微管道氣體流一向是熱門的研究議題，因此情境於微機電、真空技術和頁岩氣工程等多個領域都可能出現。本研究提出了一種新穎之非動力學方法用以分析管通道中的氣流之行為，此新提出之方法對慢速至快速之氣流均適用，且對連續流一直到自由分子流之廣泛稀薄度範圍亦均適用。

在1990年代，前人己對具有滑移邊界及可壓縮性之微管道氣流之泊肅葉流(Poiseuille flow)模型求得解析解，此方法於給定條件下可估計質量流率，從而與實驗結果相印證。然此方法需忽略統御方程式中的一個動量對流項方易於求解，該項於流速低(約低於0.15馬赫)時並不重要，而這意味著前人之解析解不適用於流速高之情況。援此，本文所提出之非動力學方法可將此動量對流項納入解算，本方法使用微通道氣流的平均壓力特性將 Navier-Stokes 方程從多維偏微分方程簡化為常微分方程，並使用數值射擊法(shooting method)可對簡化之常微分方程求解。我們將此求解法命名為簡化準二維方法(Simplified quasi-2D method, SQ2D 方法)，因此法可在平行流場之特定位置對流道中垂直流場之截面求解，故以準二維為名。SQ2D 方法對出口處速流尚未達到次音速扼流(subsonic choked flow)之條件下，約為出口處馬赫數低於0.47時，均可適用，且可同時考量一階滑移邊界條件，有助吾人解析高流速且需考量滑移效應之微管道氣體流。

此外，本研究以非平衡態狀態之機率分布函數為基礎，將諸多稀薄氣體之物理特性以距壁面距離之方程式描述，例如用於修正黏滯係數與擴散係數之調整函數，同時也推導出描述氣體壁到壁碰撞之機率方程式。本研究不僅整合了前人提出用於描述稀薄效應之方程式，而且還提出了幾項新見解並將之公式化，例如考量了氣體粒子從壁面反射如何影響非線性調整函數，以及如何將壁面碰撞現象納入滑移邊界條件等。以上這些用於描述稀薄效應之方程式，雖會讓微管道氣流的統馭方程式更複雜，然以 SQ2D 方法，仍能對之求解並克服挑戰。

綜上，本文所提出之解析方法可將 Navier-Stokes 方程的適用性擴展到自由分子流動範圍，據此對2D 微管道中泊肅葉流和庫埃特流(Couette flow)問題解算，與前人之實驗數據及動力學方法之數值模擬結果，在紐森數(Knudsen' number, Kn)低至趨近於0、高至超過Kn=100 之間均可穩合。整體而言，本研究提出之非動力學方法提供了一種更直觀、更易於理解的模式來分析微管道中的氣體流動，對稀薄氣體流動物理行為之觀察與描述亦有新見解，希能對此課題研究有所助益。

Microchannel gas flow has been an attractive research topic due to its significance in domains like micro-electro-mechanical systems (MEMS), vacuum technology, and shale gas engineering. This study introduces an innovative non-kinetic approach for assessing gas flow behavior within microchannels, applicable across slow to fast flows and encompassing a broad range of rarefaction, from hydrodynamic to free-molecular flow regimes.

In the 1990s, a Poiseuille flow model with slip boundary conditions and fluid compressibility was analytically solved, estimating mass flow rates under specific conditions and corroborating experimental outcomes. Yet, this method omitted a momentum-convection term in the governing equation, which was deemed insignificant at low velocities (approximately below 0.15 Mach), rendering it inapplicable to high-velocity scenarios.

The proposed non-kinetic approach simplifies the Navier-Stokes equation from a multi-dimensional partial differential equation into an ordinary one using the microchannel gas flow's mean pressure property. It's possible to incorporate the term related to momentum convection, and the reduced equation can be solved using numerical shooting. This technique is called the simplified quasi-2D method since this method can solve the flow field perpendicular to the streamwise direction at a specific streamwise position, abbreviated as the SQ2D method.

Using the SQ2D method, the properties of Poiseuille flow, especially the influence of flow speed and the slip boundary conditions, can be further investigated, which is applicable till the outlet Mach number is up to about 0.47, i.e., before a subsonic choke happens. This helps us understand the microchannel gas flow with significant flow velocity better.

In addition, this study formulated the non-equilibrium effects as wall-distance equations based on the probability distribution function, such as the scaling functions to correct viscosity and diffusivity for highly rarefied flow. An equation for calculating the gas's wall-to-wall collision ratio was also derived and incorporated into the proposed method. This work not only integrates earlier formulations concerning rarefication effects but also introduces novel insights, including the impact of gas molecules' specular reflections on non-linear scaling functions and the incorporation of wall-to-wall collision phenomena into slip boundary conditions. While these equations might challenge traditional analytical solutions, they can be solved seamlessly with the SQ2D method.

The proposed non-kinetic technique extends the applicability of the Navier-Stokes equations to the free-molecular flow regime. By employing this approach, the study successfully solves problems of Poiseuille flow and Couette flow in 2D microchannels, aligning closely with experimental data and numerically simulated results between the Knudsen number close to 0 to beyond 100, capable of accurately representing rarefied gas flow behavior.

Overall, the non-kinetic method proposed in this study offers a more intuitive and easier-to-understand approach to analyzing gas flow in microchannels and a valuable understanding of the physical characteristics of the flow of rarefied gases.

Abstract (Chinese) I
Acknowledgements (Chinese) III
Abstract IV
Acknowledgements VI
Contents VII
List of Figures X
List of Tables XII
Nomenclature and subscripts 1
1 Introduction 3
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.2 Literature for flows within early transitional regime . . . . . . . 7
1.2.3 Literature of diverse rarefaction effects . . . . . . . . . . . . . 9
1.3 Clarification of the use of previous publications . . . . . . . . . . . . . 13
VII
2 Background Theory 15
2.1 The Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Slow gas flow in microchannel within slip regime . . . . . . . . . . . . 19
2.3 The friction factor for compressible internal flow . . . . . . . . . . . . 23
3 Gas Flow In Microchannels/Microtubes With Considerable Velocity 26
3.1 The simplified quasi-2D (SQ2D) method . . . . . . . . . . . . . . . . . 26
3.1.1 Non-dimensional analysis . . . . . . . . . . . . . . . . . . . . 26
3.1.2 The mean-pressure-and-pressure-gradient property and the nu-
merical method . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2 Validation of the SQ2D method . . . . . . . . . . . . . . . . . . . . . . 34
3.2.1 Validation with analytical solutions and CFD methods . . . . . 34
3.2.2 Validation with previous studies via Poiseuille number . . . . . 38
3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4 Integrated Multiple Physical Properties For Highly Rarefied Gas Flow 49
4.1 Considerations from previous studies . . . . . . . . . . . . . . . . . . . 49
4.1.1 The effective mean free path . . . . . . . . . . . . . . . . . . . 51
4.1.2 The diffusion velocity . . . . . . . . . . . . . . . . . . . . . . 58
4.2 The wall-to-wall collision ratio . . . . . . . . . . . . . . . . . . . . . . 61
4.3 Integrating multiple physical properties for rarefied gas flow . . . . . . 67
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5 The Modified Slip Boundary Condition and The Modified Effective Mean
Free Path 74
5.1 The wall-to-wall slip boundary condition . . . . . . . . . . . . . . . . . 74
5.2 The modified effective mean free path . . . . . . . . . . . . . . . . . . 79
5.3 Validation for the Poiseuille flow case . . . . . . . . . . . . . . . . . . 85
5.4 Validation for the Couette flow case . . . . . . . . . . . . . . . . . . . 91
VIII
5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6 Conclusions 99
Bibliography 102

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