本文選題：Maxwell流體 + Mittag-Leffler函數(shù)　； 參考：《上海大學(xué)》2016年博士論文

【摘要】：在粘彈性流體研究領(lǐng)域,以往很多的研究者都關(guān)注于Phan-Thien-Tanner(PTT)流體,并取得了一系列富有意義的成果.在PTT流體中的研究中,研究者的主要興趣都在于線性和指數(shù)形式的PTT流體模型.實(shí)際上在PTT流體的流動模型中,除了線性和指數(shù)形式外,特別是在復(fù)雜幾何條件下的流動,會導(dǎo)致強(qiáng)粘彈性現(xiàn)象,但涉及此種情形卻鮮有研究.在第一章里,我們討論了粘彈性流的重要性,包括粘彈性流體模型控制方程的選擇,以及PTT方程的推導(dǎo).我們討論了將偏微分方程降階為常微分方程的方法,其中,相似變換扮演了重要角色.第二章里,我們開始了對Maxwell流體,即Phan-Thien-Tanner流體一種特殊情形的分析.其中關(guān)于動量和內(nèi)能的邊界層方程可以通過相似變換來化簡.所得到的結(jié)果耦合了使用解析方法所求解的非線性偏微分方程,并且用圖像化的方法來展示了問題中所出現(xiàn)的各種物理參數(shù).第三章,我們開始了對Phan-Thien-Tanner流體的研究.我們考慮了Phan-Thien-Tanner流體模型的不同形式,例如線性,二次和三次的情形.并且發(fā)展了磁流體動力流(MHD)的一系列解.所獲得的結(jié)果揭示了許多有趣的現(xiàn)象,與非牛頓流體現(xiàn)象相關(guān)的方程需要得到更深入的研究.隨后,我們指出了線性,二次以及三次模型的一些不足,為了克服這些短處,我們引入了PTT流體的Taylor形式,其中線性,一次,二次形式的PTT模型是Taylor形式的幾種特例.在第四章中,我們主要引入了一些其他的序列并且嘗試分析其效果.為此,我們引入了著名的Mittag-Leffler函數(shù).通過使用Mittag-Leffler函數(shù),我們不僅僅是重現(xiàn)了之前的結(jié)果,而且引入了其他一些對工程師和科學(xué)家都大有幫助的數(shù)學(xué)模型.我們還討論了在兩種不同幾何條件下的模型：分別是在笛卡爾坐標(biāo)下的模型和在柱坐標(biāo)下的模型.在第五章里,我們把對PTT流體模型的研究推廣到了分?jǐn)?shù)階的情形.在此章中,我們通過使用Mittag-Leffler函數(shù),提出了一些不同的并且十分有用的數(shù)學(xué)模型來消除整數(shù)階和分?jǐn)?shù)階PTT模型之間的差距.最后,在第六章里,我們提出了PTT方程的收斂準(zhǔn)則.我們引入了非牛頓邊界層流體的兩個方程,即流場的Cauchy方程和剪切流場的PTT方程.我們還分析了在半離散有限元方法下流固耦合方程的收斂性.其中我們在空間上使Galerkin有限元方法,時間上使用半隱式C-N差分格式.因此,耦合方程的收斂階可以達(dá)到O(h~2+k~2).
[Abstract]:In the field of viscoelastic fluid, many researchers have paid attention to Phan-Thien-Tanner PTT fluid, and have achieved a series of meaningful results. In the study of PTT fluid, the main interest of researchers is linear and exponential PTT fluid model. In fact, in the flow model of PTT fluid, in addition to the linear and exponential forms, especially in the complex geometric conditions, the flow will lead to strong viscoelastic phenomena, but there are few studies on this kind of case. In the first chapter, we discuss the importance of viscoelastic flow, including the selection of governing equations for viscoelastic fluid models and the derivation of PTT equations. In this paper, we discuss the method of reducing partial differential equation to ordinary differential equation, in which similarity transformation plays an important role. In the second chapter, we begin to analyze a special case of Maxwell fluid, that is, Phan-Thien-Tanner fluid. The boundary layer equation of momentum and internal energy can be simplified by similarity transformation. The results are coupled with the nonlinear partial differential equations solved by the analytic method, and the various physical parameters in the problem are shown by the method of image. In the third chapter, we begin to study the Phan-Thien-Tanner fluid. We consider different forms of Phan-Thien-Tanner fluid models, such as linear, quadratic and cubic cases. A series of solutions for MHD are developed. The obtained results reveal many interesting phenomena and the equations related to the phenomena of non-Newtonian fluids need to be further studied. Then, we point out some shortcomings of linear, quadratic and cubic models. In order to overcome these shortcomings, we introduce the Taylor form of PTT fluid, in which the linear, primary and quadratic PTT models are several special cases of Taylor form. In Chapter 4, we mainly introduce some other sequences and try to analyze their effects. For this reason, we introduce the famous Mittag-Leffler function. By using the Mittag-Leffler function, we not only recreate previous results, but also introduce other mathematical models that are of great benefit to engineers and scientists alike. We also discuss the models under two different geometric conditions: one in Cartesian coordinates and the other in cylindrical coordinates. In chapter 5, we extend the study of PTT fluid model to fractional order case. In this chapter, by using the Mittag-Leffler function, we propose some different and useful mathematical models to close the gap between integer order and fractional order PTT model. Finally, in chapter 6, we propose the convergence criterion of PTT equation. We introduce two equations of non-Newtonian boundary layer fluid, namely, the Cauchy equation of the flow field and the PTT equation of the shear flow field. We also analyze the convergence of the fluid-solid coupling equation under the semi-discrete finite element method. We make the Galerkin finite element method in space and use the semi-implicit C-N difference scheme in time. Therefore, the convergence order of the coupled equations can reach O(h~2 Ke ~ 2 ~ (2).
【學(xué)位授予單位】：上海大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2016
【分類號】：O357

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本文編號：1936057