PTT流体的粘弹性流动研究

发布时间：2018-05-26 05:08

本文选题：Maxwell流体 + Mittag-Leffler函数　；参考：《上海大学》2016年博士论文

【摘要】：在粘弹性流体研究领域,以往很多的研究者都关注于Phan-Thien-Tanner(PTT)流体,并取得了一系列富有意义的成果.在PTT流体中的研究中,研究者的主要兴趣都在于线性和指数形式的PTT流体模型.实际上在PTT流体的流动模型中,除了线性和指数形式外,特别是在复杂几何条件下的流动,会导致强粘弹性现象,但涉及此种情形却鲜有研究.在第一章里,我们讨论了粘弹性流的重要性,包括粘弹性流体模型控制方程的选择,以及PTT方程的推导.我们讨论了将偏微分方程降阶为常微分方程的方法,其中,相似变换扮演了重要角色.第二章里,我们开始了对Maxwell流体,即Phan-Thien-Tanner流体一种特殊情形的分析.其中关于动量和内能的边界层方程可以通过相似变换来化简.所得到的结果耦合了使用解析方法所求解的非线性偏微分方程,并且用图像化的方法来展示了问题中所出现的各种物理参数.第三章,我们开始了对Phan-Thien-Tanner流体的研究.我们考虑了Phan-Thien-Tanner流体模型的不同形式,例如线性,二次和三次的情形.并且发展了磁流体动力流(MHD)的一系列解.所获得的结果揭示了许多有趣的现象,与非牛顿流体现象相关的方程需要得到更深入的研究.随后,我们指出了线性,二次以及三次模型的一些不足,为了克服这些短处,我们引入了PTT流体的Taylor形式,其中线性,一次,二次形式的PTT模型是Taylor形式的几种特例.在第四章中,我们主要引入了一些其他的序列并且尝试分析其效果.为此,我们引入了著名的Mittag-Leffler函数.通过使用Mittag-Leffler函数,我们不仅仅是重现了之前的结果,而且引入了其他一些对工程师和科学家都大有帮助的数学模型.我们还讨论了在两种不同几何条件下的模型：分别是在笛卡尔坐标下的模型和在柱坐标下的模型.在第五章里,我们把对PTT流体模型的研究推广到了分数阶的情形.在此章中,我们通过使用Mittag-Leffler函数,提出了一些不同的并且十分有用的数学模型来消除整数阶和分数阶PTT模型之间的差距.最后,在第六章里,我们提出了PTT方程的收敛准则.我们引入了非牛顿边界层流体的两个方程,即流场的Cauchy方程和剪切流场的PTT方程.我们还分析了在半离散有限元方法下流固耦合方程的收敛性.其中我们在空间上使Galerkin有限元方法,时间上使用半隐式C-N差分格式.因此,耦合方程的收敛阶可以达到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).
【学位授予单位】：上海大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O357

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