在這篇論文中，我們考察亞音速流體流進流出一給定的有限長管道的問題，目的在於找到管道入口和出口處內蘊的（物理上可接受的）邊界條件。我們首先刻劃一組物理邊界條件，借此可以確定長方形管道中亞音速無旋流的存在性和唯一性。在給定管道入口處的流量及水平來流方向和出口處的適當壓力條件下，存在兩個正的常數m₀和m₁(m₀ < m₁)，使得當流量m ∈ [m₀,m₁)，在長方形管道中存在唯一的亞音速無旋流。流體的水平速度是恒正的，並且當流量m趨於m₁時，流體的最大速度也將趨於音速。問題的困難來自于由流量蘊含的非局部項和出口處的壓力條件。我們首先引入伯努利常數作參數的附屬問題来處理非局部项，然後建立流量和伯努利常數之間的單調關係。為處理壓力條件引起的斜導數缺失問題，我們利用角速度和壓力來重新描述這一問題。我們利用Moser迭代來得到角速度的最大模估計，以確保流體的水平速度恒為正。 / 我們接下來考察彎曲管道中一般來流方向和管道壁的幾何結構對亞音速無旋流的影響。我們發現來流方向和管道壁的傾斜角度和出口壓力起著相似的作用。管道壁的曲率也起著很重要的作用。我們的結果也可以推廣到二維亞音速歐拉流和三維軸對稱亞音速歐拉流的情形。 / 接下來我們考慮三維有限長管道中的亞音速歐拉流的情形，這是最有趣和最困難的情形，也是論文的核心部份。我們在二維管道中給定的邊界條件在三維有一個自然的推廣。這些重要的提示對我們尋求歐拉方程組新的分解及借此理解其中的雙曲與橢圓耦合的結構是至關重要的。我們的新的分解的關鍵想法在於利用伯努利定律來約化速度場。具體的做法是通過定義新的變量 [附圖] 及通過 [附圖] ，利用伯努利函數B 來代替u₁。這樣我們可以更深入地挖掘伯努利定律的作用，期望借此可以稍微簡化一下複雜的歐拉方程組。對伯努利函數為常數的流體，我們找到了一個新的守恆量，這跟二維的約化的旋度的情形相似。讓人驚奇的是，我們還可以找到一組新的守恆律，這一情況即使在二維也從未被人注意到。我們利用這一分解來證明長方體管道中靠近某些特殊亞音速流并滿足給定的入口處的來流方向及伯努利函數和出口處的壓力條件的亞音速歐拉流的存在性和唯一性。同樣的想法可以用於不可壓歐拉方程組、自相似歐拉方程組、帶阻尼項的歐拉方程組、定常歐拉泊松方程組和定常的歐拉麥克斯韋方程組。 / 最後我們考慮歐拉泊松方程組某些定常亞音速解的結構穩定性。如果帶亞音速背景電荷的背景解的馬赫數和電場都比較小的話，那麼背景解關於背景電荷、來流方向、伯努利函數、出口壓力的小撓動是結構穩定的。在我們的數學分析中新的元素在於求解帶斜導數邊界條件和Dirichlet邊界條件的混合型的二階強耦合的橢圓型方程組。 / In this thesis, we investigate an inflow-outflow problem for subsonic gas flows in a nozzle with finite length, aiming at finding intrinsic (physically acceptable) boundary conditions on upstream and downstream. We first characterize a set of physical boundary conditions that ensure the existence and uniqueness of a subsonic irrotational flow in a rectangle. Our results show that suppose we prescribe the horizontal incoming flow angle at the inlet and an appropriate pressure at the exit, there exists two positive constants m₀ and m₁ with m₀ < m₁, such that a global subsonic irrotational flow exists uniquely in the nozzle, provided that the incoming mass flux m ∈ [m₀,m₁). The maximum speed will approach the sonic speed as the mass flux m tends to m₁. The new difficulties arise from the nonlocal term involved in the mass flux and the pressure condition at the exit. We first introduce an auxiliary problem with the Bernoulli’s constant as a parameter to localize the nonlocal term and then establish a monotonic relation between the mass flux and the Bernoulli’s constant to recover the original problem. To deal with the loss of obliqueness induced by the pressure condition at the exit, we employ the formulation in terms of the angular velocity and the density. A Moser iteration is applied to obtain the L∞ estimate of the angular velocity, which guarantees that the flow possesses a positive horizontal velocity in the whole nozzle. / As a continuation, we investigate the influence of the incoming flow angle and the geometry structure of the nozzle walls on subsonic flows in a finitely long curved nozzle. It turns out to be interesting that the incoming flow angle and the angles of inclination of nozzle walls play the same role as the end pressure. The curvatures of the nozzle walls play an important role. We also extend our results to subsonic Euler flows in the 2-D and 3-D asymmetric cases. / Then it comes to the most interesting and difficult casethe 3-D subsonic Euler flow in a bounded nozzle, which is also the essential part of this thesis. The boundary conditions we have imposed in the 2-D case have a natural extension in the 3-D case. These important clues help us a lot to develop a new formulation to get some insights on the coupling structure between hyperbolic and elliptic modes in the Euler equations. The key idea in our new formulation is to use the Bernoulli’s law to reduce the dimension of the velocity field by defining new variables [with formula] and replacing u₁ by the Bernoulli’s function B through [with formula] In this way, we can explore the role of the Bernoulli’s law in greater depth and hope that may simplify the Euler equations a little bit. We find a new conserved quantity for flows with a constant Bernoulli’s function, which behaves like the scaled vorticity in the 2-D case. More surprisingly, a system of new conservation laws can be derived, which is never been observed before, even in the two dimensional case. We employ this formulation to construct a smooth subsonic Euler flow in a rectangular cylinder by assigning the incoming flow angles and the Bernoulli’s function at the inlet and the end pressure at the exit, which is also required to be adjacent to some special subsonic states. The same idea can be applied to obtain similar information for the incompressible Euler equations, the self-similar Euler equations, the steady Euler equations with damping, the steady Euler-Poisson equations and the steady Euler-Maxwell equations. / Last, we are concerned with the structural stability of some steady subsonic solutions for the Euler-Poisson system. A steady subsonic solution with subsonic background charge is proven to be structurally stable with respect to small perturbations of the background charge, the incoming flow angles and the end pressure, provided the background solution has a low Mach number and a small electric field. The new ingredient in our mathematical analysis is the solvability of a new second order elliptic system supplemented with oblique derivative conditions at the inlet and Dirichlet boundary conditions at the exit of the nozzle. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Weng, Shangkun. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 176-187). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese. / Chapter 1 --- Introduction --- p.11 / Chapter 2 --- Subsonic irrotational flows in a rectangular nozzle --- p.26 / Chapter 2.1 --- Introduction --- p.27 / Chapter 2.2 --- Reduction of the problem and main results --- p.33 / Chapter 2.2.1 --- An auxiliary problem --- p.34 / Chapter 2.2.2 --- Restrictions on the end pressure --- p.34 / Chapter 2.2.3 --- Main results --- p.36 / Chapter 2.3 --- Unique solvability of the auxiliary problem --- p.38 / Chapter 2.3.1 --- Reformulation of the auxiliary problem --- p.38 / Chapter 2.3.2 --- Proof of Theorem 2.2.2 --- p.40 / Chapter 2.4 --- The relationship between the mass flux m and the Bernoulli’s constant B --- p.56 / Chapter 3 --- Subsonic irrotational flows in a 2-D finitely long curved nozzle --- p.61 / Chapter 3.1 --- Introduction --- p.62 / Chapter 3.2 --- Reduction of the problem and main results --- p.65 / Chapter 3.2.1 --- An auxiliary problem --- p.65 / Chapter 3.2.2 --- Main results --- p.66 / Chapter 3.3 --- Unique solvability of the auxiliary problem --- p.68 / Chapter 3.3.1 --- Reformulation of the auxiliary problem --- p.68 / Chapter 3.3.2 --- Proof of Theorem 3.2.1 --- p.69 / Chapter 4 --- Subsonic Euler flows in a divergent nozzle --- p.85 / Chapter 4.1 --- Introduction --- p.86 / Chapter 4.2 --- Subsonic Euler flows in a 2-D divergent nozzle --- p.87 / Chapter 4.2.1 --- Formulation of the problem and main results . --- p.87 / Chapter 4.2.2 --- Proof of Theorem 4.2.2 --- p.92 / Chapter 4.3 --- Subsonic Euler flows in a three-dimensional divergent conic nozzle with an asymmetric end pressure --- p.97 / Chapter 4.3.1 --- Formulation of the problem and main results . --- p.97 / Chapter 4.3.2 --- Proof of Theorem 4.3.2 --- p.102 / Chapter 5 --- A new formulation for the 3-D compressible Euler equations --- p.108 / Chapter 5.1 --- Introduction --- p.109 / Chapter 5.2 --- A new formulation for the 3-D compressible Euler equations --- p.114 / Chapter 5.3 --- The 3-D compressible Euler equations with a constant Bernoulli’s function --- p.117 / Chapter 5.3.1 --- A new conserved quantity --- p.117 / Chapter 5.3.2 --- A system of new conservation laws --- p.122 / Chapter 5.4 --- Subsonic Euler flows in a rectangular cylinder --- p.128 / Chapter 5.4.1 --- Extension to the domain Ωe = [0, 1] × T² --- p.130 / Chapter 5.4.2 --- Main results --- p.130 / Chapter 5.4.3 --- Proof of Theorem 5.4.1 --- p.132 / Chapter 5.5 --- A new formulation for the 3-D incompressible Euler equations --- p.139 / Chapter 5.5.1 --- A new formulation for the 3-D incompressible Euler equations --- p.139 / Chapter 5.5.2 --- The 3-D incompressible Euler equations with a constant Bernoulli’s function --- p.142 / Chapter 5.6 --- Appendix: The verification of (5.3.4)-(5.3.6) --- p.145 / Chapter 6 --- On steady subsonic flows for the Euler-Poisson models --- p.148 / Chapter 6.1 --- Introduction --- p.149 / Chapter 6.2 --- Preliminary --- p.152 / Chapter 6.2.1 --- A new formulation for the Euler-Poisson equations --- p.152 / Chapter 6.2.2 --- Background solutions --- p.153 / Chapter 6.3 --- Structural stability of background solutions --- p.156 / Chapter 6.3.1 --- Extension to the domain Ωe = [0, 1] × T² --- p.158 / Chapter 6.3.2 --- Main results --- p.159 / Chapter 6.3.3 --- Proof of Theorem 6.3.1 --- p.162 / Chapter 7 --- Discussions and Future works --- p.170 / Chapter 7.1 --- Subsonic flows in a finitely long nozzle --- p.170 / Chapter 7.2 --- The transonic shock problem in a 3-D divergent nozzle --- p.172 / Chapter 7.3 --- Dynamical stability of a transonic shock in nozzles --- p.174 / Bibliography --- p.175
| Identifer | oai:union.ndltd.org:cuhk.edu.hk/oai:cuhk-dr:cuhk_327736 |
| Date | January 2012 |
| Contributors | Weng, Shangkun., Chinese University of Hong Kong Graduate School. Division of Mathematics. |
| Source Sets | The Chinese University of Hong Kong |
| Language | English, Chinese |
| Detected Language | English |
| Type | Text, bibliography |
| Format | electronic resource, electronic resource, remote, 1 online resource (187 leaves) : ill. (some col.) |
| Rights | Use of this resource is governed by the terms and conditions of the Creative Commons “Attribution-NonCommercial-NoDerivatives 4.0 International” License (http://creativecommons.org/licenses/by-nc-nd/4.0/) |
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