Computational Study of a Droplet Migration on a Horizontal Solid Surface with Temperature Gradients

Computational Study of a Droplet Migration on a Horizontal Solid Surface with Temperature Gradients 國 立 中 央 大 學 NATIONAL CENTRAL UNIVERSITY 液滴於具溫度梯度的水平固體表面上遷移行為 之數值研究 Computational Study of a Droplet Migration on a Horizontal Solid Surface with Temperature Gradients HuyBich Nguyen December, 2010 COMPUTATIONAL STUDY OF A DROPLET MIGRATION ON A HORIZONTAL SOLID SURFACE WITH TEMPERATURE GRADIENTS By HUY BICH NGUYEN A dissertation presented to the Department of Mechanical Engineering of the National Central University in partial fulfillment of the requirements for the degree of Doctor of Philosophy Dissertation Advisor: Chair Prof. JYH CHEN CHEN Department of Mechanical Engineering NATIONAL CENTRAL UNIVERSITY December, 2010 國 立 中 央 大 學 機械工程學系博士班 博 士 論 文 液滴於具溫度梯度的水平固體表面上遷移行為 之數值研究 Computational Study of a Droplet Migration on a Horizontal Solid Surface with Temperature Gradients 研 究 生：阮揮碧 指導教授：陳志臣 教授 中 華 民 國 九十九 年 十二 月 Dedicated to the love of my life, Le Thi Tuyet Huyen, and my loving daughters Nguyen Le Thu Truc and Nguyen Le Quynh Truc. NATIONAL CENTRAL UNIVERSITY DEPARTMENT OF MECHANICAL ENGINEERING We hereby approve the dissertation of Huy Bich Nguyen ______________________________________________________ for the degree of Doctor of Philosophy (Signed) Prof. TsingFa Lin Prof. JingTang Yang (National ChiaoTung University) (National Taiwan University) ______________________ ______________________ (Chair of the committee) Prof. JyhChen Chen Prof. JiinYuh Jang (National Central University) (National ChengKung University) ___________________ ______________________ Prof. LihWu Hourng Prof. ShuSan Hsiau (National Central University) (National Central University) _____________________ ______________________ Assoc. Prof. MingTsung Hung Prof. JerLiang Yeh (National Central University) (National TsingHua University) ______________________ ______________________ Approved on the Twenty first day of December, Two thousand and Ten Abstract of Dissertation Presented to the Department of Mechanical Engineering of the National Central University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy COMPUTATIONAL STUDY OF A DROPLET MIGRATION ON A HORIZONTAL SOLID SURFACE WITH TEMPERATURE GRADIENTS By HUY BICH NGUYEN Recently, the migration of a liquid droplet on a horizontal solid surface has attracted widespread attention from many researchers and engineers because of its promising prospective in a variety of applications in biology, chemistry, and industry. In this dissertation, a proper computational model is developed for investigating the transient migration of a liquid droplet on a horizontal solid surface subjected to uniform temperature gradients. Numerical calculations are carried out by solving the Navier Stokes equations coupled with the energy equation through the finite element method (FEM). The conservative level set method, the arbitrary Lagrangian Eulerian (ALE) method, and the continuum surface force (CSF) method are employed to treat the movement and deformation of the dropletair interface and the surface tension force during the motion process. Some physical properties of fluids dependent on temperature are also considered. The study indicates that when a liquid droplet is of small size, two asymmetric thermocapillary vortices are generated inside the droplet. The thermocapillary vortex on the hot side is always larger in size than that appearing on the cold one. The net momentum of the thermocapillary convection inside droplet pushes the droplet moves from the larger vortex (hot side) to the smaller one (cold side). The variation of the size of the thermocapillary ii vortex during the movement causes the speed of the droplet to initially increase and then decrease slowly until approaching a constant value. A higher imposed temperature gradient leads the droplet velocity to reach the maximal value earlier and have a higher final speed. If the static contact angle of the droplet is less (or higher) than 90 degrees, the droplet speed is lower (or higher) since the net thermocapillary momentum in the horizontal direction is diminished (or enhanced) by the presence of capillary force. The lower slip length leads to the smaller droplet speed. In addition, the quasisteady migration speed of a small droplet is linearly proportional to its size due to the stronger net thermocapillary momentum. The effect of gravity is insignificant and the thermocapillary convection is dominated. The computational model is verified by comparing to the previous experimental results. When the droplet turns into larger, the influence of gravity becomes important. The combined thermocapillary and buoyancy force driven convection produce complex dynamic behavior of fluid motion inside the droplet. In the middle size regime, the quasisteady migration speed of the droplet reaches a maximum, but this is gradually reduced as the droplet size increases due to the suppression of the net thermocapillary momentum by the buoyancy force. In the large droplet size regime, two pairs of convection vortices exist inside the droplet as a result of the appearance of the buoyancydriven convection accompanying the thermocapillary convection. The quasisteady migration speed quickly diminishes, mainly due to the reduction of the net thermocapillary momentum from the stronger buoyancy convection. The droplet speed tendency is found to be a good agreement with the experimental results. iii 摘要 由於液滴在水平之固體表面的移動之不同應用，具備在工業、化學與生物學上的前景，最近正吸引許多學者與工程師的注意。本研究為使用數值模擬，探討固體表面溫度梯度所形成液滴與空氣界面表面張力梯度，而促成液滴移動之物 理機制。運用有限元素法、等位函數法 ( Level Set Method )、 ALE 運 動 描 述 法（arbitrary Lagrangian Eulerian method, ALE method ） 與 連 續 表 面 力 學 法（continuum surface force method, CFS method），求解 NavierStokes 方 程 式 與能量方程式。 本研究指出當液滴為小尺寸時，液滴內會產生兩個不對稱的熱毛細力(thermocapillary force） 渦 漩。 在 液 滴 熱 邊 的 熱 毛 細 力 渦 漩 尺 寸 總 是 大 於 在 冷邊者 。液滴內 熱毛細力 渦漩之淨 動量驅動 液滴由熱 邊（大 渦 漩） 向冷邊（小渦漩）移動。移動過程中熱毛細力渦漩尺寸的變化，造成液滴的速度於初期增加而後緩慢降低，直至接近一個近穩定速度之定值。增加溫度梯度，將導致液滴較早達到近穩定速度，及較高近穩定速度。在水平方向的熱毛細力與毛細力相互作用下，當液滴之靜接觸角小於（或大於）90度時，熱毛細力與毛細力具有抵制（或相乘）效果，使液滴的近穩定速度降低（或增加）。較短的滑動長度導致較低的液滴移動速度。而淨熱毛細力使液滴之近穩定移動速度與其尺寸呈線性關係。重力的效果不明顯而熱毛細力對流才是主導移動的主因。本研究模擬結果可驗證其他學者已發表之實驗結果。 iv 當液滴尺寸變大，重力的影響愈重要。液滴內熱毛細力與浮力生成之對流，產生複雜的流體動力行為。對於中尺寸的液滴，它的近穩定移動速度達到最大，但隨液滴尺寸增大，浮力抑制熱毛細力淨動量，使近穩定移動速逐漸降低。對於大尺寸液滴，液滴內存在兩對對流渦漩，這是浮力對流與熱毛細力對流所產生的結果。由於較強浮力對流使淨熱毛細力動量降低，液滴的近穩定移動速度很快減低。上述模擬之液滴速度趨勢與其他學者已發表實驗結果有很好的一致性。 v ACKNOWLEDGEMENTS My sincerely appreciation and deeply respect go to my advisor, Chair Professor JyhChen Chen, Dean of College of Engineering of the National Central University (NCU), for his unlimited support with continuous and patient advice during my studies. Without his advice, my research could not to be achieved. I would like to thank the members of my dissertation committee: Prof. LihWu Hourng (NCU), Prof. ShuSan Hsiau (NCU), Assoc. Prof. MingTsung Hung (NCU), Prof. JiinYuh Jang (National ChengKung University), Prof. TsingFa Lin (National ChiaoTung University), Prof. JingTang Yang (National Taiwan University), and Prof. JerLiang Yeh (National TsingHua University), for their support. I thank Prof. Bruce A. Finlayson (University of Washington, USA) for suggestion using the ALE method, and Prof. William BJ Zimmerman (University of Sheffield, UK) and Assoc. Prof. NamTrung Nguyen (Nanyang Technological University, Singapore) for discussions about contact line and others. I express my sincere acknowledgements to the National Science Council (NSC) of the Taiwan government for granting me a threeyear doctoral fellowship, to the National Central University for awarding me the highest NCU’s PhD scholarship, and to Vietnam Ministry of Education and Training and my workplace, Nong Lam University, Hochiminh city, Vietnam, for approving me to study in Taiwan. It is my pleasure to thank people in my LHPG Lab who are formerly Ph.D. candidates: Dr. ChungWei Lu, Dr. ChengWei Chien, Dr. HsuehI Chen, Dr. ChangHung Chiang, Dr. GowJiun Sheu, Dr. ChengLin Chung, and Dr. F. S. Hwu; recently Ph.D. candidates: HungLin Hsieh, ChunHung Chen, YingYang Teng, Heng Chiou, and MingTe Lin; and several Master students, and specially my Prof.’s assistant Ms. Jenyin Tien for helping me one way vi or another during my studies at the National Central University, Taiwan. I also thank the individuals in the Mechanical Department, the Office of International Affairs, and my friends for supporting me in NCU. Finally, I dedicate this dissertation to my lovely wife, TuyetHuyen Thi Le, to my two nice daughters, Thu Truc Le Nguyen (Listar Nguyen) and Quynh Truc Le Nguyen (Wendy Nguyen), and to my relatives for their sympathy, sharing my difficulties, continuous support, and warm encouragement over the years of my studies in Taiwan. vii TABLE OF CONTENTS ABSTRACT ...............................................................................................................................i 摘要 ........................................................................................................................................ iii ACKNOWLEDGEMENTS ..................................................................................................... v TABLE OF CONTENTS ...................................................................................................... vii LIST OF FIGURES ................................................................................................................xi LIST OF TABLES ............................................................................................................. xviii NOMENCLATURE ..............................................................................................................xix Chapter I Introduction .......................................................................................................... 1 1.1 Motivation ...................................................................................................................... 1 1.2 Objectives ....................................................................................................................... 3 1.3 Dissertation structure ...................................................................................................... 4 Chapter II Background ......................................................................................................... 6 2.1 Driving force .................................................................................................................. 6 2.2 Thermocapillary convection ........................................................................................... 9 2.3 Buoyancy convection ................................................................................................... 10 2.4 Contact line and contact angle ...................................................................................... 12 2.5 Contact angle hysteresis ............................................................................................... 13 2.6 Slippage and contact line motion ................................................................................. 14 viii 2.7 Capillary length ............................................................................................................ 17 Chapter III Literature review ............................................................................................. 19 3.1 Theoretical works ......................................................................................................... 19 3.2 Experimental works ...................................................................................................... 22 3.3 Numerical works .......................................................................................................... 27 Chapter IV Theoretical Formulations and Computational Methods ............................. 29 4.1 Validity of continuum model ....................................................................................... 29 4.2 Physical Model ............................................................................................................. 30 4.3 Theoretical formulations .............................................................................................. 30 4.3.1 Governing equations .............................................................................................. 30 4.3.2 Boundary conditions .............................................................................................. 33 4.3.3 Initial conditions .................................................................................................... 34 4.4 Computational methods ................................................................................................ 35 4.4.1 Conservative level set method ............................................................................... 37 4.4.2 Arbitrary Lagrangian Eulerian (ALE) method ...................................................... 39 4.5 Computational Solving Process .................................................................................... 39 4.6 Convergence test .......................................................................................................... 41 4.6.1 Meshing ................................................................................................................. 41 4.6.2 Numerical Errors.................................................................................................... 41 4.6.3 Wall boundary test ................................................................................................. 42 ix Chapter V A Droplet Migration Induced by Thermocapillary Convection ................... 48 5.1 Droplet thermocapillary migration mechanism ............................................................ 52 5.2 Effect of temperature gradient ...................................................................................... 52 5.3 Effect of contact angle .................................................................................................. 58 5.4 Effect of initial condition.............................................................................................. 64 5.5 Effect of slip behavior .................................................................................................. 71 5.6 Validation of model ...................................................................................................... 75 5.7 Summary ...................................................................................................................... 78 Chapter VI A Droplet Motion Induced by Combined Thermocapillary – Buoyancy Convection .............................................................................................................................. 80 61 The deformation of the droplet .................................................................................... 80 62 The droplet migration in different sizes ....................................................................... 81 6.2.1 The small sized droplet regime .............................................................................. 82 6.2.2 The middle sized droplet regime ........................................................................... 82 6.2.3 The large sized droplet regime............................................................................... 89 6.3 Summary ...................................................................................................................... 89 Chapter VII Concluding remarks and future works ........................................................ 96 7.1 Conclusions .................................................................................................................. 96 7.2 Main contributions ....................................................................................................... 98 7.3 Recommendations and future works ............................................................................ 99 x Bibliographies ....................................................................................................................... 100 Appendix ............................................................................................................................... 109 A. Calculating height profiles of a droplet ................................................................... 109 B. Publications during Ph.D. studies ............................................................................ 112 xi LIST OF FIGURES Figure 2 1. A static droplet on a hydrophilic (partially wetting) solid surface (. ... 12 Figure 2 2. A static droplet on a hydrophobic (nonwetting) solid surface (. .......... 13 Figure 2 3. Definition of the slip length. a) No slip: b = 0; b) Partial slip: the linear velocity ux profile in the flowing which is characterized by a viscosity and a constant shear stress does not vanish at the liquidsolid interface. Extrapolation into the solid at a depth b is needed to obtain a vanishing velocity as supposed by the noslip boundary condition; and c) Perfect slip: b = ∞. ................................................................................................................... 17 Figure 3 1. Droplet velocity versus footprint radius with (a) experiment and (b) and (c) analytical results . ..................................................................................................................... 21 Figure 3 2. Velocity V of a droplet of PDMS versus its initial radius R, for various thermal gradientsT . The thick solid straight lines fitting the points converge at a point A (R = 0, V0). They intersect the V = 0 axis at R = Rc. For R > 6 mm, a saturation of the velocity is observed versus viscosity. ........................................................................................................ 24 Figure 3 3. Displacement histories of the silicone oil droplets of (a) 0.1 l and (b) 1l 24. .................................................................................................................................................. 25 Figure 3 4. (a) (d) Droplet of silicone oil (PDMS) moves along a heater array. A thin residue trailed the wetting droplet. The images sequence represents times (a) t = 0s, (b) 44 s, (c) 88 s, and (d) 132 s . ............................................................................................................. 26 xii Figure 3 5. A time sequence of side view images of the moving droplet with two different contact angles: a) smaller than 900 and b) larger than 90°. The dashed line indicates the position of the laser sheet . ....................................................................................................... 27 Figure 4 1. Schematic representation used for computation. The value of the level set function Φ is equal to 0.5 at the airdroplet interface. The air phase (subdomain Ω1) and the droplet phase (subdomain Ω2) are represented by 1 ≥ Φ > 0.5 and 0.5 > Φ ≥ 0, respectively.31 Figure 4 2. Flow chart of computational solving process ................................................... 40 Figure 4 3. Typical mesh used in the computational domain (W = 10 mm and H = 1.5 mm) for a droplet with static contact angle φ = 900, footprint radius L = 0.55 mm. ............... 42 Figure 4 4. Typical mesh used in the computational domain (W = 35 mm and H = 10 mm) for a droplet with static contact angle φ = 600, footprint radius L = 4 mm, and maximum height hm = 2.31 mm. ............................................................................................................... 42 Figure 4 5. Flow fields for the domain with W = 1.5 mm: a) in whole domain; b) inside droplet. ..................................................................................................................................... 44 Figure 4 6. Flow fields for the domain with W = 3 mm: a) whole domain; b) inside droplet. .................................................................................................................................................. 45 Figure 4 7. Flow field for domain with W = 4 mm: a) whole domain; b) inside droplet. .. 46 Figure 4 8. Velocity of droplet for different domain sizes ................................................. 47 xiii Figure 5 1. Isotherms for a droplet with static contact angle φ = 900, footprint radius L = 0.55 mm and temperature gradient G = 40 Kmm between case of a) Constant viscosity and b) Viscosity dependent temperature. ........................................................................................ 50 Figure 5 2. Flowfield inside droplet with static contact angle φ = 900, footprint radius L = 0.55 mm and temperature gradient G = 40 Kmm between case of a) Constant viscosity and b) Viscosity dependent temperature. ........................................................................................ 51 Figure 5 3. Migration velocity of droplet with static contact angle φ = 900, footprint radius L = 0.55 mm and temperature gradient G = 40 Kmm between case of a) Constant viscosity and b) Viscosity dependent temperature. ................................................................................. 51 Figure 5 4. Displacement of the silicone oil droplet with φ = 900 and L = 0.55 mm for the applied temperature gradient G = 40 Kmm and b = 1 nm. ..................................................... 53 Figure 5 5. Velocity of the silicone oil droplet with φ = 900 and L = 0.55 mm for the applied temperature gradient G = 40 Kmm and b = 1 nm. ..................................................... 53 Figure 5 6. Isotherms inside and outside the silicone oil droplet with φ = 900 and L = 0.55 mm from t = 0.06 to 2.91 seconds for G = 40 Kmm (Ma = 587.3) and b = 1 nm. The red curve shows the dropletair interface. Isothermal contour lines are plotted every 7 K. .......... 55 Figure 5 7. Streamlines for the silicone oil droplet with φ = 900 and L = 0.55 mm for G = 40 Kmm (Ma = 587.3) and b = 1 nm at 0.01 sec. ................................................................... 56 Figure 5 8. Temperature distribution along the silicone oil dropletair interface at different times: a) From 0.01 sec to 1.11 sec; and b) from 2.61 sec to 2.91 sec for G = 40 Kmm (Ma = 587.3) and b = 1 nmθ is defined as the angle from the droplet center to the interface; θ = 00 xiv corresponds to the rear contact point; and θ = 1800 indicates the relation to the front contact point. ........................................................................................................................................ 57 Figure 5 9. Flowfield inside the silicone oil droplet with φ = 900 and L = 0.55 mm from t = 0.06 to 2.91 seconds for G = 40 Kmm (Ma = 587.3) and b = 1 nm. The red curve shows the dropletair interface. ........................................................................................................... 58 Figure 5 10. Displacement of the silicone oil droplet with φ = 900 and L = 0.55 mm for b = 1 nm and different applied temperature gradient G = 10, 20.5, and 40 Kmm corresponding to Ma = 146.8, 301, and 587.3, respectively. ........................................................................... 60 Figure 5 11. Velcoity of the silicone oil droplet with φ = 900 and L = 0.55 mm for b = 1 nm and different applied temperature gradient G = 10, 20.5, and 40 Kmm corresponding to Ma = 146.8, 301, and 587.3, respectively against time. ........................................................... 61 Figure 5 12. Streamline of the silicone oil droplet with φ = 900 and L = 0.55 mm for b = 1 nm and different applied temperature gradient G = 10, 20.5, and 40 Kmm corresponding to Ma = 146.8, 301, and 587.3, respectively. ............................................................................... 62 Figure 5 13. Isotherms for the silicone oil droplet with φ = 900 and L = 0.55 mm at different temperature gradients G = 10, 20.5 and 40 Kmm coressponding to Ma = 146.8, 301 and 587.3, respectively, and b = 1 nm at a) t = 0.06sec; b) t = 0.51sec; and c) t = 1.11 sec. Isothermal contours lines are plotted every 7 K....................................................................... 63 Figure 5 14. Flowfield inside the silicone oil droplet with φ = 900 and L = 0.55 mm at different temperature gradients G = 10, 20.5 and 40 Kmm coressponding to Ma = 146.8, 301 and 587.3, respectively, and b = 1 nm at a) t = 0.06sec; b) t = 0.51sec; and c) t = 1.11 sec. ... 64 xv Figure 5 15. Dynamic contact angles of the silicone oil droplet at G = 10 Kmm and b = 1 nm versus time with a) φ = 700 and 900, and L = 0.648 mm and 0.55mm, respectively; b) φ = 1000 and L = 0.536 mm. ........................................................................................................... 66 Figure 5 16. Contact angle hysteresis of the silicone oil droplet with φ = 700, 900, and 1000 and L = 0.648 mm, 0.55mm and 0.536 mm, respectively, at G = 10 Kmm and b = 1 nm versus time. .............................................................................................................................. 67 Figure 5 17. a) Isotherms and b) flow field inside the silicone oil droplet with φ = 700, 900, and 1000 and L = 0.648 mm, 0.55mm and 0.536 mm, respectively, for G = 10 Kmm and b = 1 nm at t = 0.51 sec. ................................................................................................................. 69 Figure 5 18. Velocity of the silicone oil droplet with φ = 700, 900, and 1000 and L = 0.648 mm, 0.55mm and 0.536 mm, respectively, at G = 10 Kmm and b = 1 nm versus time. ........ 70 Figure 5 19. a) Isotherms and b) flow field inside the silicone oil droplet with φ = 900 and L = 0.55 mm for G = 10 Kmm and b = 1 nm for different initial conditions at different times. Isothermal contours lines are plotted every 7 K....................................................................... 70 Figure 5 20. Velocity of the silicone oil droplet with φ = 900 and L = 0.55 mm at G = 10 Kmm and b = 1 nm versus time for different initial conditions. ............................................. 71 Figure 5 21. Velocity of the squalane droplet with L = 2.6 mm, SCA = 410, and G = 3.12 Kmm for different slip lengths. ............................................................................................... 72 Figure 5 22. Flow field inside the squalane droplet with a footprint radius L = 2.6 mm, static contact angle SCA = 410, and temperature gradient G = 3.12 Kmm for different slip lengths. The red arrows denote the stagnation point at the dropletair interface. .................... 73 xvi Figure 5 23. CAH of the squalane droplet with L = 2.6 mm, SCA = 410, and G = 3.12 Kmm for different slip lengths. ............................................................................................... 74 Figure 5 24. The speed of the squalane droplet with L = 2.6 mm, SCA = 410, and G = 3.12 Kmm for different slip lengths in comparison between analytical and numerical method. ... 75 Figure 5 25. Velocity of the squalane droplet with φ = 410 and L = 2.6 mm at G = 3.6 Kmm and b = 3 nm versus time. ............................................................................................. 76 Figure 5 26. Isotherms inside and outside the squalane droplet with φ = 410 and L = 2.6 mm at G = 3.6 Kmm and b = 3 nm for different times. Isothermal contours lines are plotted every 6 K. ................................................................................................................................. 77 Figure 5 27. a) The dynamic contact angles and b) the contact angle hysteresis against time for the squalane droplet with φ = 410 and L = 2.6 mm at G = 3.6 Kmm and b = 3 nm. . 78 Figure 6 1. Deformation of droplet versus droplet size. The black and red curves designate the initial and deformed dropletair interface, respectively. .................................................... 82 Figure 6 2. Quasisteady migration velocity of droplet versus droplet size with G = 3.6 Kmm and b = 1nm. ................................................................................................................. 83 Figure 6 3. Droplet velocity for different small sized droplets versus time with G = 3.6 Kmm and b = 1nm. ................................................................................................................. 83 xvii Figure 6 4. (a) Flow field and (b) isotherms inside droplet for different sizes L = 1.4, 1.8, and 2 mm with G = 3.6 Kmm and b = 1nm. The solid arrows denote the moving direction. Isothermal contour lines are plotted every 5 K. ....................................................................... 85 Figure 6 5. Different advancing and receding dynamic contact angles for different droplet sizes with G = 3.6 Kmm and b = 1nm. ................................................................................... 87 Figure 6 6. (a) Flow field and (b) isotherms inside droplet for different size L = 2.2, 3 and 3.4 mm with G = 3.6 Kmm and b = 1nm. The solid arrows denote the moving direction. Isothermal contour lines are plotted every 5 K. ....................................................................... 88 Figure 6 7. (a) Flow field and (b) isotherms inside droplet with L = 4 mm, G = 3.6 Kmm, and b = 1nm at different moving times. The solid arrow denotes the moving direction. Isothermal contour lines are plotted every 5 K. ....................................................................... 91 Figure 6 8. Flow field inside droplet with L = 5 mm, G = 3.6 Kmm, and b = 1nm from t = 0.5 to 10 seconds. The solid arrow denotes the moving direction. ......................................... 92 Figure 6 9. Isotherms inside droplet with L = 5 mm, G = 3.6 Kmm, and b = 1nm from t = 0.5 to 10 seconds. The solid arrow denotes the moving direction. Isothermal contour lines are plotted every 5 K. ............................................................................................................... 93 Figure 6 10. (a) Flow field and (b) isotherms inside droplet with L = 5.5 mm, G = 3.6 Kmm, and b = 1nm at different moving times. The solid arrow denotes the moving direction.................................................................................................................................... 94 Figure A 1. A simple model used to count the height profiles of a spherical droplet. 110 xviii LIST OF TABLES Table 1. Forces and external fields used to manipulate motion of fluids. It is also possible to use external means to manipulate particles embedded in flows. ............................................... 6 Table 2. Physical properties of the fluids (at 250C)................................................................ 49 xix NOMENCLATURE English symbols Aab Absolute tolerance Art Relative tolerance b Slip length CLS Liquidsolid coefficient of the LennardJones potential Cp Heat capacity Dcm Collective molecular diffusion coefficient dmol Molecular diameter E Error estimate of the solver in solution S Efr Free energy FGR Gravity FSV Surface tension force G Temperature gradient g Gravitational acceleration g Gravitational acceleration constant H Height of model geometry hm Maximum height of droplet h(x) Height profiles of droplet i Unit vector in xdirection j Unit vector in zdirection k Thermal conductivity L Footprint radius of droplet xx L Mean free path length M Molecular weight Mm Molecular scale N Number of degrees of freedom n Normal vector unit NA Avogadro number NE Number of the elements p Pressure R Length scale S Solution vector sec Second Sst Structure factor for first molecular layer S(x) Dropletair interface T Temperature t Time Teq Equilibrium temperature U Contact line speed u Velocity component in xdirection Uref Reference velocity uTangential velocity v Velocity component in zdirection V Speed of droplet V Velocity vector W Width of model geometry xxi x Horizontal direction in spatial coordinate X Horizontal direction in reference coordinate x mesh Mesh velocity in xdirection Y Dependent variable z Vertical direction in spatial coordinate Z Vertical direction in reference coordinates z mesh Mesh velocity in zdirection Greek symbols    Thermal diffusivity    Thermal expansion coefficient of liquid    Laplace length or capillary length    Dirac delta function  Difference  Parameter controlling thickness of interface    Level set function T Surfacetension temperature coefficient    Droplet size    Static contact angle  Droplet volume  Mean curvature  Reinitialization parameter  Dynamic viscosity xxii  Kinematic viscosity    Order θ Angle represents for position at the dropletair interface  Density of fluid  Surface tension    Tangential vector unit s Integration of shear stress on the liquidsolid wall interface Ω Domain    Representative physical properties of fluids Operators  Gradient  Divergence Subscripts a Air A Advancing c Critical C Cold side d Degrees of freedom F Front side H Hot side i Phase xxiii l Liquid LG Liquidgas ref Reference state R Receding s Solid surface SG Solid gas SL Solid liquid VP Vapor phase Nondimensional Numbers BoD Dynamic Bond number T 2 m D g h Ma Ra Bo      Ca Capillary number    ref U Ca Kn Knudsen number Kn = L R Ma Marangoni number     2 T GR Ma Pe Pelect number   ref RU Pe Pr Prandtl number Pr C k p     Ra Rayleight number    GR g Ra 4 Re Reynolds number Pr U R Ma Re ref     xxiv Abbreviations ALE Arbitrary Lagrangian Eulerian CA Contact angle CAH Contact angle hysteresis CSF Continuum surface force DCA Dynamic contact angle DOF Degrees of freedom FEM Finite element method LOC Lap on Chip MEMS MicroElectroMechanical System μTAS MicroTotalAnalysis System SCA Static contact angle VOF Volume of fluid method 1 Chapter I Introduction 1.1 Motivation From the last two decades until now, the migration of a liquid droplet on a horizontal solid surface has attracted widespread attention from many researchers and engineers due to its promising prospective in various applications in biology, industry, and chemistry 113. It can potentially be applied to the process of material removing or recovering in a microgravity environment space, the condensation heat transfer, the Labonachip (LOC), the fields of MicroElectroMechanical System (MEMS), the field of MicroTotalAnalysis System (μTAS), and the vision of developing entire biochemical laboratories on the surface of silicon or polymer chips. The successful applications of the migration of a liquid droplet require effective controlling and manipulating of droplet movement behavior. It presses forward the demand for better physical insight towards understanding this phenomenon. Most of the existing theoretical studies 1419 of the thermocapillary migration of a liquid droplet on a horizontal solid surface have used the lubrication theory to predict the steady speed of a droplet in which some physical phenomena are neglected. This could lead to the problem of oversimplifying, with results that may be only valid for certain flow regimes. In detail, some problems that may be encountered include, but are not limited to: firstly, the lubrication approximation which has been used in several previous theoretical works cannot solve the transient migration of a droplet with a large static contact angle (SCA), since the aspect ratio of maximum height of the droplet to its footprint is not small enough to fulfill this theory. For the thermocapillary migration of a droplet on a horizontal 2 solid surface, the temperature gradients along the droplet surface depend strongly on the heat and fluid transport behaviors among the substrate, droplet, and air. Therefore, the assumption that the temperature gradient along the dropletair interface is the same as that on the solid surface might be not reasonable, except for a droplet with a very small SCA. A droplet with higher SCA needs more time to reach the equilibrium or a quasisteady state, due to the larger diffusion time coming from the greater height of a droplet which has the same footprint. Moreover, the contact angle (CA) can be modified by the thermocapillary flow inside the droplet 16. These first problems are impossible to be solved by theoretical model. Secondly, under terrestrial condition and temperature differences, two flows always occur inside a droplet: thermocapillary convection and buoyancy convection. The flow should become more complicated if the droplet size is large enough so that the effects of gravity become more important. Finally, it is wellknown that droplet shape strongly influences on the droplet motion behavior; with the enhancement of the hydrostatic force to the increase of the droplet size, the droplet could be deformed in different geometries which are not easy to predict or solve by theoretical investigation. Hence, up to date, the migration mechanism and the flow behavior of such droplet movements are still unclear and remain open questions. From an experimental aspect, it is very hard to conduct precise experimental measurements of the temperature and the flow velocity during the transport process of a droplet at the millimeter scale. In addition, the available equipments and devices for monitoring, probing, and sensing the droplet motion process without interfering with the motion physics limit the experimental works. In the context of numerical simulation studies, there are several difficult factors that the model should deal with in describing and modeling the complex behavior of thermocapillary 3 migration of a droplet under transient regime. They are (i) the strong nonlinear coupling of several factors, such as the dropletair surface tension and the fluid properties to temperature, (ii) the momentum inertia in droplet and air phase, (iii) the evolution of deformable droplet, (iv) the slip behavior at liquidsolid interface, and (v) the motion of boundary with time dependence. However, with the rapid advance of computing technology and the development of computational methods, numerical simulations can solve these above problems. They have also grown to give new results and better understanding of the complex transport processes and have opened new fields of investigation. In cases of study of the transient migration of a liquid droplet on a horizontal solid surface caused by temperature gradients, numerical simulations promise great potential, and provide a valuable research platform for extending our knowledge and insight towards understanding these phenomena. To date, to our best knowledge, there is only one related numerical simulation 20 that was performed to investigate the flow and temperature fields inside droplets. However, the numerical model is not represented in detail and the quantitative agreement between the numerical and experimental results is not achieved. All above interested subjects have motivated the present dissertation. 1.2 Objectives The main objective in the present dissertation is to develop an appropriate numerical means for investigation dynamic behavior of a liquid droplet migration on a horizontal solid surface induced by temperature gradients under the transient regime. Specifically, the goals of this dissertation are to investigate: 4  the physical mechanism of a liquid droplet migration on a horizontal solid surface with temperature gradients;  the effects of imposed temperature gradients, initial conditions, slippage, and static contact angles which are represented for different wetting surface conditions, on the thermocapillary migration behavior of a liquid droplet;  the interplay between thermocapillary convection and capillary flow during the transient motion of a liquid droplet;  the role of dynamic contact angles on the droplet motion process; and  the migration behavior of a droplet under the influences of combined thermocapillarybuoyancy convection and free surface deformation induced by gravity. 1.3 Dissertation structure The novel conceptual thermocapillary convection, buoyancy convection, slippage, and other factors relating to droplet manipulation and actuation are presented and analyzed in chapter 2. In chapter 3, a literature review is presented separately based on problem solving methods, namely theoretical analyzing, numerical investigating, and experimental performing. Only works on the thermocapillary migration of a droplet on horizontal solid surface which relate to our studies are presented and analyzed. The physical problems with the proper theoretical formulations and the computational methods are shown in chapter 4. The continuum for modeling the fluids flow, the physical assumptions, the governing equations, the boundary conditions, and the initial conditions applied for the numerical model to solve problems at the transient regime through the finite element method (FEM) are presented in detail. The conservative level set method, the 5 arbitrary Lagrangian Eulerian (ALE) method, and the continuum surface force (CSF) method employed to treat the movement and deformation of the dropletair interface and the surface tension force during the motion process are shown. The convergence test which focuses on meshing, numerical errors, and effects of wall boundaries on the motion behavior of the droplet is also illustrated in detail. Chapter 5 shows the results of a submillimeter scale droplet with large contact angle migration induced by thermocapillary convection only. The driving mechanism of thermocapillary migration of a liquid droplet and the effect of temperature gradients, static contact angles showing the solid surface wettability, initial conditions, and slippage on the droplet motion behavior are analyzed in detail and discussed. The migration of large sized droplets in different sizes induced by combined thermocapillary convection and buoyant convection and free surface deformation induced by gravity is investigated in chapter 6. The deformation of a droplet due to the effect of hydrostatic force is examined. The interplay between these two convections, which play an important role in discovering the complicated behavior of migration of a liquid droplet in a wide range sizes, is explored. The dissertation ends with chapter 7 in which the contributions and remarks from the present research are highlighted. Recommendations for future activities dedicated to this field are also given. 6 Chapter II Background 2.1 Driving force It is well known that the capillary phenomena might occur in cases where the surface tension varies from point to point on the surface. The nature of motion in each of the fluid phases can be changed by forces near the interphase boundary. Therefore, the condition of a balance of the forces exerting in each phase should be performed on the changeable phase boundary which can be described in general form: i j i 1 j j i 1 i 2 j j i i 2 1 2 2 1 x x V x V x V x V r 1 r 1 p p                                                            n n , (21) where p, V, and r are respectively pressure, dynamic viscosity, surface tension, velocity, and principal radii of curvature of the surface. Subscript 1 and 2 denote phase 1 and 2. ni with i = 1, 2, 3 are the components of the unit vector normal to the surface and directed into the interior of phase 1, and summary over a repeated index with j = 1, 2, 3 is assumed. By taking the projections of the Eq. (21) in the directions normal and tangential to the interface boundary, two scalar conditions correspondingly can be obtained: 1 n 1 2 n 2 1 2 2 1 n V 2 n V 2 r 1 r 1 p p                             , (22) and 7                           1 n 1 2 n 2 n V V n V V , (23) where n and  denote the direction normal and tangential to the interface, respectively. Equation (22) and (23) evidence that when the term of dependence of surface tension are not small compared to the other terms, surface tension will, in a necessary approach, influence the velocity distribution in each of the phases. Following surface tension driving flow mechanism as above, it can be inferred that a liquid droplet migration on a horizontal solid surface might be achieved if the variation of surface tension is generated to play the role of driving force. This might be created by using several kinds of external fields as shown in Table 1 3. When a gasliquid interface is present, the migration of fluid can be generated by controlling spatial variations of surface tension. These variations can be produced by various techniques including thermal 14, chemical 14, 21, electrical 22, electrochemical 23, electrowetting24, 25, magnetic 26, 27, vibrating 28, and photoirradiative29 methods. One of the most popular methods of generating the surface tension gradient is using the thermal fields 30, as surface tension experiences sensitive linear change with a wide range of temperature. The idea of using a temperature difference as a driving force to move a droplet has been recognized first by Bouasse 31. In his study, a metal wire with a heated lower end, tilted slightly upward, was used to move a droplet upward against the force from gravity. Under thermal field, fluid flow can be induced through local heating that can vary liquid surface tension, producing thermocapillary shear stress. In the present studies in this dissertation, the temperature gradient is used as an external field to create driving force for moving a liquid droplet on a horizontal solid surface. 8 Table 1. Forces and external fields used to manipulate motion of fluids. It is also possible to use external means to manipulate particles embedded in flows 3. Driving force Subcategorization Remarks; representative references Pressure gradient P   Familiar case as in pipe flow Capillary effects Surface tension, σ Capillary pressure difference Thermal Electrical (electrocapillarity)  Surface tension gradient,    Chemical   Thermal   Electrical   Optical  Electric fields E DC electroosmosis Uniform velocity field   AC electroosmosis Rectified flows   Dielectrophoresis Response 2 E  Magnetic field Lorentz forces Magnetohydrodynamic stirring  Rotation Centrifugal forces  Sound Acoustic streaming  9 2.2 Thermocapillary convection For most liquidgas interfaces, it is well known that there is a variation in the surface tension at the interface if the temperature at the interface is not uniform. This surface tension variation due to temperature can be described as (T T ) ref T ref       , (24) where ref denotes the surface tension at the reference temperature Tref, and the surface tension temperature coefficient T is the rate of change of the surface tension to the change of temperature. In the present study, for the liquids of interest here, the surface tension of a liquid against its own gas will decrease with temperature so that exhibits an inverse relationship with temperature which is represented as T T      . (25) For most liquidair interfaces, is almost constant over a wide range of temperatures in several tens of Kelvin 30 and is taken by positive value. Therefore, the fluids can be driven by the variation of surface tension through a temperature variation along the liquidair interface. The flow motion of fluids caused by such surface tension gradients is normally termed as thermocapillary convection. It is noted that there is no critical temperature difference in thermocapillary convection, and this convection exists at any nonzero temperature gradient. For a liquidair interface, equation (23) can be rewritten as                           a n a l n l V n V V n V , (26) 10 where subscript l and a denote liquid and air phase, respectively. From Eqs. (24) (26), it is straightforward to see that the direction of the thermocapillary flow is towards the lower temperature or colder side of the interface due to higher tension on that side. This means that at the liquidair interface, the fluid flows in the direction opposite to that of the surfacetemperature gradient. The strength of thermocapillary convection is characterized by the Marangoni number,     2 T GR Ma (27) where R is the length scale, G is the imposed temperature gradient,  and  respectively denote the thermal diffusivity and dynamic viscosity of the fluid. When a liquid droplet surrounded with air is placed on a nonuniform temperature solid surface, the gradient of surface tension  induced by the temperature gradient S G T S    expressed as S T S     G , (28) may push it along. Here, it can be seen that the driving force for the flow is GS. Such droplet motion is termed the thermocapillary migration. 2.3 Buoyancy convection Under terrestrial condition, when a liquid droplet is placed on a horizontal solid surface with a temperature gradient, accompanied by the thermocapillary convection, the liquid within the droplet could experience motion caused by the buoyancy force which is generated 11 from variation of density to temperature T. The dependence of fluid density on temperature can be expressed as (1 (T T )) ref ref      . (29) The thermal expansion coefficient of fluid  is described, T 1       , (210) which is positive for most fluids excluding for water in the range 0 to 40C with negative magnitude. Such flow of fluids is referred to as buoyant convection or free or natural convection, and buoyancy force is the driving mechanism of the flow. This flow can be characterized by the Rayleigh number,     GR g Ra 4 , (211) where  and g denote the kinematic viscosity and the gravitational acceleration constant, respectively. From Eqs. (27) and (211), it can be seen that the ratio between the Ra and Ma numbers might represent a measure of the relative strength of buoyancy forces to thermocapillary forces which can be shown as T 2 D g R Ma Ra Bo      . (212) This number can be quantified as the effects of buoyancy convection in comparison to the thermocappillary convection and is termed the dynamic Bond number. 12 2.4 Contact line and contact angle When a small amount of liquid is placed on a horizontal solid surface, it might form a droplet. The droplet shape is determined by the volume of liquid and the solid surface condition, which is characterized by three interphases: liquid gas, liquid solid and solid gas. The intersection of three phases is a closed line which is termed the contact line. When the droplet is axisymmetric, this line is the basic circle of the droplet. When seen in two dimensions, this line is a point and called the contact point or trijunction point (Figs. 21 and 22). Figure 2 1. A static droplet on a hydrophilic (partially wetting) solid surface (. The quantity of the energetic affinity between liquid and solid surface is directly related to free energy change as a liquid wets a surface. Hence, the wettability of a liquid reflects its affinity for solid surface and can be quantified by an angle termed contact angle φ (CA) that is defined as the angle formed between the dropletair interface and the dropletsolid interface (Figs. 21 and 22). The size of the CA is directly related to the change in free energy which 13 is determined by the force balance between the surface tensions at the contact line. If the solid surface is ideal (i.e. smooth, planar, and homogeneous), the CA is equal to the equilibrium value which leads to the well known Young equation 32 LG SG SL  cos     , (213) where σLG, σSG and σSL denote the surface tension of the liquidgas, solidgas and solidliquid interfaces, respectively. The contact angle φ in the Young relation is called the equilibrium or static contact angle (SCA). The liquid is said to be on a hydrophilic (wetting) (Fig. 21) or hydrophobic (nonwetting) surface (Fig. 22) if the value of CA is less or greater than 900. Figure 2 2. A static droplet on a hydrophobic (nonwetting) solid surface (. 14 2.5 Contact angle hysteresis It is noted that the SCA as in Eq. (213) exists only under the equilibrium conditions. When a droplet is exerted on by external forces, the contact line is in motion and the problem becomes dynamic. The contact line is advancing when its speed U > 0 and receding when U < 0. The extrapolated magnitude of the CA in the limit as U  0 with U > 0 is termed the advancing contact angle A and with U < 0 is called the receding contact angle R 33, 34. The moving contact line could lead the contact angles to vary, and the droplet shape might become asymmetric. The CAs under dynamic regime are named dynamic contact angles (DCA). The difference between advancing contact angle and receding contact angle is referred to as the contact angle hysteresis (CAH). A more detail explanation about the CAH can be found in Ref. 35. The dynamic contact angle difference plays a very important role in the case of liquid droplet motion on a solid surface. This role will be discussed in detail in chapters 5 and 6. For instance, when CAH occurs, the variation in the interfacial curvature produces a pressure difference across the droplet. As a result, the liquid inside the droplet moves from high to low pressure, which is called the capillary flow. If this flow is in the same direction as the thermocapillary flow, the droplet moves faster 36. In contrast, the droplet will only move if it is exceeded by the thermocapillary force 37. 2.6 Slippage and contact line motion When a liquid droplet moves on a horizontal solid surface, the molecular interaction between liquid and solid surfaces is very complicated. Several years ago, the studies of fluid dynamics for Newtonian liquids relied on a common assumption in continuum fluid dynamics which supposes that fluid molecules in the immediate vicinity of the solid surface 15 move with exactly the same velocity as that surface. This means that the relative fluidsolid velocity equals to zero, which is generally called the no slip boundary condition for the flow of Newtonian liquids near a solid surface (Fig. 2.3a). A longtime agreement for assuming the no slip boundary condition had the consequence that many present works and textbooks of fluid dynamics fail to mention that the no slip boundary condition remains an assumption 38. Indeed, the nature of the boundary condition for fluid flow past a solid surface has been revived by several experimental investigations, molecular simulations, and new theoretical approaches 3951. These works strongly show evidence of slip occurring at the liquidsolid surface interface. Recently, the idea of “slip length” or “slip coefficient”, first proposed by Navier 52, has been commonly used to determine the slip behavior of a liquid at a solid surface. This idea states that at a solid boundary, the normal component of the fluid velocity should vanish at the impermeable solid wall because of kinematic reasons and the tangential velocity u is proportional to the shear rate (or the rate of strain), z u u b     , (214) where u is the velocity, z is the normal direction to the solid surface, and the constant of proportionality b is called the slip length. The slip length is defined as the distance beyond the liquidsolid wall interface where the liquid velocity extrapolates to zero (Fig. 23b). It mainly depends on wettability and roughness of the solid surface 53. It is also found that the shear rate normal to the surface and the presence of gaseous layers at the liquidsolid wall interface might affect the slip behavior 42, 54. Theoretically, the slip length magnitude justified by Barrat and Bocquet 55 is equal 16 st SL l m cm S C M D b   (215) where Dcm is a collective molecular diffusion coefficient, Sst is the structure factor for first molecular layer, CSL is solid liquid coefficient of the LennardJones potential, and Mm is a molecular scale. In practice, several works 45, 50, 51, 5659 indicate that the value of the slip length could normally be less than a few nanometers for flows over flat hydrophilic surfaces, while it is as large as tens of micrometers for superhydrophobic surfaces. The structural and dynamical properties of the liquid solid wall interface, therefore, might be represented by the slip coefficient magnitude. From the definition of the slip length, it also can be interpreted that when b = ∞, the relative fluidsolid velocity uux(z) which is normally termed as the perfect slip (Fig. 23c). In the context of a Newtonian liquid droplet moving on a solid surface, the no slip boundary condition hypothesis for solidliquid interface could contrast to the moving contact lines 33, 34. When the contact lines move, a singularity force occurs at the neighborhood of the liquidsolid interface. In order to overcome this problem, it has been suggested that the slip boundary condition be imposed at the solidliquid interface as a way to remove singularities arising in the motion of contact lines as reviewed in 33, 34. 17 Figure 2 3. Definition of the slip length. a) No slip: b = 0; b) Partial slip: the linear velocity ux profile in the flowing which is characterized by a viscosity and a constant shear stress does not vanish at the liquidsolid interface. Extrapolation into the solid at a depth b is needed to obtain a vanishing velocity as supposed by the noslip boundary condition; and c) Perfect slip: b = ∞. 2.7 Capillary length The equilibrium shape of the liquid droplet is determined by minimizing the free energy Efr consisting of the surface energy and the gravitational potential energy of the bulk droplet 12 as                E min d g dr z min fr , (216) 18 under the constant volume constraint: const r d   , where  is the droplet size,  and denote the droplet volume and dropletair interface, respectively. From Eq. (216), it is seen that the shape droplet problem can be governed by a characteristic length,  g     where the parameter  is called Laplace length or capillary length.Therefore, when the gravity includes for a liquid droplet system, the droplet shape might be formed which depends on the ratio of the surface tension and the gravity. In the context of the effects from thermocapillary stress, the deformation of a droplet is normally evaluated by the Capillary number defined as    ref U Ca , (218) where    GR U T ref (219) is the reference velocity. The magnitude of the Capillary number determines the relative importance of the normal stress imbalance in deforming a dropletair interface. If the difference in normal stress across the dropletair interface is uniform over the surface of droplet, the curvature would be uniform leading the droplet forming in a spherical shape. Oppositely, variations in this difference could lead to deformation from the spherical shape.  19 Chapter III Literature review This chapter is a literature review on some of the existing theoretical, experimental, and numerical studies investigating the motion of a liquid droplet on a horizontal solid surface induced by temperature gradients. 3.1 Theoretical works Several theoretical works 1419 have tried to find out the final steady speed of a droplet motion by using the lubrication theory. In order to validate the application of this theory, some assumptions are given, including that the flow velocity is slow and the aspect ratio between the height profile (h(x)) and footprint radius (L) of a droplet (hL) is small. The Pelect number,   ref RU Pe , (31) is also assumed to be small so that the temperature field within the droplet is mainly dominated by conduction. The temperature gradient along the dropletair interface is assumed to be the same as that on the solid surface. From this assumption, it can be inferred that there is only one thermocapillary convection vortex inside the droplet. The effects from the convective term are neglected. The effects of gravity are not to be considered in most studies with assumptions that the droplet is of small size. Brochard 14 analyzes a droplet migration on a horizontal chemical inhomogeneous and nonuniform temperature surface. The contact angle is assumed to be small so that the liquid is in thermal equilibrium with the solid surface, Tl(x, z) = Ts(x). The thermal convection 20 within the system is also neglected. A speed equation of the droplet has been developed which showed that the advancing velocity depends on the surface energies and temperature gradients. The study also points out that a droplet forms as a spherical cap or a “thick pancake” if its footprint l is smaller than or bigger than the capillary lengthThe large droplet is flattened mainly due to gravity. Ford and Nadim 15 conducted a theoretical analysis of the migration of a droplet with an infinitely long footprint and arbitrary height profiles induced by thermocapillary forces. It should be noted that the conventional description of flow in a droplet on a solid surface supposes the no slip boundary condition fails due to the appearance of the infinity stresses at the contact line. The authors overcame this problem by using the slip condition for the dynamic boundary condition in the neighborhood of the contact line. The prediction for the migration speed of a droplet is obtained which shows that it depends on the dynamic contact angles, the height profile, and the slip length. Smith 16 described how the motion of the contact lines changes with the apparent contact angles, under the assumption of constant thermocapillary stress. This study also showed that the thermocapillary cell inside the droplet distorts the free surface and changes the dynamic contact angles. Following the work of Ford and Nadim 15, Chen et al. 17 used a hydrodynamic model to describe the steady motion of a two dimensional droplet driven by thermocapillary forces including the CAH. The study exhibited that the speed of a droplet V is very sensitive to the change of dynamic contact angles described as                L (cos cos ) 2L T (1 2cos ) 6 J 1 V R R A A T , (32) 21 where J is defined as     L L h(x) 3b dx 2L 1 J . (33) By adjusting two parameters, the CAH and the slip length, they demonstrated that the droplet speed is more significantly influenced by the CAH than th

COMPUTATIONAL STUDY OF A DROPLET MIGRATION

ON A HORIZONTAL SOLID SURFACE WITH

TEMPERATURE GRADIENTS

By

HUY BICH NGUYEN

A dissertation presented to the Department of Mechanical Engineering of the National Central University in partial fulfillment of the requirements for the

degree of Doctor of Philosophy

Dissertation Advisor: Chair Prof JYH - CHEN CHEN

Department of Mechanical Engineering NATIONAL CENTRAL UNIVERSITY

December, 2010

研 究 生：阮揮碧 指導教授：陳志臣 教授

Dedicated to the love of my life, Le Thi Tuyet Huyen, and my loving daughters Nguyen Le Thu Truc and Nguyen Le Quynh Truc

NATIONAL CENTRAL UNIVERSITY

DEPARTMENT OF MECHANICAL ENGINEERING

We hereby approve the dissertation of

Huy Bich Nguyen

for the degree of Doctor of Philosophy

(Signed)

Prof Tsing-Fa Lin Prof Jing-Tang Yang

(National Chiao-Tung University) (National Taiwan University)

(Chair of the committee)

Prof Jyh-Chen Chen Prof Jiin-Yuh Jang

(National Central University) (National Cheng-Kung University)

_

Prof Lih-Wu Hourng Prof Shu-San Hsiau

(National Central University) (National Central University)

_

Assoc Prof Ming-Tsung Hung Prof Jer-Liang Yeh

(National Central University) (National Tsing-Hua University)

Abstract of Dissertation Presented to the Department of Mechanical

Engineering of the National Central University in Partial Fulfillment of the

Requirements for the Degree of Doctor of Philosophy

COMPUTATIONAL STUDY OF A DROPLET MIGRATION

ON A HORIZONTAL SOLID SURFACE WITH

TEMPERATURE GRADIENTS

By

HUY BICH NGUYEN

Recently, the migration of a liquid droplet on a horizontal solid surface has attracted widespread attention from many researchers and engineers because of its promising prospective in a variety of applications in biology, chemistry, and industry In this dissertation, a proper computational model is developed for investigating the transient migration of a liquid droplet on a horizontal solid surface subjected to uniform temperature gradients Numerical calculations are carried out by solving the Navier - Stokes equations coupled with the energy equation through the finite element method (FEM) The conservative level set method, the arbitrary Lagrangian Eulerian (ALE) method, and the continuum surface force (CSF) method are employed to treat the movement and deformation

of the droplet/air interface and the surface tension force during the motion process Some physical properties of fluids dependent on temperature are also considered

The study indicates that when a liquid droplet is of small size, two asymmetric thermocapillary vortices are generated inside the droplet The thermocapillary vortex on the hot side is always larger in size than that appearing on the cold one The net momentum of the thermocapillary convection inside droplet pushes the droplet moves from the larger vortex (hot side) to the smaller one (cold side) The variation of the size of the thermocapillary

ii

vortex during the movement causes the speed of the droplet to initially increase and then decrease slowly until approaching a constant value A higher imposed temperature gradient leads the droplet velocity to reach the maximal value earlier and have a higher final speed If the static contact angle of the droplet is less (or higher) than 90 degrees, the droplet speed is lower (or higher) since the net thermocapillary momentum in the horizontal direction is diminished (or enhanced) by the presence of capillary force The lower slip length leads to the smaller droplet speed In addition, the quasisteady migration speed of a small droplet is linearly proportional to its size due to the stronger net thermocapillary momentum The effect of gravity is insignificant and the thermocapillary convection is dominated The computational model is verified by comparing to the previous experimental results

When the droplet turns into larger, the influence of gravity becomes important The combined thermocapillary and buoyancy force driven convection produce complex dynamic behavior of fluid motion inside the droplet In the middle size regime, the quasisteady migration speed of the droplet reaches a maximum, but this is gradually reduced as the droplet size increases due to the suppression of the net thermocapillary momentum by the buoyancy force In the large droplet size regime, two pairs of convection vortices exist inside the droplet as a result of the appearance of the buoyancy-driven convection accompanying the thermocapillary convection The quasisteady migration speed quickly diminishes, mainly due to the reduction of the net thermocapillary momentum from the stronger buoyancy convection The droplet speed tendency is found to be a good agreement with the experimental results

iii

摘要

由於液滴在水平之固體表面的移動之不同應用，具備在工業、化學與生物學上的前景，最近正吸引許多學者與工程師的注意。本研究為使用數值模擬，探討固體表面溫度梯度所形成液滴與空氣界面表面張力梯度，而促成液滴移動之

物 理機制。運用有限元素法、等位函數法 ( Level Set Method )、 ALE 運 動 描

述 法（arbitrary Lagrangian Eulerian method, ALE method ） 與 連 續 表 面 力 學 法（continuum surface force method, CFS method），求解 Navier-Stokes 方 程 式 與能量方程式。

mocapillary force） 渦 漩。 在 液 滴 熱 邊 的 熱 毛 細 力 渦 漩 尺 寸 總 是 大

本研究指出當液滴為小尺寸時，液滴內會產生兩個不對稱的熱毛細力(ther-於 在 冷邊者 。液滴內 熱毛細力 渦漩之淨 動量驅動 液滴由熱 邊（大 渦 漩） 向冷邊（小渦漩）移動。移動過程中熱毛細力渦漩尺寸的變化，造成液滴的速度於初期增加而後緩慢降低，直至接近一個近穩定速度之定值。增加溫度梯度

，將導致液滴較早達到近穩定速度，及較高近穩定速度。在水平方向的熱毛細力與毛細力相互作用下，當液滴之靜接觸角小於（或大於）90度時，熱毛細力與毛細力具有抵制（或相乘）效果，使液滴的近穩定速度降低（或增加）。較短的滑動長度導致較低的液滴移動速度。而淨熱毛細力使液滴之近穩定移動速度與其尺寸呈線性關係。重力的效果不明顯而熱毛細力對流才是主導移動的主因。本研究模擬結果可驗證其他學者已發表之實驗結果。

iv

當液滴尺寸變大，重力的影響愈重要。液滴內熱毛細力與浮力生成之對流，產生複雜的流體動力行為。對於中尺寸的液滴，它的近穩定移動速度達到最大

，但隨液滴尺寸增大，浮力抑制熱毛細力淨動量，使近穩定移動速逐漸降低。對於大尺寸液滴，液滴內存在兩對對流渦漩，這是浮力對流與熱毛細力對流所產生的結果。由於較強浮力對流使淨熱毛細力動量降低，液滴的近穩定移動速度很快減低。上述模擬之液滴速度趨勢與其他學者已發表實驗結果有很好的一致性。

v

ACKNOWLEDGEMENTS

My sincerely appreciation and deeply respect go to my advisor, Chair Professor Chen Chen, Dean of College of Engineering of the National Central University (NCU), for his unlimited support with continuous and patient advice during my studies Without his advice, my research could not to be achieved

Jyh-I would like to thank the members of my dissertation committee: Prof Lih-Wu Hourng (NCU), Prof Shu-San Hsiau (NCU), Assoc Prof Ming-Tsung Hung (NCU), Prof Jiin-Yuh Jang (National Cheng-Kung University), Prof Tsing-Fa Lin (National Chiao-Tung University), Prof Jing-Tang Yang (National Taiwan University), and Prof Jer-Liang Yeh (National Tsing-Hua University), for their support I thank Prof Bruce A Finlayson (University of Washington, USA) for suggestion using the ALE method, and Prof William

BJ Zimmerman (University of Sheffield, UK) and Assoc Prof Nam-Trung Nguyen (Nanyang Technological University, Singapore) for discussions about contact line and others

I express my sincere acknowledgements to the National Science Council (NSC) of the Taiwan government for granting me a three-year doctoral fellowship, to the National Central University for awarding me the highest NCU’s PhD scholarship, and to Vietnam Ministry of Education and Training and my workplace, Nong Lam University, Hochiminh city, Vietnam, for approving me to study in Taiwan

It is my pleasure to thank people in my LHPG Lab who are formerly Ph.D candidates:

Dr Chung-Wei Lu, Dr Cheng-Wei Chien, Dr Hsueh-I Chen, Dr Chang-Hung Chiang, Dr Gow-Jiun Sheu, Dr Cheng-Lin Chung, and Dr F S Hwu; recently Ph.D candidates: Hung-Lin Hsieh, Chun-Hung Chen, Ying-Yang Teng, Heng Chiou, and Ming-Te Lin; and several Master students, and specially my Prof.’s assistant Ms Jenyin Tien for helping me one way

vi

or another during my studies at the National Central University, Taiwan I also thank the individuals in the Mechanical Department, the Office of International Affairs, and my friends for supporting me in NCU

Finally, I dedicate this dissertation to my lovely wife, Tuyet-Huyen Thi Le, to my two nice daughters, Thu Truc Le Nguyen (Listar Nguyen) and Quynh Truc Le Nguyen (Wendy Nguyen), and to my relatives for their sympathy, sharing my difficulties, continuous support, and warm encouragement over the years of my studies in Taiwan

vii

TABLE OF CONTENTS

ABSTRACT i

摘要 iii

ACKNOWLEDGEMENTS v

TABLE OF CONTENTS vii

LIST OF FIGURES xi

LIST OF TABLES xviii

NOMENCLATURE xix

Chapter I Introduction 1

1.1 Motivation 1

1.2 Objectives 3

1.3 Dissertation structure 4

Chapter II Background 6

2.1 Driving force 6

2.2 Thermocapillary convection 9

2.3 Buoyancy convection 10

2.4 Contact line and contact angle 12

2.5 Contact angle hysteresis 13

2.6 Slippage and contact line motion 14

viii

2.7 Capillary length 17

Chapter III Literature review 19

3.1 Theoretical works 19

3.2 Experimental works 22

3.3 Numerical works 27

Chapter IV Theoretical Formulations and Computational Methods 29

4.1 Validity of continuum model 29

4.2 Physical Model 30

4.3 Theoretical formulations 30

4.3.1 Governing equations 30

4.3.2 Boundary conditions 33

4.3.3 Initial conditions 34

4.4 Computational methods 35

4.4.1 Conservative level set method 37

4.4.2 Arbitrary Lagrangian Eulerian (ALE) method 39

4.5 Computational Solving Process 39

4.6 Convergence test 41

4.6.1 Meshing 41

4.6.2 Numerical Errors 41

4.6.3 Wall boundary test 42

ix

Chapter V A Droplet Migration Induced by Thermocapillary Convection 48

5.1 Droplet thermocapillary migration mechanism 52

5.2 Effect of temperature gradient 52

5.3 Effect of contact angle 58

5.4 Effect of initial condition 64

5.5 Effect of slip behavior 71

5.6 Validation of model 75

5.7 Summary 78

Chapter VI A Droplet Motion Induced by Combined Thermocapillary – Buoyancy Convection 80

6-1 The deformation of the droplet 80

6-2 The droplet migration in different sizes 81

6.2.1 The small sized droplet regime 82

6.2.2 The middle sized droplet regime 82

6.2.3 The large sized droplet regime 89

6.3 Summary 89

Chapter VII Concluding remarks and future works 96

7.1 Conclusions 96

7.2 Main contributions 98

7.3 Recommendations and future works 99

x

Bibliographies 100

Appendix 109

A Calculating height profiles of a droplet 109

B Publications during Ph.D studies 112

xi

LIST OF FIGURES

Figure 2 - 1 A static droplet on a hydrophilic (partially wetting) solid surface ( 12

Figure 2 - 2 A static droplet on a hydrophobic (nonwetting) solid surface ( 13

Figure 2 - 3 Definition of the slip length a) No slip: b = 0; b) Partial slip: the linear

velocity ux profile in the flowing which is characterized by a viscosity and a constant shear stress does not vanish at the liquid/solid interface Extrapolation into the solid at a depth b is needed to obtain a vanishing velocity as supposed by the no-slip boundary condition; and c) Perfect slip: b = ∞ 17

Figure 3 - 1 Droplet velocity versus footprint radius with (a) experiment and (b) and (c)

analytical results 21

Figure 3 - 2 Velocity V of a droplet of PDMS versus its initial radius R, for various thermal

gradientsT The thick solid straight lines fitting the points converge at a point A (R = 0,

V0) They intersect the V = 0 axis at R = Rc For R > 6 mm, a saturation of the velocity is

observed versus viscosity 24

Figure 3 - 3 Displacement histories of the silicone oil droplets of (a) 0.1 l and (b) 1l [24] 25

Figure 3 - 4 (a) - (d) Droplet of silicone oil (PDMS) moves along a heater array A thin

residue trailed the wetting droplet The images sequence represents times (a) t = 0s, (b) 44 s, (c) 88 s, and (d) 132 s 26

xii

Figure 3 - 5 A time sequence of side view images of the moving droplet with two different

contact angles: a) smaller than 900 and b) larger than 90° The dashed line indicates the position of the laser sheet 27

Figure 4 - 1 Schematic representation used for computation The value of the level set

function Φ is equal to 0.5 at the air/droplet interface The air phase (subdomain Ω1)and the droplet phase (subdomain Ω2) are represented by 1 ≥ Φ > 0.5 and 0.5 > Φ ≥ 0, respectively.31

Figure 4 - 2 Flow chart of computational solving process 40

Figure 4 - 3 Typical mesh used in the computational domain (W = 10 mm and H = 1.5

mm) for a droplet with static contact angle φ = 900

, footprint radius L = 0.55 mm 42

Figure 4 - 4 Typical mesh used in the computational domain (W = 35 mm and H = 10 mm)

for a droplet with static contact angle φ = 600

, footprint radius L = 4 mm, and maximum height hm = 2.31 mm 42

Figure 4 - 5 Flow fields for the domain with W = 1.5 mm: a) in whole domain; b) inside

droplet 44

Figure 4 - 6 Flow fields for the domain with W = 3 mm: a) whole domain; b) inside droplet.

45

Figure 4 - 7 Flow field for domain with W = 4 mm: a) whole domain; b) inside droplet 46

Figure 4 - 8 Velocity of droplet for different domain sizes 47

xiii

Figure 5 - 1 Isotherms for a droplet with static contact angle φ = 900, footprint radius L = 0.55 mm and temperature gradient G = 40 K/mm between case of a) Constant viscosity and b) Viscosity dependent temperature 50

Figure 5 - 2 Flowfield inside droplet with static contact angle φ = 900, footprint radius L = 0.55 mm and temperature gradient G = 40 K/mm between case of a) Constant viscosity and b) Viscosity dependent temperature 51

Figure 5 - 3 Migration velocity of droplet with static contact angle φ = 900, footprint radius

L = 0.55 mm and temperature gradient G = 40 K/mm between case of a) Constant viscosity and b) Viscosity dependent temperature 51

Figure 5 - 4 Displacement of the silicone oil droplet with φ = 900 and L = 0.55 mm for the applied temperature gradient G = 40 K/mm and b = 1 nm 53

Figure 5 - 5 Velocity of the silicone oil droplet with φ = 900 and L = 0.55 mm for the applied temperature gradient G = 40 K/mm and b = 1 nm 53

Figure 5 - 6 Isotherms inside and outside the silicone oil droplet with φ = 900 and L = 0.55

mm from t = 0.06 to 2.91 seconds for G = 40 K/mm (Ma = 587.3) and b = 1 nm The red curve shows the droplet/air interface Isothermal contour lines are plotted every 7 K 55

Figure 5 - 7 Streamlines for the silicone oil droplet with φ = 900 and L = 0.55 mm for G =

40 K/mm (Ma = 587.3) and b = 1 nm at 0.01 sec 56

Figure 5 - 8 Temperature distribution along the silicone oil droplet/air interface at different

times: a) From 0.01 sec to 1.11 sec; and b) from 2.61 sec to 2.91 sec for G = 40 K/mm (Ma = 587.3) and b = 1 nmθ is defined as the angle from the droplet center to the interface; θ = 00

xiv

corresponds to the rear contact point; and θ = 1800 indicates the relation to the front contact point 57

Figure 5 - 9 Flowfield inside the silicone oil droplet with φ = 900 and L = 0.55 mm from t

= 0.06 to 2.91 seconds for G = 40 K/mm (Ma = 587.3) and b = 1 nm The red curve shows the droplet/air interface 58

Figure 5 - 10 Displacement of the silicone oil droplet with φ = 900 and L = 0.55 mm for b

= 1 nm and different applied temperature gradient G = 10, 20.5, and 40 K/mm corresponding

to Ma = 146.8, 301, and 587.3, respectively 60

Figure 5 - 11 Velcoity of the silicone oil droplet with φ = 900 and L = 0.55 mm for b = 1

nm and different applied temperature gradient G = 10, 20.5, and 40 K/mm corresponding to

Ma = 146.8, 301, and 587.3, respectively against time 61

Figure 5 - 12 Streamline of the silicone oil droplet with φ = 900 and L = 0.55 mm for b = 1

nm and different applied temperature gradient G = 10, 20.5, and 40 K/mm corresponding to

Ma = 146.8, 301, and 587.3, respectively 62

Figure 5 - 13 Isotherms for the silicone oil droplet with φ = 900 and L = 0.55 mm at different temperature gradients G = 10, 20.5 and 40 K/mm coressponding to Ma = 146.8, 301 and 587.3, respectively, and b = 1 nm at a) t = 0.06sec; b) t = 0.51sec; and c) t = 1.11 sec Isothermal contours lines are plotted every 7 K 63

Figure 5 - 14 Flowfield inside the silicone oil droplet with φ = 900 and L = 0.55 mm at different temperature gradients G = 10, 20.5 and 40 K/mm coressponding to Ma = 146.8, 301 and 587.3, respectively, and b = 1 nm at a) t = 0.06sec; b) t = 0.51sec; and c) t = 1.11 sec 64

xv

Figure 5 - 15 Dynamic contact angles of the silicone oil droplet at G = 10 K/mm and b = 1

nm versus time with a) φ = 700 and 900, and L = 0.648 mm and 0.55mm, respectively; b) φ =

1000 and L = 0.536 mm 66

Figure 5 - 16 Contact angle hysteresis of the silicone oil droplet with φ = 700, 900, and 1000and L = 0.648 mm, 0.55mm and 0.536 mm, respectively, at G = 10 K/mm and b = 1 nm versus time 67

Figure 5 - 17 a) Isotherms and b) flow field inside the silicone oil droplet with φ = 700, 900,

1 nm at t = 0.51 sec 69

Figure 5 - 18 Velocity of the silicone oil droplet with φ = 700, 900, and 1000 and L = 0.648

mm, 0.55mm and 0.536 mm, respectively, at G = 10 K/mm and b = 1 nm versus time 70

Figure 5 - 19 a) Isotherms and b) flow field inside the silicone oil droplet with φ = 900 and

L = 0.55 mm for G = 10 K/mm and b = 1 nm for different initial conditions at different times Isothermal contours lines are plotted every 7 K 70

Figure 5 - 20 Velocity of the silicone oil droplet with φ = 900 and L = 0.55 mm at G = 10 K/mm and b = 1 nm versus time for different initial conditions 71

Figure 5 - 21 Velocity of the squalane droplet with L = 2.6 mm, SCA = 410, and G = 3.12 K/mm for different slip lengths 72

Figure 5 - 22 Flow field inside the squalane droplet with a footprint radius L = 2.6 mm,

static contact angle SCA = 410, and temperature gradient G = 3.12 K/mm for different slip lengths The red arrows denote the stagnation point at the droplet-air interface 73

Figure 5 - 26 Isotherms inside and outside the squalane droplet with φ = 410 and L = 2.6

mm at G = 3.6 K/mm and b = 3 nm for different times Isothermal contours lines are plotted every 6 K 77

Figure 5 - 27 a) The dynamic contact angles and b) the contact angle hysteresis against

time for the squalane droplet with φ = 410

and L = 2.6 mm at G = 3.6 K/mm and b = 3 nm 78

Figure 6 - 1 Deformation of droplet versus droplet size The black and red curves designate

the initial and deformed droplet/air interface, respectively 82

Figure 6- 2 Quasisteady migration velocity of droplet versus droplet size with G = 3.6

K/mm and b = 1nm 83

Figure 6 - 3 Droplet velocity for different small sized droplets versus time with G = 3.6

K/mm and b = 1nm 83

xvii

Figure 6 - 4 (a) Flow field and (b) isotherms inside droplet for different sizes L = 1.4, 1.8,

and 2 mm with G = 3.6 K/mm and b = 1nm The solid arrows denote the moving direction Isothermal contour lines are plotted every 5 K 85

Figure 6 - 5 Different advancing and receding dynamic contact angles for different droplet

sizes with G = 3.6 K/mm and b = 1nm 87

Figure 6 - 6 (a) Flow field and (b) isotherms inside droplet for different size L = 2.2, 3 and

3.4 mm with G = 3.6 K/mm and b = 1nm The solid arrows denote the moving direction Isothermal contour lines are plotted every 5 K 88

Figure 6 - 7 (a) Flow field and (b) isotherms inside droplet with L = 4 mm, G = 3.6 K/mm,

and b = 1nm at different moving times The solid arrow denotes the moving direction Isothermal contour lines are plotted every 5 K 91

Figure 6 - 8 Flow field inside droplet with L = 5 mm, G = 3.6 K/mm, and b = 1nm from t =

0.5 to 10 seconds The solid arrow denotes the moving direction 92

Figure 6 - 9 Isotherms inside droplet with L = 5 mm, G = 3.6 K/mm, and b = 1nm from t =

0.5 to 10 seconds The solid arrow denotes the moving direction Isothermal contour lines are plotted every 5 K 93

Figure 6 - 10 (a) Flow field and (b) isotherms inside droplet with L = 5.5 mm, G = 3.6

K/mm, and b = 1nm at different moving times The solid arrow denotes the moving direction 94

Figure A - 1 A simple model used to count the height profiles of a spherical droplet 110

xviii

LIST OF TABLES

Table 1 Forces and external fields used to manipulate motion of fluids It is also possible to

use external means to manipulate particles embedded in flows 6

Table 2 Physical properties of the fluids (at 250C) 49

xix

NOMENCLATURE English symbols

xx

xxi

xmesh Mesh velocity in x-direction

z mesh Mesh velocity in z-direction

Greek symbols

M a

RaBo







xxiv

Abbreviations

1

Chapter I Introduction

1.1 Motivation

From the last two decades until now, the migration of a liquid droplet on a horizontal solid surface has attracted widespread attention from many researchers and engineers due to its promising prospective in various applications in biology, industry, and chemistry [1-13]

It can potentially be applied to the process of material removing or recovering in a microgravity environment space, the condensation heat transfer, the Lab-on-a-chip (LOC), the fields of Micro-Electro-Mechanical System (MEMS), the field of Micro-Total-Analysis System (μTAS), and the vision of developing entire bio/chemical laboratories on the surface

of silicon or polymer chips The successful applications of the migration of a liquid droplet require effective controlling and manipulating of droplet movement behavior It presses forward the demand for better physical insight towards understanding this phenomenon Most of the existing theoretical studies [14-19] of the thermocapillary migration of a liquid droplet on a horizontal solid surface have used the lubrication theory to predict the steady speed of a droplet in which some physical phenomena are neglected This could lead

to the problem of oversimplifying, with results that may be only valid for certain flow regimes In detail, some problems that may be encountered include, but are not limited to: firstly, the lubrication approximation which has been used in several previous theoretical works cannot solve the transient migration of a droplet with a large static contact angle (SCA), since the aspect ratio of maximum height of the droplet to its footprint is not small enough to fulfill this theory For the thermocapillary migration of a droplet on a horizontal

2

solid surface, the temperature gradients along the droplet surface depend strongly on the heat and fluid transport behaviors among the substrate, droplet, and air Therefore, the assumption that the temperature gradient along the droplet/air interface is the same as that on the solid surface might be not reasonable, except for a droplet with a very small SCA A droplet with higher SCA needs more time to reach the equilibrium or a quasi-steady state, due to the larger diffusion time coming from the greater height of a droplet which has the same footprint Moreover, the contact angle (CA) can be modified by the thermocapillary flow inside the droplet [16] These first problems are impossible to be solved by theoretical model

Secondly, under terrestrial condition and temperature differences, two flows always occur inside a droplet: thermocapillary convection and buoyancy convection The flow should become more complicated if the droplet size is large enough so that the effects of gravity become more important

Finally, it is well-known that droplet shape strongly influences on the droplet motion behavior; with the enhancement of the hydrostatic force to the increase of the droplet size, the droplet could be deformed in different geometries which are not easy to predict or solve by theoretical investigation Hence, up to date, the migration mechanism and the flow behavior

of such droplet movements are still unclear and remain open questions

From an experimental aspect, it is very hard to conduct precise experimental measurements of the temperature and the flow velocity during the transport process of a droplet at the millimeter scale In addition, the available equipments and devices for monitoring, probing, and sensing the droplet motion process without interfering with the motion physics limit the experimental works

In the context of numerical simulation studies, there are several difficult factors that the model should deal with in describing and modeling the complex behavior of thermocapillary

3

migration of a droplet under transient regime They are (i) the strong nonlinear coupling of several factors, such as the droplet/air surface tension and the fluid properties to temperature, (ii) the momentum inertia in droplet and air phase, (iii) the evolution of deformable droplet, (iv) the slip behavior at liquid/solid interface, and (v) the motion of boundary with time dependence However, with the rapid advance of computing technology and the development

of computational methods, numerical simulations can solve these above problems They have also grown to give new results and better understanding of the complex transport processes and have opened new fields of investigation In cases of study of the transient migration of a liquid droplet on a horizontal solid surface caused by temperature gradients, numerical simulations promise great potential, and provide a valuable research platform for extending our knowledge and insight towards understanding these phenomena To date, to our best knowledge, there is only one related numerical simulation [20] that was performed to investigate the flow and temperature fields inside droplets However, the numerical model is not represented in detail and the quantitative agreement between the numerical and experimental results is not achieved

All above interested subjects have motivated the present dissertation

1.2 Objectives

The main objective in the present dissertation is to develop an appropriate numerical means for investigation dynamic behavior of a liquid droplet migration on a horizontal solid surface induced by temperature gradients under the transient regime Specifically, the goals

of this dissertation are to investigate:

4

 the physical mechanism of a liquid droplet migration on a horizontal solid surface

with temperature gradients;

 the effects of imposed temperature gradients, initial conditions, slippage, and static contact angles which are represented for different wetting surface conditions, on the

thermocapillary migration behavior of a liquid droplet;

 the interplay between thermocapillary convection and capillary flow during the

transient motion of a liquid droplet;

 the role of dynamic contact angles on the droplet motion process; and

 the migration behavior of a droplet under the influences of combined

thermocapillary-buoyancy convection and free surface deformation induced by gravity

1.3 Dissertation structure

The novel conceptual thermocapillary convection, buoyancy convection, slippage, and other factors relating to droplet manipulation and actuation are presented and analyzed in chapter 2

In chapter 3, a literature review is presented separately based on problem solving methods, namely theoretical analyzing, numerical investigating, and experimental performing Only works on the thermocapillary migration of a droplet on horizontal solid surface which relate to our studies are presented and analyzed

The physical problems with the proper theoretical formulations and the computational methods are shown in chapter 4 The continuum for modeling the fluids flow, the physical assumptions, the governing equations, the boundary conditions, and the initial conditions applied for the numerical model to solve problems at the transient regime through the finite element method (FEM) are presented in detail The conservative level set method, the

5

arbitrary Lagrangian Eulerian (ALE) method, and the continuum surface force (CSF) method employed to treat the movement and deformation of the droplet/air interface and the surface tension force during the motion process are shown The convergence test which focuses on meshing, numerical errors, and effects of wall boundaries on the motion behavior of the droplet is also illustrated in detail

Chapter 5 shows the results of a sub-millimeter scale droplet with large contact angle migration induced by thermocapillary convection only The driving mechanism of thermocapillary migration of a liquid droplet and the effect of temperature gradients, static contact angles showing the solid surface wettability, initial conditions, and slippage on the droplet motion behavior are analyzed in detail and discussed

The migration of large sized droplets in different sizes induced by combined thermocapillary convection and buoyant convection and free surface deformation induced by gravity is investigated in chapter 6 The deformation of a droplet due to the effect of hydrostatic force is examined The interplay between these two convections, which play an important role in discovering the complicated behavior of migration of a liquid droplet in a wide range sizes, is explored

The dissertation ends with chapter 7 in which the contributions and remarks from the present research are highlighted Recommendations for future activities dedicated to this field are also given

6

Chapter II Background

2.1 Driving force

It is well known that the capillary phenomena might occur in cases where the surface tension varies from point to point on the surface The nature of motion in each of the fluid phases can be changed by forces near the inter-phase boundary Therefore, the condition of a balance of the forces exerting in each phase should be performed on the changeable phase boundary which can be described in general form:

i j 1 i j j

i 1 2 i j j

i 2 i 2 1 1

2

xx

Vx

Vx

Vx

Vr

1r

1p

where p, V, and r are respectively pressure, dynamic viscosity, surface tension, velocity,

and principal radii of curvature of the surface Subscript 1 and 2 denote phase 1 and 2 ni

with i = 1, 2, 3 are the components of the unit vector normal to the surface and directed into the interior of phase 1, and summary over a repeated index with j = 1, 2, 3 is assumed By taking the projections of the Eq (2-1) in the directions normal and tangential to the interface boundary, two scalar conditions correspondingly can be obtained:

1

n 1 2

n 2 2

1 1

2

n

V2n

V2

r

1r

1p

2

n

VVn

VV

, (2-3)

where n and  denote the direction normal and tangential to the interface, respectively Equation (2-2) and (2-3) evidence that when the term of dependence of surface tension are not small compared to the other terms, surface tension will, in a necessary approach, influence the velocity distribution in each of the phases Following surface tension driving flow mechanism as above, it can be inferred that a liquid droplet migration on a horizontal solid surface might be achieved if the variation of surface tension is generated to play the role

of driving force This might be created by using several kinds of external fields as shown in Table 1 [3] When a gas-liquid interface is present, the migration of fluid can be generated

by controlling spatial variations of surface tension These variations can be produced by various techniques including thermal [14], chemical [14, 21], electrical [22], electrochemical [23], electro-wetting[24, 25], magnetic [26, 27], vibrating [28], and photo-irradiative[29] methods One of the most popular methods of generating the surface tension gradient is using the thermal fields [30], as surface tension experiences sensitive linear change with a wide range of temperature The idea of using a temperature difference as a driving force to move a droplet has been recognized first by Bouasse [31] In his study, a metal wire with a heated lower end, tilted slightly upward, was used to move a droplet upward against the force from gravity Under thermal field, fluid flow can be induced through local heating that can vary liquid surface tension, producing thermocapillary shear stress In the present studies in this dissertation, the temperature gradient is used as an external field to create driving force for moving a liquid droplet on a horizontal solid surface

8

Table 1 Forces and external fields used to manipulate motion of fluids It is also possible to

use external means to manipulate particles embedded in flows [3]

Capillary effects Surface tension, σ Capillary pressure difference

- Thermal

- Electrical (electrocapillarity)

Electric fields E - DC electro-osmosis Uniform velocity field 

9

2.2 Thermocapillary convection

For most liquid/gas interfaces, it is well known that there is a variation in the surface tension at the interface if the temperature at the interface is not uniform This surface tension variation due to temperature can be described as

)TT

T ref  





 , (2-4) where ref denotes the surface tension at the reference temperature Tref, and the surface tension temperature coefficient T is the rate of change of the surface tension to the change of temperature In the present study, for the liquids of interest here, the surface tension of a liquid against its own gas will decrease with temperature so that exhibits an inverse relationship with temperature which is represented as

by the variation of surface tension through a temperature variation along the liquid/air interface The flow motion of fluids caused by such surface tension gradients is normally termed as thermocapillary convection It is noted that there is no critical temperature difference in thermocapillary convection, and this convection exists at any nonzero temperature gradient For a liquid/air interface, equation (2-3) can be rewritten as

l

n l

Vn

VV

nV

, (2-6)

10

where subscript l and a denote liquid and air phase, respectively From Eqs (2-4) - (2-6), it

is straightforward to see that the direction of the thermocapillary flow is towards the lower temperature or colder side of the interface due to higher tension on that side This means that

at the liquid/air interface, the fluid flows in the direction opposite to that of the temperature gradient The strength of thermocapillary convection is characterized by the Marangoni number,

When a liquid droplet surrounded with air is placed on a non-uniform temperature solid

surface, the gradient of surface tension   induced by the temperature gradient

ST

expressed as

S T

11

from variation of density to temperature T The dependence of fluid density on temperature

can be expressed as

))TT(1

which is positive for most fluids excluding for water in the range 0 to 40C with negative

magnitude Such flow of fluids is referred to as buoyant convection or free or natural

convection, and buoyancy force is the driving mechanism of the flow This flow can be

characterized by the Rayleigh number,

From Eqs (2-7) and (2-11), it can be seen that the ratio between the Ra and Ma numbers

might represent a measure of the relative strength of buoyancy forces to thermocapillary

forces which can be shown as

T

2 D

Rg

M a

RaBo

This number can be quantified as the effects of buoyancy convection in comparison to the

thermocappillary convection and is termed the dynamic Bond number