Stability Analysis of Thin Liquid Film in a Micro-Pipette

本文旨在透過理論分析探討當蒸氣或氣體流經一微管時，黏附在微管內壁薄液膜因受到氣液兩相基本流場、氣液界面的滑動邊界條件、液膜厚度以及界面變形和表面張力等因素的影響，產生的不穩定現象。此現象統稱為界面不穩定性。線性穩定性分析的方法將被本文所採用，主要的分析步驟為：(1) 以解析方式求得氣、液兩相的穩態基本解，(2) 引進微擾量，並推導線性微擾方程式及對應的邊界條件，(3) 無因次化以求得相關的無因次參數，(4) 透過normal mode的分析方法，將擾動方程式化為eigenvalue problem，(5) 以解析方式探討液膜的長波不穩定性，(6) 以數值計算分析液膜不穩定性發生的條件及物理機制。研究的結果發現：(1) 氣液兩相的基本流場、液膜厚度以及滑動邊界條件確實會影響液膜不穩定現象的發生，(2) 由長波不穩定性的分析可知氣液兩相的基本流場以及滑動邊界條件均會使得液膜從stationary mode變成為oscillatory mode，但不影響不穩定發生的波數 (wavenumber)，(3) 數值計算不僅印證了長波不穩定性分析的解析結果，更發現基本流場，即雷諾數的效應，會使得不穩定現象發生的擾動波的波數範圍增大，但不穩定現象的最大成長速率 (maximum growth rate) 則因流場的慣性效應相對減緩，(4) 使得液膜發生不穩定的擾動波的波長和液膜厚度成正比關係，(5) 滑動邊界條件使得發生不穩定現象的波數有所改變，界於一般無滑動邊界條件 (no-slip boundary condition) 和靜止液膜發生不穩定的波數之間，同時也改變了液膜振盪不穩定 (oscillatory mode) 發生的頻率。The present study is aimed at investigating theoretically the instability of the thin liquid film attached to the inside wall of a micro-pipette through which there exists a steady vapor or gaseous flow. Such an instability, usually called “interfacial instability”, is mainly due to the interfacial deformation and surface tension effect accompanied with the liquid-vapor flow, slip boundary condition at the liquid-vapor interface, liquid film thickness, etc.The linear stability analysis is employed to accomplish the task. The main procedures of analysis are described briefly as follows: (1) Obtain the basic solutions for the liquid-vapor flow analytically. (2) Derive the disturbance equations and the related boundary conditions for small disturbances. (3) Obtain the dimensionless parameters by introducing appropriate non-dimensionalization scheme. (4) With the aid of normal mode analysis, the disturbance equations is reduced to an eigenvalue problem. (5) Solve the long-wave instability analytically. (6) Obtain the onset conditions for the interfacial instability of the thin liquid film with numerical computations and discuss the physical mechanism.The results indicate that the interfacial deformation, basic liquid-vapor flows, slip boundary condition, and liquid film thickness really affect the interfacial instability of the thin liquid film. In the long-wave instabilities, although the liquid-vapor flow and the interfacial slip boundary condition do not change the onset wavenumber, they make the instability shift from the stationary mode to the oscillatory mode. The numerical computation not only demonstrates such a phenomenon but also finds that, in general situations, the basic liquid-vapor flow, or Reynolds number effect, will increase the range of the onset wavenumber of instability. However, the maximum growth rate of instability decreases due to the inertia effect. In addition, it is also found that the wavelength of the unstable disturbances depends proportionally on the film thickness. The interfacial slip boundary condition also changes the onset wavenumber, varying between the one with no-slip boundary condition and that for a stationary liquid film, the classical problem. The frequency of the oscillatory mode of instability has been changed as well.Table of Contents文摘要 IBSTRACT IIABLE OF CONTENTS IVIGURE CAPTIONS VIABLE CAPTIONS VIIOMENCLATURE VIIIHAPTER ONE 1NTRODUCTION 1-1 Two-Phase Flows 1-2 The Micro-Channel Flows 2-3 The Slip Boundary Conditions 6-4 Literature Review 11-5 Scope of Present Study 13HAPTER TWO ASIC SOLUTIONS FOR CO-AXIAL VAPOR-THIN-LIQUID-FILM FLOWS 15-1 Assumptions and Governing Equations 16-2 Basic Solutions for Flows with No-Slip Boundary Conditions 17-3 Basic Solutions for Flows with Slip Boundary Conditions 19-4 Results and Discussion 21HAPTER THREE ISTURBANCE EQUATIONS AND BOUNDARY CONDITIONS 25-1 Linearized Equations for Small Disturbances 26-2 Linearized Boundary Conditions for Small Disturbances 29-3 Normal Mode Analysis 33-4 Negligence of Disturbances in Vapor Phase 35HAPTER FOUR ONG-WAVE STABILITY ANALYSIS 37-1 Long-wave Stability Analysis 37-2 Results and Discussion 40HAPTER FIVE INEAR STABILITY ANALYSIS WITH NUMERICAL COMPUTATIONS 42-1 The Numerical Formulation 42-1-1 The Reduced Disturbance Equation and Boundary Conditions in terms of 42-1-2 The Numerical Formulation 44-2 The Numerical Computation 47-3 Results and discussion 49-3-1 Justification of Numerical Method 49-3-2 Instability of the Thin Liquid Film with Basic Flows 50HAPTER SIX 58ONCLUSIONS AND FUTURE WORK 58EFERENCES 6