臺灣博碩士論文加值系統

本研究利用線性與非線性理論，探討當流體薄膜沿直立旋轉圓柱之外、內表面流下時，流動系統穩定性之現象。當以線性模式探討流動狀況時，發現不足以顯現出整個流動特性，因此，推導出廣義非線性運動模式以代表整個物理系統。本文利用長波微擾法對於所得到之自由面運動方程式進行之研究，以探討流體薄膜於穩態平衡時，受微小擾動後所表現之穩定性，首先為利用正模分析法來探討液膜之線性穩定性，進而得出線性中立穩定曲線、線性振幅增長率及線性波速。其次應用時間和空間之多重尺度法研究液膜的非線性穩定性，並得知Ginzburg-Landau 型方程式為解存在之必要條件，利用此方程式可研究液膜表面波之非線性行為，並定出無條件穩定、亞臨界不穩定、超臨界穩定及超臨界爆炸解研究之內容，主要探討代表旋轉效應之旋轉數(Rotation number)及圓柱半徑大小對流動系統穩定性的影響，選擇之流體為牛頓流體、黏彈流體及微極流體，最後探討具有相變化之問題，為簡化分析之參數，研究非牛頓流體時僅選擇代表性之流體，其中黏彈性流體之黏彈性參數為k =0.05、微極流體之微極參數為K =1。
研究結果發現，旋轉數及圓柱半徑大小對於四種流動系統之穩定性具有相同之影響，現將結論歸納如下：
（一） 圓柱半徑大小對流動系統穩定性的影響
半徑越大流動系統越穩定，反之流動系統不穩定。造成此種現象的原因為，圓柱表面波之波峰半徑較波谷半徑大，且表面張力將會於較小曲率半徑處產生較大之毛細壓力(capillary pressure)，因此毛細力會將流體從波谷推向波峰，導致波幅增加，故較小半徑之圓柱將增加對流之流動和表面波之波動，即干擾後之振幅受到曲率之影響而較易增長。隨半徑值減小，流場強度愈大，系統更加不穩定。
（二） 液膜沿相同半徑圓柱內、外表面流下之穩定性分析
當研究液膜沿相同半徑圓柱之內、外表面流下時，發現由於所推導出之管外流自由面運動方程式，其特徵長度為圓柱半徑加上液膜厚度(R+h)，較管內流之圓柱半徑減掉液膜厚度(R-h)為大，因此根據結論(一)可知，於相同半徑之條件下，管外流較管內流穩定。
（三） 液膜沿旋轉圓柱外表面流下之穩定性分析
當討論相同圓柱半徑，但是不同旋轉速度下之情形時，發現旋轉會使得具較大半徑之波峰比較小半徑之波谷承受更大之離心力，因此導致流動系統不穩定，此不穩定之現象隨旋轉速度增加更加明顯。而當討論相同旋轉速度，但是不同圓柱半徑時，必須同時考慮代表穩定因素之圓柱半徑與不穩定因素之離心力間的交互影響，於低雷諾數區域內，圓柱半徑對於穩定性有較高之貢獻度，隨著雷諾數增加，離心力之影響漸漸大過圓柱半徑，因此原先具有較高穩定性之大半徑圓柱，會因為離心力之作用導致穩定性大大降低。
（四）液膜沿旋轉圓柱內表面流下之穩定性分析
當討論相同圓柱半徑，但是不同旋轉速度下之情形時，發現旋轉所產生之離心力，使得流體具有一個向外並貼附在圓柱內表面上的力量，導致流動系統穩定，此穩定之現象隨旋轉速度增加更加明顯。因此圓柱半徑及旋轉均扮演穩定流場之角色。

The paper presents both the linear and nonlinear stability theories for characterization of film flows down on the outer and inner surface of a rotating cylinder. After showing the insufficiency of the linear model in charactering certain flow behaviors, a generalized nonlinear kinematic model is then derived to represent the physical system. The long-wave perturbation method is employed to explore the stability of the steady state flow system, which is subject to minute disturbance, by studying the derived evolution equations for
interfacial waves. In the first step, the normal mode method is used to characterize the linear behaviors. The threshold conditions, the linear growth rate of the amplitudes and the linear wave speeds are obtained subsequently as the by-products of linear solution. In the second step, an elaborated nonlinear film flow model is solved by using the method of multiple scales to characterize flow behaviors. It is shown that the necessary condition for the existence of such a solution us governed by the Ginzburg-Landau equation. The various states of sub-critical stability, sub-critical instability, supercritical stability, and supercritical explosion are obtained from the nonlinear analysis.
The purpose of this study is to discuss the effect of rotating parameter, Rotation number, and size of cylinder on the stability of the flow system. The
chosen fluids are the Newtonian fluid, the viscoelastic fluid and the micropolar fluid. And at last, the problem of phase transformation is taken into consideration. For simplifying the problem, the representative fluid is selected. i.e. the viscoelastic fluid with the viscoelastic parameter k=0.05, the micropolar fluid with micropolar parameter K=1.
The results of this study are found that the effect of Rotation number and the size of cylinder on the four different flow systems are the same. And the
conclusions are summarized and drawn as following:
(1) The effect of cylinder size on the stability:
The larger the cylinder size and the more the stability is. Therefore, the curvature has a destabilizing effect. This destabilizing effect occurs because the radius of the trough of wave have a smaller value than that at the crest of waves, and the surface tension will produce large capillary pressure at the smaller radius of curvature. This induces the capillary pressure force tending to move
the fluid trough to crest, thus increasing the amplitude of wave.
(2) The stability of film flows down on the inner and outer surface of cylinder with the same radius:
When studying the film falling down on the outer and inner surface of the cylinder with the same radius, the characteristic length of external flow, R+h, is
larger than the internal flow, R-h. According to the conclusion (1), the external flow is more stable than the internal flow under the condition of the same cylinder radius.
(3) The stability of film flows down on the outer surface of rotating cylinder:
When discussing the same cylinder size but subject to different rotating speed, the induced centrifugal force is found. Its direction is toward the positive
radius axis and serving as the destabilizing factor. This is because the crests of wave feel a larger centrifugal force than the troughs, which have smaller radius. The flow system will become less stable as the rotation speed increased. When studying the same rotating speed but subject to different cylinder size, the interaction between the stabilizing factor, radius, and the destabilizing factor, rotation, should be considered. In the low Reynolds number region, the contribution of radius to the stability is higher. As the Reynolds number is gradually increased, the centrifugal force will destroy the stability drastically.
(4) The stability of film flows down on the inner surface of rotating cylinder:
When discussing the same cylinder size but subject to different rotating speed, the induced centrifugal force is found to let the fluid sticking to the wall
of cylinder. Therefore, the centrifugal force serves as the stabilizing factor. The faster the rotation speed and the more the stability is. In this case, the cylinder size and the rotation all play important role in stabilizing the flow system.

中文提要................................................................................................................I
英文提要..............................................................................................................IV
誌謝......................................................................................................................VII
目錄.....................................................................................................................VIII
表目錄.................................................................................................................XIII
圖目錄.................................................................................................................XIV
符號...............................................................................................................XXXIV
第一章 緒論............................................................................................................1
1-1 研究動機............................................................................................................1
1-2 研究目的............................................................................................................4
1-3 研究方法...........................................................................................................5
1-4 論文架構...........................................................................................................7

第二章 文獻回顧..............................................................................................11
2-1自由面演展方程式之推導............................................................................11
2-2液動穩定性分析..............................................................................................13
2-3 薄膜流穩定性分析之文獻回顧.....................................................................15
第三章 牛頓流體沿旋轉垂直圓柱內外表面流下之非線性穩定分析..........19
3-1統御方程式與邊界條件..................................................................................19
3-1-1牛頓流體薄膜沿旋轉圓柱外表面流下..............................................20
3-1-2牛頓流體薄膜沿旋轉圓柱內表面流下............................................24
3-2牛頓流體薄膜流之長波微擾解.....................................................................25
3-2-1牛頓流體薄膜沿旋轉圓柱外表面流下之長波微擾解......................25
3-2-2牛頓流體薄膜沿旋轉圓柱內表面流下之長波微擾解......................29
3-3牛頓流體薄膜流之線性與非線性穩定性分析............................................30
3-3-1牛頓流體薄膜沿旋轉圓柱外表面流下................................................31
3-3-2牛頓流體薄膜沿旋轉圓柱內表面流下................................................37
第四章 黏彈性薄膜沿旋轉垂直圓柱內外表面流下之非線性穩定分析........64
4-1統御方程式與邊界條件....................................................................................65
4-1-1黏彈流體薄膜沿旋轉圓柱外表面流下.................................................67
4-1-2黏彈流體薄膜沿旋轉圓柱內表面流下...................................................71
4-2黏彈流體薄膜流之長波微擾解.......................................................................72
4-2-1黏彈流體薄膜沿旋轉圓柱外表面流下之長波微擾解.........................72
4-2-2黏彈流體薄膜沿旋轉圓柱內表面流下之長波微擾解.........................76
4-3黏彈流體薄膜流之線性與非線性穩定性分析................................................78
4-3-1黏彈流體薄膜沿旋轉圓柱外表面流下....................................................78
4-3-2黏彈流體薄膜沿旋轉圓柱內表面流下....................................................85
第五章 微極薄膜沿旋轉垂直圓柱內外表面流下之非線性穩定分析….........114
5-1統御方程式與邊界條件......................................................................................115
5-1-1微極流體薄膜沿旋轉圓柱外表面流下...................................................116
5-1-2微極流體薄膜沿旋轉圓柱內表面流下...................................................120
5-2 微極薄膜流之長波微擾解................................................................................122
5-2-1微極流體薄膜沿旋轉圓柱外表面流下之長波微擾解...........................122
5-2-2微極流體薄膜沿旋轉圓柱內表面流下之長波微擾解...........................127
5-3微極流體薄膜流之線性與非線性穩定性分析................................................129
5-3-1微極流體薄膜沿旋轉圓柱外表面流下.....................................................129
5-3-2微極流體薄膜沿旋轉圓柱內表面流下......................................................135
第六章 凝結薄膜流沿旋轉垂直圓柱內外表面流下之非線性穩定分析..........163
6-1統御方程式與邊界條件.........................................................................................164
6-1-1凝結薄膜流沿旋轉圓柱外表面流下........................................................164
6-1-2凝結薄膜流沿旋轉圓柱內表面流下........................................................168
6-2 凝結薄膜流之長波微擾解..................................................................................169
6-2-1凝結薄膜流沿旋轉圓柱外表面流下之長波微擾解.................................170
6-2-2凝結薄膜流沿旋轉圓柱內表面流下之長波微擾解.................................176
6-3凝結薄膜流之線性與非線性穩定性分析...........................................................179
6-3-1凝結薄膜流沿旋轉圓柱外表面流下..........................................................179
6-3-2凝結薄膜流沿旋轉圓柱內表面流下..........................................................184
第七章 結論與建議.......................................................................................................207
7-1 結論.......................................................................................................................207
7-2 建議......................................................................................................................209
參考文獻.........................................................................................................................211
附錄A：自由面演展方程式之穩定性分析理論.....................................................221
A-1 基本流動................................................................................................................221
A-2 線性液膜穩定性分析..........................................................................................222
A-3 非線性液膜穩定性分析......................................................................................223
附錄B：直立圓柱上薄膜流自由面邊界條件之推導.............................................231
附錄C：高斯曲率半徑.................................................................................................237
附錄D：直立圓柱上薄膜流之穩態平衡解..............................................................240
自述..................................................................................................................................242
歷年發表著作.................................................................................................................243

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