三維晶格波茲曼法結合親/疏水性邊界模擬液氣兩相流之流場

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研究生：	吳承隆
論文名稱：	三維晶格波茲曼法結合親/疏水性邊界模擬液氣兩相流之流場 3D Lattice Boltzmann Simulations of Incompressible Liquid-Gas system on Partial Wetting Surface
指導教授：	林昭安
口試委員:
學位類別：	碩士 Master
系所名稱：	工學院 - 動力機械工程學系 Department of Power Mechanical Engineering
論文出版年：	2010
畢業學年度：	98
語文別：	英文
論文頁數：	62
中文關鍵詞：	三維、晶格波茲曼、高密度比、兩相流、親水性、疏水性、液滴
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In this thesis, a 3D lattice Boltzmann multiphase fluid model based on D3Q19 which is capable of dealing two phase flow with large density ratio and partial wetting surface with a given contact angle is investigated. Multiphase fluid behavior can be simulated using the Navier-Stokes
equation coupling with convective Cahn-Hilliard equation [35], where the latter equation is used to capture the interface of multiphase fluid in terms of chemical potential. By adopting the free-energy model and adding an additional term into the total free energy to describe surface energy, a droplet rests on a surface with given contact angle can be simulated.
The capability of the present model, which is to compute two phase flow with wettability controllable surface, is validated by simulating a droplet that rests on a surface with given contact angle. The contact angles calculated from the results agree with theory. The effect of gravitational force on droplet shape is also discussed in terms of Bond number.
The present model is applied to simulate liquid lens cases. The simulation result shows good compatibility with the experiment done by Hsieh et al. [70], where Bond number is about 1.3.

1 Introduction                                                                                                                                                 1
     1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
            1.1.1 Lattice Boltzmann method  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
            1.1.2 Multiphase and multicomponent fluid systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
            1.1.3 Partial wetting boundary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
     1.2 Literature survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
            1.2.1 Lattice Boltzmann multiphase fluid model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
            1.2.2 Lattice Boltzmann multiphase fluid model with large density ratio . . . . . . . . . . . . . . 7
            1.2.3 Wettability control on the solid surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
     1.3 Electrowetting and its applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
     1.4 Droplet manipulating mechanism and its applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
     1.5 Objective and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 Theory and governing equations                                                                                                          14
     2.1 The Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
     2.2 The BGK and the low-Mach-number approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
            2.2.1 The BGK approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
            2.2.2 The low-Mach-number approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
     2.3 Discretization of the Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
            2.3.1 Discretization of phase space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18
            2.3.2 Dicretization of time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
     2.4 The free-energy model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
            2.4.1 The free-energy function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
            2.4.2 Analytical solution of interface profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
     2.5 A Lattice Boltzmann model for multiphase flows with large density ratio . . . . . . . . . . . . . 23
            2.5.1 The governing equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23
            2.5.2 Lattice Boltzmann Equaiton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
            2.5.3 Interface capturing equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25
     2.6 Non-dimensionalized governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
     2.7 Wetting theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28
            2.7.1 Basic aspect of wetting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
            2.7.2 Electrowetting theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
            2.7.3 Wettability Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3 Numerical algorithm                                                                                                                                    32
     3.1 Simulation procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
     3.2 Boundary conditions for the computational domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
            3.2.1 Velocity boundary conditions for solving fluid flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
            3.2.2 Bounce back boundary conditions for solving interface capturing equation . . . . . 37
     3.3 Treatment on the gradient of the order parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
            3.3.1 Gradient of the order parameter: Inner points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
            3.3.2 Gradient of the order parameter: Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
     3.4 Treatment on the laplace order parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
            3.4.1 Laplace order parameter: Inner points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
            3.4.2 Laplace order parameter: Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4 Numerical results                                                                                                                                          41
     4.1 Wettability control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
     4.2 Effect of Bond number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
     4.3 Applications: Lens fabrication technique by electrowetting . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5 Conclusions and Future Works                                                                                                                52
     5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
     5.2 Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

                                

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