臺灣博碩士論文加值系統

In this thesis, a lattice Boltzmann multiphase fluid model capable to deal with large density ratio and with partial wetting surface is introduced. Multiphase fluid behavior can be simulated using the the Navier-Stokes equation coupling with convective Cahn-Hilliard equation [1], where the latter equation is used to capture the interface of multiphase fluid in terms of chemical potential. By adopting the multiphase model and adding an additional
term describing surface energy, a droplet rests on a surface with given contact angle can be computed.

The capability of the present model, which is to compute two phase flows 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. Besides, phenomena caused by different wetting condition of the surface are presented in this thesis, such as droplet on heterogeneous surface and transient response of a droplet with changing wettability.

The present model is applied to simulate liquid lens cases. The simulation result shows good compatibility with the experiment done by Hsieh et al [58], where Bond number is about $0.05$. Further improvement of the lens fabrication technique is suggested with observations in the present work. Optical properties of the lens such as back focal length of the liquid lens can be changed by controlling Bond number to reach wider FOV or longer BFL, which are
important specifications in optics.

1 Introduction 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Lattice Boltzmann method . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Multiphase and multicomponent
uid 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 . . . . . . . 6
1.2.3 Wettability control on the solid surface . . . . . 8
1.3 Electrowetting and its applications . . . . . . . . 10
1.4 Objective and motivation . . . . . . . . . . . . . . 11
2 Theory and governing equations 12
2.1 The Boltzmann equation . . . . . . . . . . . . . . . 12
2.2 The BGK and the low-Mach-number approximation . . . 13
2.2.1 The BGK approximation . . . . . . . . . . . . . . 13
2.2.2 The low-Mach-number approximation. . . . . . . . . 15
2.3 Discretization of the Boltzmann equation . . . . . . 16
2.3.1 Discretization of phase space . . . . . . . . . . 16
2.3.2 Dicretization of time . . . . . . . . . . . . . . 18
2.4 The free-energy model . . . . . . . . . . . . . . . 19
2.4.1 The free-energy function . . . . . . . . . . . . . 19
2.4.2 Analytical solution of interface profile . . . . . 20
2.5 A Lattice Boltzmann model for multiphase
flows with large densityratio . . . . . . . . . . . 21
2.5.1 The governing equation . . . . . . . . . . . . . . 21
2.5.2 Lattice Boltzmann equaiton . . . . . . . . . . . . 22
2.5.3 Interface capturing equation . . . . . . . . . . . 23
2.6 Non-dimensionalized governing equations . . . . . . 24
2.7 Wetting theory . . . . . . . . . . . . . . . . . . . 25
2.7.1 Basic aspect of wetting . . . . . . . . . . . . . 25
2.7.2 Electrowetting theory . . . . . . . . . . . . . . 26
2.7.3 Wettability Modelling . . . . . . . . . . . . . . 27
3 Numerical algorithm 29
3.1 Simulation procedure . . . . . . . . . . . . . . . . 29
3.2 Boundary conditions for the computational domain . . 30
3.2.1 Velocity boundary conditions for solving
fluid flow . . . . . . . . . . . . . . . . . . . . 30
3.2.2 Bounce back boundary conditions for solving
interface capture equation . . . . . . . . . . . . 32
3.3 Partial wetting condition (PWC). . . . . . . . . . . 32
4 Numerical results 34
4.1 Numerical validation . . . . . . . . . . . . . . . . 34
4.1.1 Interface profile . . . . . . . . . . . . . . . . 34
4.1.2 Wettability control. . . . . . . . . . . . . . . . 37
4.2 Effect of Bond number. . . . . . . . . . . . . . . . 39
4.3 Droplet on a heterogeneous surface . . . . . . . . . 43
4.4 Transient response of a droplet . . . . . . . . . . 45
4.5 Applications:
Lens fabrication technique by electrowetting . . . . 51
5 Conclusions 63

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