臺灣博碩士論文加值系統

本論文改善及平行化一有限差分的高階數值方法來求解非正交座標複雜外型的PNP-NS方程組，此方程組包含了描述外加電場電位勢的Laplace方程、描述壁面電位勢的Poisson方程、描述離子濃度分布的Nernst-Planck方程以及由庫倫力驅動的不可壓縮Navier-Stokes方程組。論文內容主要描述因電位差開啟的各種不同邊界外型離子通道，其內部流體經由通道留出至外部的物理行為，以及模擬完整細胞膜上多離子通道的傳輸行為。

Abstract
In this study, a high-order numerical method of finite difference is improved and parallelized to solve the PNP-NS equations of complex external shape with non-orthogonal coordinates. The equations under investigation includes the Laplace equation for potential of the applied electric field and the Poisson equation describing the potential of the wall potential. The Nernst-Planck equation describing the ion concentration distribution and the incompressible Navier-Stokes equations for the Coulomb force. The content of the paper mainly describes the various boundary-shaped ion channels opened by the potential difference, the physical behavior of the internal fluid leaving the channel through the channel, and the transmission behavior of the multi-ion channel on the intact cell membrane.

Table of Contents i
List of Figures v
中文摘要 vii
Abstract viii
Explanation of Symbols ix
Chapter 1. Introduction 1
1.1 Preface 1
1.2 Research motivation 2
1.3 Literature Review 4
1.4 Thesis outline 8
Chapter 2.Theoretical Background 9
2.1 Introduction to Cells and Cell Structure 9
2.2 Introduction of Cell membrane 11
2.3 Introduction of Transphospholipid bilayer transport 15
2.4 Introduction of Ion channels 16
2.5 Introduction of Action potential 17
Chapter 3.The Establishment of Mathematical Models 20
3.1 Basic assumptions 20
3.2 The governing equation 21
3.2.1 Laplace equation describing the potential of the applied electric field 21
3.2.2 Poisson equation describing wall electric potential 22
3.2.3 Nernst-Planck equation describing positive and negative ion distribution 23
3.2.4 Navier-Stokes equation for incompressible viscous flow 24
3.3 Coordinate Transformation Equation 25
Chapter 4.Numerical Model 29
4.1 Discrete Time 30
4.2 Discrete Space 31
4.3 Discreteness of the pressure equation 36
4.4 Numerical verification 38
4.5 Calculation procedures 40
4.6 Parallelization 42
Chapter 5.Numerical Simulation of Ion Channels 45
5.1 Description of the problem 45
5.1.1 Parameter setting 46
5.2 Flow field analysis of rectangular non-periodic boundary 46
5.2.1 Initial and boundary conditions of the calculation model 46
5.3 Flow field analysis of rectangular periodic boundary 50
5.3.1 Initial and boundary conditions of the calculation model 50
5.4 Flow field analysis of sector-shaped periodic boundary conditions 52
5.4.1 Initial and boundary conditions of the calculation model 53
5.5 Discussion 55
Chapter 6.Conclusion 64
6.1 Research results and discussion 64
6.2 Future work and outlook 66
References 67

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