Modeling and simulation of droplet dynamics for microfluidic applications

저작자표시-비영리-변경금지 2.0 대한민국 이용자는 아래의 조건을 따르는 경우에 한하여 자유롭게 l 이 저작물을 복제, 배포, 전송, 전시, 공연 및 방송할 수 있습니다 다음과 같은 조건을 따라야 합니다: 저작자표시 귀하는 원저작자를 표시하여야 합니다 비영리 귀하는 이 저작물을 영리 목적으로 이용할 수 없습니다 변경금지 귀하는 이 저작물을 개작, 변형 또는 가공할 수 없습니다 l l 귀하는, 이 저작물의 재이용이나 배포의 경우, 이 저작물에 적용된 이용허락조건 을 명확하게 나타내어야 합니다 저작권자로부터 별도의 허가를 받으면 이러한 조건들은 적용되지 않습니다 저작권법에 따른 이용자의 권리는 위의 내용에 의하여 영향을 받지 않습니다 이것은 이용허락규약(Legal Code)을 이해하기 쉽게 요약한 것입니다 Disclaimer Ph.D Thesis Modeling and simulation of droplet dynamics for microfluidic applications Graduate School of Yeungnam University Department of Mechanical Engineering Major in Mechanical Engineering Van Thanh Hoang Advisor: Professor Jang Min Park, Ph.D August 2019 Ph.D Thesis Modeling and simulation of droplet dynamics for microfluidic applications Advisor: Professor Jang Min Park, Ph.D Presented as Ph.D Thesis August 2019 Graduate School of Yeungnam University Department of Mechanical Engineering Major in Mechanical Engineering Van Thanh Hoang ACKNOWLEDGMENTS I would like to dedicate this thesis for my late father who highly encouraged me to pursue a master and a doctoral program when he left this world almost nine years ago The thesis also is dedicated to the author’s mother who is seventy six years old and living far from me now I really would like to express my deepest gratitude to my thesis advisor, Professor Jang Min Park for dedicated help, valuable and devoted instructions, and everything he has done for me in academic direction and in my life as well over the last three years of my doctoral program I am so grateful to the committee members, Prof Jiseok Lim, Prof Jungwook Choi, Prof Kisoo Yoo, and Prof Kyoung Duck Seo for attending my presentation as well as providing pieces of advice for my doctoral thesis completion During my doctoral program, I wish to express my thanks to the Yeungnam University for supporting the scholarship and providing an excellent academic environment I also thank all of Lab members, Mr Gong Yao, Mr Liu Wankun, Mr Wu Yue, Mr Heeseung Lee, Mr Seung-Yeop Lee, always gave me encouragement and support during my doctoral program Finally, I would like to thank my family, especially my wife for their constant support and encouragement Date: May 15th, 2019 Van Thanh Hoang (호앙반탄) Multiphase Materials Processing Lab., ME/YU I ABSTRACT Design of microchannel geometry plays a key role for transport and manipulation of liquid droplets and contraction microchannel has been widely used for many applications in droplet-based microfluidic systems This study first aims to investigate droplet dynamics in contraction microchannel for more details and then to propose a simplified model used for microfluidic systems to describe droplet dynamics In particular, for contraction microchannel, three regimes of droplet dynamics, including trap, squeeze and breakup are characterized, which depends on capillary number (Ca) and contraction ratio (C) Theoretical models have been also proposed to describe transitions from one to another regime as a function of capillary number and contraction ratio The critical capillary number of transition from trap to squeeze has been found as a function of contraction ratio expressed as CaIc=a(CM-1), whereas critical capillary number CaIIc = c1C-1 depicts the transition from squeeze to breakup Additionally, the deformation, retraction and breakup along downstream of the contraction microchannel have been explored for more details To describe dynamics of droplet in microfluidic system, one-dimensional model based a Taylor analogy has been proposed to predict droplet deformation at steady state and transient behavior accurately The characteristic time for droplet reaching steady state is dependent on viscosity ratio and the droplet deformation at steady state is significantly influenced by viscosity ratio of which the order of II magnitude ranges from -1 to Finally, theoretical estimation of condition for droplet breakup was also proposed in the present study, which shows a good agreement with experimental result in the literature Keywords: Droplet dynamics, Microfluidics, Numerical simulation, Taylor analogy model III Contraction microchannel, TABLE OF CONTENTS ACKNOWLEDGMENTS .I ABSTRACT II TABLE OF CONTENTS IV LIST OF FIGURES VI NOMENCLATURES VIII CHAPTER INTRODUCTION 1.1 Droplet-based microfluidic system 1.2 Contraction microchannel in microfluidic system 1.3 Dynamics of droplet in contraction microchannel 1.4 Droplet dynamics in extensional flow 1.5 Problem statement 1.6 Dissertation overview CHAPTER PROBLEM DESCRIPTION 2.1 Problem description of contraction microchannel 2.2 Problem description for proposed model 2.3 Dimensionless numbers 11 CHAPTER TAYLOR ANALOGY MODELING 12 3.1 Damped spring-mass model 12 3.2 Taylor analogy breakup (TAB) model 13 3.3 Proposed model 15 3.4 Condition for droplet breakup 17 CHAPTER COMPUTATIONAL MODEL AND VALIDATION 18 4.1 Computational model and methods 18 4.2 Computational domain of contraction microchannel 19 4.3 Computational domain for the proposed model 22 4.4 Validation of simulation results in planar extensional flow 25 CHAPTER RESULTS AND DISCUSSIONS 27 5.1 Droplet dynamics in the contraction microchannel 27 5.1.1 Three regimes of the droplet dynamics 27 IV 5.1.2 Droplet dynamics along downstream of contraction microchannel 34 5.2 Performance of the proposed model 41 5.2.1 Steady behavior of droplet deformation 42 5.2.2 Transient behavior of droplet deformation 44 5.2.3 Critical capillary number for droplet breakup 45 CHAPTER CONSCLUSIONS AND RECOMMENDATIONS 47 6.1 Conclusions 47 6.2 Recommendations 48 REFERENCES 50 요약 59 CURRICULUM VITAE 61 V 2.3 Dimensionless numbers The droplet and medium viscosities are denoted as μd and μm respectively The droplet and medium densities are denoted as ρd and ρm respectively The denotation of σ is the surface tension coefficient between the droplet and medium phases The droplet dynamics is characterized by dimensionless parameters A capillary number (Ca) is defined as Ca = 𝜇𝑚 𝜀̇𝑅/𝜎 where the extension rate is defined as 𝜀̇ = 𝑣/𝑅 [23], v is a characteristic velocity Depending on droplet position in the contraction microchannel, two kinds of the characteristic velocities were employed to define capillary numbers The capillary number CaI defined as 𝐶𝑎𝐼 = 𝜇𝑚 𝑣𝑖 𝜎 is used for the large microchannel within the length of Li, where the inlet velocity (vi) is considered as the characteristic velocity, whereas the average velocity in the contraction microchannel (vc) is utilized to defined the capillary number CaII defined as 𝐶𝑎𝐼𝐼 = 𝜇𝑚 𝑣𝑐 𝜎 in the contraction microchannel within the length of 15D Values of viscosity, velocity and interfacial tension can be determined thanks to the definitions of capillary number above Reynolds number (Re) is defined as Re = 𝜌𝑚 𝜀̇𝑅 /𝜇𝑚 A viscosity ratio (λ) is defined as 𝜆 = 𝜇𝑑 /𝜇𝑚 , and λ of 0.15 was employed to study droplet dynamics in contraction microchannel In proposed model based on Taylor analogy, droplet dynamics was investigated for a wide range of viscosity ratio and capillary number A density ratio is defined as 𝜅 = 𝜌𝑑 /𝜌𝑚 which is fixed as unity in this study [37] 11 CHAPTER TAYLOR ANALOGY MODELING Taylor [54] first used an analogy between a spring-mass system and droplet to investigate droplet deformation in a high-speed air flow As stated by this model, the spring force is analogous to the surface tension force and the pressure drag force on the droplet represents an external force In regard to the analogy, next, O’Rourke and Amsden [55] introduced a damping component for describing the viscous behavior of the droplet and damped spring-mass system was employed to calculate droplet breakup in a spray at high Reynolds number The model was called Taylor Analogy Breakup (TAB) model [55] In the present study, Taylor analogy will be used to depict the droplet dynamics in planar extensional flow at low Re regime Specifically, viscous drag force will be operated as an external force term, and damping component will be empirically considered to capture the droplet dynamics for a wide range of capillary number and viscosity ratio 3.1 Damped spring-mass model A simple oscillatory system consists of a mass, as spring and a damper The damped spring-mass model is expressed in Equation (3.1) where x is the displacement of the spring, m is the mass, F is the external force, k is the spring coefficient, d is the damping coefficient 𝑚𝑥̈ = 𝐹 − 𝑘𝑥 − 𝑑𝑥̇ 12 (3.1) According to the damped spring-mass system, there can be three different cases of motions depending on damping ratio ξ = 𝑑 2𝑚√𝑘/𝑚 When ξ = 1, the system is critical damping, so any slight decrease in the damping force leads to oscillatory motion When ( ξ > ), the system is overdamping, in this case the damping coefficient d is large in comparing with the spring constant k When (0 < ξ < 1), the system is underdamping, the damping coefficient is small in comparing with the spring constant The solutions for each case can be shown as Equations (3.2), (3.3), and (3.4), where the displacement x of spring is non-dimensionalized by 𝑑 𝑑 𝑘 𝑑 𝑑 𝑘 𝑑 setting 𝑦 = 𝑥/𝑅 , 𝑟1 = − 2𝑚 + √(2𝑚) − 𝑚 , 𝑟2 = − 2𝑚 − √(2𝑚) − 𝑚 , α = 2𝑚 , 𝑘 𝑑 𝜔 = √𝑚 − (2𝑚) , and b1, b2 are constants defined based on initial conditions [56] It can be seen that the displacement at steady behavior is given as y (𝑡 → ∞) = 1𝐹 𝑅𝑘 1𝐹 when ξ = (3.2) 1𝐹 when ξ > (3.3) 𝑦(𝑡) = 𝑅 𝑘 + 𝑏1 𝑒 𝑟1 𝑡 + 𝑏2 𝑡𝑒 𝑟1 𝑡 , 𝑦(𝑡) = 𝑅 𝑘 + 𝑏1 𝑒 𝑟1 𝑡 + 𝑏2 𝑒 𝑟2 𝑡 , 1𝐹 𝑦(𝑡) = 𝑅 𝑘 + 𝑒 −𝛼𝑡 (𝑏1 cos 𝜔𝑡 + 𝑏2 sin 𝜔𝑡), when < ξ < (3.4) 3.2 Taylor analogy breakup (TAB) model Taylor analogy breakup (TAB) model was developed to depict the droplet breakup in a spray model This TAB model was found to have advantages in terms of simplicity and accuracy, so it has been used in several applications [57-61] 13 According to the TAB model, the displacement x in Equation (3.1) corresponds to the displacement of the droplet equator x described in Fig 2.2(b), m is the mass of the droplet, F is the pressure drag force, k is the surface tension component, d is the viscosity component More specifically, the physical coefficients in Equation (3.1) can be expressed as Equations (3.5), (3.6), and (3.7), where CF, Ck, and Cd are the dimensionless constants and v is the relative velocity between the droplet and the medium 𝐹 = 𝐶𝐹 𝑚 𝑘 𝑚 𝜌𝑚 𝑣 = 𝐶𝑘 𝜌 𝑑 𝑚 (3.5) 𝜌𝑑 𝑅 𝜎 𝑑𝑅 (3.6) 𝜇𝑑 = 𝐶𝑑 𝜌 𝑑𝑅 (3.7) In the engine sprays, the surface tension is more dominant than the viscosity, it means the damping ratio ξ is less than unity, thus it can be regarded as an underdamped case Thus, the solution of Equation (3.1) in the TAB model can be written as Equation (3.8) 𝐶 𝐶 𝐶 𝑦(𝑡) = 𝐶𝐹 We + 𝑒 −α𝑡 [{𝑦0 − 𝐶𝐹 We} cos 𝜔𝑡 + 𝜔 {𝑦0̇ + α (𝑦0 − 𝐶𝐹 We)} sin 𝜔𝑡] (3.8) 𝑘 𝑘 𝑘 where We is the Weber number defined as We = 𝜌𝑚 𝑣 𝑅 𝜎 , y0 and 𝑦̇ are initial displacement and velocity, respectively, which are assumed to be zero in the TAB model [55] 14 3.3 Proposed model This section presents a theoretical model to describe the droplet dynamics at low Reynolds regime by using Taylor analogy Theoretical models for the external force (F) and the damping coefficient (d) in Equation (3.1) have been proposed, while the surface tension force component is assumed to be the same with TAB model as shown in Equation (3.6) Effect of Reynolds number was neglected in this theoretical models At the low Re regime, the drag is dominated by viscous friction The viscous drag force applying on a liquid droplet is given as Equation (3.9) [62] In addition, the viscous drag force is dependent on the droplet shape which is changed during deformation in the present case, so the external force can be proposed as Equation (3.10) where C1 is a constant The mass of droplet, m is given as 𝑚 = 𝜌𝑑 𝜋𝑅 Therefore, F/m is given as Equation (3.11) It can be noted that the present model including Equations (3.6) and (3.11) assumes at low Ca and low Re regimes 𝐹𝑑 = 2𝜋𝜇𝑚 𝑣𝑅 𝐹 = 2𝜋𝜇𝑚 𝑣𝑅 𝐹 𝑚 = 1.5 3𝜆+2 𝜆+1 3𝜆+2 𝜆+1 3𝜆+2 (3.9) 𝜆+1 (1 + 𝐶1 𝑦) 𝜇 𝑣 (1 + 𝐶1 𝑦) 𝜌 𝑚𝑅2 𝑑 (3.10) (3.11) For the effect of damping coefficient (d) in the TAB model, it should be realized that only the droplet viscosity is taken into account because the air viscosity is negligible However, in the present study, both viscosities of droplet and medium 15 are dominant and they should be considered in viscosity effect Hence, the component d/m is empirically proposed as Equation (3.12), where Q is a constant 𝑑 = 𝐶𝑑 𝑚 𝑄 1−𝑄 𝜇𝑑 𝜇 𝑚 (3.12) 𝜌𝑑 𝑅 Finally, by substituting Equations (3.6), (3.11), (3.12) into Equation (3.1) and by using a dimensionless displacement as 𝑦 = 𝑥/𝑅, the Equation (3.1) can be nondimensionalized as Equation (3.13) 𝑦̈ = 1.5 3𝜆+2 𝜆+1 𝜇 𝑣 (1 + 𝐶1 𝑦) 𝜌 𝑚𝑅3 − 𝐶𝑘 𝜌 𝑑 𝜎 𝑑 𝑦 − 𝐶𝑑 𝑅3 𝑄 1−𝑄 𝜇𝑑 𝜇 𝑚 𝜌𝑑 𝑅 𝑦̇ (3.13) In this study, the viscosity is much more dominant than the surface tension, it means the damping ratio ξ is larger than unity, therefore the system can be operated as an overdamped case Initial conditions of the position y0 and the velocity 𝑦̇ are set at zero and the solution of Equation (3.13) can be expressed as Equation (3.14), where 𝑦𝑠 is the displacement of the droplet equator at steady state and defined as Equation (3.15), r1 and r2 are defined in the section 3.1, and Ca is defined as Ca = 𝜇𝑚 𝑣/𝜎 [52] 𝑦(𝑡) = 𝑦𝑠 + 𝑟 𝑦𝑠 −𝑟1 𝑦𝑠 = (𝑟1 𝑒 𝑟2 𝑡 − 𝑟2 𝑒 𝑟1 𝑡 ) 𝐶𝑘 3𝜆+2 −𝐶1 1.5 𝐶𝑎 𝜆+1 (3.14) (3.15) The performance of the proposed model was verified by comparing with the previous experimental data in the literature [37,40], the droplet deformation Df shown in Equation (2.2) should be evaluated It can be seen that there is only one parameter of one-dimensional displacement y in the present model, it is assumed 16 that the cross-sectional area of the droplet at XY plane keeps constant during the droplet deformation Thus, it should be noted this assumption is acceptable for low Capillary number flow Then, the deformation parameter of droplet Df is rewritten as Equation (3.16) The droplet deformation at steady state Df, which occurs at an infinite time, is denoted as Ds (𝑦(𝑡)+1)2 −1 𝐷𝑓 (𝑡) = (𝑦(𝑡)+1)2 +1 (3.16) 3.4 Condition for droplet breakup Droplet dynamics of breakup is one of the main objectives of this study The purpose of this part is to Figure out critical capillary number for droplet breakup and critical droplet deformation which are a function of viscosity ratio Droplet will be broken as long as the dimensionless displacement 𝑦𝑠 is larger than a critical coefficient Cb, and the condition is expressed as inequation (3.17) Therefore, at the critical limit, the critical capillary number Cac for breakup of droplet can be derived as Equation (3.18) 𝐶𝑘 3𝜆+2 −𝐶1 1.5 𝐶𝑎 𝜆+1 𝐶𝑎𝑐 = ≥ 𝐶𝑏 𝐶𝑘 3𝜆+2 ( +𝐶1 )1.5 𝐶𝑏 𝜆+1 17 (3.17) (3.18) CHAPTER COMPUTATIONAL MODEL AND VALIDATION 4.1 Computational model and methods Stokes flow arises from Navier-Stokes equations where inertial forces are assumed to be negligible In microfluidic system, laminar flow is applied and droplet dynamics is governed by the mathematical models of the Stokes flow that consists of conservation of momentum and conservation of mass The medium and the droplet phases are assumed incompressible Newtonian liquids A volume of fluid (VOF) model is used to capture the interface between the droplet and medium phases [63], and the continuum surface force model is employed to handle the surface tension as a body force [64] No slip condition is used at the microchannel wall, the droplet is assumed to be not wetting on the wall, so the contact angle between droplet phase and wall is 180o Tool of ANSYS Fluent is used for the numerical simulation of droplet deformation in the contraction microchannel and planar extensional flow The solution methods include the coupled scheme for pressure-velocity, a second order upwind scheme for momentum conservation equation, the PRESTO! scheme for pressure interpolation, and the Geo-Reconstruct scheme for interface interpolation For the time discretization, a variable time step method is employed for run calculation The Courant number of 0.05 was used to capture the transient behavior accurately and a larger Courant number of 0.25 can be used for recording droplet deformation at steady state 18 4.2 Computational domain of contraction microchannel In order to reduce computational cost, a symmetric contraction microchannel for simulation is shown by the grey color domain as shown in Fig 2.1(a) The computational domain is discretized uniformly by hexahedral elements which have an element size of W/30 [63] Up to our knowledge, there was not experimental data in the literature for verification Therefore, in this case, the validation of the computational model was performed for a different problem, i.e T-junction microchannel Schematic diagram of the T-junction microchannel is shown in Fig 4.1 [63] In this problem, the study focused on if the computational model can capture different regimes of droplet generation, and also a quantitative verification was performed by measuring the droplet length along the microchannel Fig 4.2 shows three regimes of droplet generation corresponding to different velocities of dispersed and medium phases Fig 4.3 presents droplet length as a function of capillary number for two kinds of flow rates Both qualitative and quantitative comparisons show that the present simulation results was found to have good agreement with previous experimental data 19 Fig 4.1 Schematic diagram of T-junction used in validation: (a) a full geometry and (b) side view of the geometry Dimensions unit is in micrometer Wc and Wd are the inlet widths for the continuous phase and dispersed phase, respectively (WT = Wc = Wd) and LT is droplet length in the downstream 20 Fig 4.2 Three regimes of droplet generation (1) Experimental results from Li et al., (2012) and (2) present simulation (a) vct=0.83mm/s, vd=0.69mm/s, (b) vct=3.47mm/s, vd=0.28mm/s, (c) vct=10.0mm/s, vd=5.0mm/s, (d) vct=20.0mm/s, vd=10.0mm/s, where vct and vd represent the continuous phase inlet velocity and the dispersed phase inlet velocity, respectively 21 Fig 4.3 Dimensionless droplet length as a function of Ca for two different flow rates (8.06μL/h and 20μL/h) of the dispersed phase Li et al.’s experiment and present simulation applied the disperse phase flow rate of 8.06μL/h, and Li et al.’s experiment and present simulation applied the disperse phase flow rate of 20μL/h 4.3 Computational domain for the proposed model For computation of the droplet dynamics under planar extensional flow, a computational domain of cube shape is applied To save computational cost, a symmetric model which corresponds to a one-eighth of the full three-dimensional geometry was used as shown in Fig 4.4 As boundary conditions, surfaces 1, and are applied a given velocity field described as Equation (2.1), while the other three surfaces are solved as symmetric boundaries as shown in Fig 4.4 In the present investigation, the domain edge length is times the droplet radius With 22 this scale, boundary effect of the computational domain on droplet dynamics can be negligible [65] After performing mesh convergence tests, the computational domain is discretized by 100100100 uniform hexahedral elements The mesh convergence tests were carried out by using four types of mesh, including 40×40×40, 75×75×75, 100×100×100, and 150×150×150 In the mesh convergence tests, the droplet deformation at steady state and transient behavior as shown in Fig 4.5 and the mesh type of 100×100×100 is found to be reasonable solution with a reduced computational cost Tải FULL (77 trang): bit.ly/2Ywib4t Dự phòng: fb.com/KhoTaiLieuAZ Fig 4.4 A one-eighth of the full model used for the computational domain in planar extensional flow [53] 23 Tải FULL (77 trang): bit.ly/2Ywib4t Dự phòng: fb.com/KhoTaiLieuAZ Fig 4.5 Mesh convergence test for λ=1 and Ca=0.067; (a) steady state, (b) transient behavior of the droplet deformation 24 4.4 Validation of simulation results in planar extensional flow Present simulation of droplet deformation at steady state (Ds) in planar extensional flow was verified by experimental data [37,40] depending on capillary number and viscosity ratio, which is shown in Fig 4.6 It can be seen that the simulation results is found to have good agreement with the previous experimental data Generally, the droplet deformation at steady state increases with the capillary number and the viscosity ratio In addition, critical capillary number for the breakup of droplet decreases when the viscosity ratio increases [37] In transient behavior, the droplet deformation is illustrated as Fig 4.7, where simulation results was compared to the previous experimental data [40] with various capillary number for viscosity ratio of unity The dimensionless time is defined as 𝑡 ∗ = 𝑡𝜀̇ In Fig 4.7, the dimensionless characteristic time for droplet reaching steady state is independent from capillary number However, in the present research of the simulation and the proposed theoretical model, it is found that viscosity ratio strongly affect the dimensionless characteristic time 6831ef11 25 ...Ph.D Thesis Modeling and simulation of droplet dynamics for microfluidic applications Graduate School of Yeungnam University Department of Mechanical Engineering Major in... Van Thanh Hoang Advisor: Professor Jang Min Park, Ph.D August 2019 Ph.D Thesis Modeling and simulation of droplet dynamics for microfluidic applications Advisor: Professor Jang Min Park, Ph.D... three-dimensional numerical simulation and theoretical modeling In droplet- based microfluidic systems, dynamics of droplet in microfluidic systems is determined by the strength of the flow type which