 11, 371(1979) [39] R Borcia and M Bestehern, Phase-field simulation for evaporation with convection in liquid-vapor systems, Eur Phys J B 44, 101-108(2005) [40] J W Cahn and J E Hilliard, Free Energy of a Nonuniform System I Interfacial Free Energy, J Chem Phys 28, 258(1958) [41] L M Pismen and Y Pomeau, Disjoining potential and spreading of thin liquid layers in the diffuse-interface model coupled to hydrodynamics,Phys Rev E 62, 2480 (2000) 65 BIBLIOGRAPHY BIBLIOGRAPHY [42] S Marella, S Krishnan, H Liu, HS Udaykumar, Sharp interface Cartesian grid method I: an easily implemented technique for 3D moving boundary computations, Journal of Computational Physics 210 (1), 1-31 [43] R Borcia, and M Bestehorn, Phase-field model for Marangoni convection in liquid-gas systems with a deformable interface, Phys Rev E., 67, 066307(2003) [44] Y Yang, J.W Garvin, H.S Udaykumar, Sharp interface simulation of interaction of a growing dendrite with a stationary solid particle, International journal of heat and mass transfer 48 (25), 5270-5283 [45] A Karma, D.A Kessler, and H Levine, Phase-field model of mode III dynamic fracture, Phys Rev Lett 87, 045501(2001) [46] A Kapahi, J Mousel, S Sambasivan, and H.S UdaykumarParallel, sharp interface Eulerian approach to high-speed multi-material flows, Computers & Fluids 83, 144-156 [47] W.E., W.Q Ren, and E Vanden-Eijnden, String method for the study of rare events, Physical Review B 66, 052301(2002) [48] W Ren and E Vanden-Eijnden, Simplified and improved string method for computing the minimum energy paths in barrier-crossing events, The Journal of Chemical Physics, 126, 164103(2007) [49] C Zhang and Z Wang, Nucleation of membrane adhesions, Physics Review E 77, 021906(2008) [50] W Ren and E Vanden-Eijnden, A climbing string method for saddle point search, Journal of Chemical Physics 138, 134105 (2013) [51] H Jonsson, G Mills, and K W Jacobsen, Classical and Quantum Dynamics in Condensed Phase Simulations, edited by B J Berne, G Ciccoti, and D F Coker, World Scientific, Singapore(1998) 66 BIBLIOGRAPHY BIBLIOGRAPHY [52] J.J Xu, Y Yang, J Lowengrub, A level-set continuum method for two-phase flows with insoluble surfactant, J Comput Phys.231 5897C5909(2012) [53] Z Zhang, S Xu and W Ren, Derivation of a continuum model and the energy law for moving contact lines with insoluble surfactants, Physics of Fluids 26, 062103(2014) [54] A Ulitsky and R Elber, A new technique to calculate steepest descent paths in flexiable polyatomic systems, J Chem Phys 96, 1510(1990) [55] R Olender and R Elber, Calculation of classical trajectoies with a very large time step: Formalism and numerical examples, J Chem Phys 105, 9299(1996) [56] G Henkelman and H Jonsson, A dimer method for finding saddle points on high dimensional potential surface using only the first derivative, J Chem Phys 111, 7010(1999) [57] G Henkelman, B P Uberuaga, and H Jonsson, Methods for finding saddle points and minimum energy pahts, J Chem Phys 113, 9901(2000) 67 [...]... moving contact line using molecular dynamics and continuum mechanics [17] Besides establishing physical models for moving contact line problems, a number of numerical methods have been put forward to verify the accuracy of these models An efficient numerical scheme for the two phase moving contact line problem with variable density, viscosity, and slip length is developed by Gao and Wang [18] The numerical. .. line and static contact angle While slip boundary conditions induce the definition of the moving contact line and moving contact angle Contact line problems have wide applications, e.g., industrial emulsification, liquid/liquid extraction and hydrodesulfurization of crude oil, polymer blending and plastic production, cleaning using detergent, micro fluids, etc [10–13] Meanwhile, contact line problems are... fewer iterative steps are needed for the evolution of final steady state using implicit difference methods when the same stopping criterion is used 9 Chapter 2 Numerical Methods 2.2 2.2 The dimensionless equations of the phase field model The dimensionless equations of the phase field model As discussed in chapter 1, the fundamental equations of the phase field model in liquid-vapor system are as follows, ∂ρ... applied to deal with thermodynamically multi -phase systems with complicate and time-dependent interfaces By introducing the phase field, the multi -phase systems can be treated continuously from one medium to another The theory of the phase field model has been carefully 1 1 Chapter 1 Introduction 1.1 The phase field model investigated in [3] and [4] The phase field model has been applied to a liquid-vapor... two phases around the interfacial region, the interface in the level set method is non-diffuse [43] 1.2 The contact line and contact angle When two immiscible fluids are placed on a substrate or one kind of liquid with its own vapor is put in a container, the line where the interface of the two fluid phases or the interface of liquid and its own vapor intersects the substrate is named as the contact line. .. line The contact angle is the tangent angle along the contact line intersecting the substrate Figure 1.1 presents an example of droplet-vapor system and 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 of two fluids or the liquid with its own vapor is zero, we refer to the static contact line and... depending on the phase field and a diffusive field (variational formulations) Equations of the model are then obtained by using general relations of Statistical Physics Such a functional is constructed from physical considerations, but contains a parameter or combination of parameters related to the interface width Parameters of the model are then chosen by studying the limit of the model with this width... scaling regimes [5] Therefore, modeling and simulation of contact line problems have attracted a lot of interests and much effort has been made to research on address these difficulties The equilibrium configuration of the static contact line was researched 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... vapor kinematic viscosity, respectively, and ρl , ρv are liquid density and vapor density, respectively [39] 10 Chapter 2 Numerical Methods 2.3 2.3 Numerical methods for the phase field model Numerical methods for the phase field model Before presenting a detailed description of numerical methods, we should introduce mesh into the domain we are interested in The computational domain is a cube in three... and the solid substrate is obtained by Borcia [15] The sharp interface model and diffuse interface model for the moving contact line problem based on thermodynamic principles are derived by Ren [5] Continuum models in cases of partial wetting and complete wetting is derived for the moving contact line problem through a combination of macroscopic and microscopic considerations by Ren, Hu and E [16] Ren .. .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... refer to the static contact line and static contact angle While slip boundary conditions induce the definition of the moving contact line and moving contact angle Contact line problems have wide... respectively [39] 10 Chapter Numerical Methods 2.3 2.3 Numerical methods for the phase field model Numerical methods for the phase field model Before presenting a detailed description of numerical methods,