【作者】 李良；

【导师】 马天；

【作者基本信息】 四川大学 ， 应用数学， 2022， 博士

【摘要】 本学位论文从非线性算子的动态分歧与跃迁角度研究了外加均匀磁场和带有旋转效应的导电流体的稳定性与相变现象.通过分析该导电流体控制模型的稳定性、跃迁以及分歧,对该类流体的相变动力学过程给出了完整的数学分析,并且分析了磁场和旋转效应对导电流体相变的影响.全文的内容共分为四章.在第一章中,我们首先介绍了研究所使用的方法-相变动力学理论的发展和应用情况,接着介绍了本文所研究模型的背景,研究的主要内容和创新点.在第二章中,我们介绍了一些相关的预备知识,如线性全连续场的谱理论,相变动力学理论以及平均法等理论工具.接着,我们通过无量纲化过程建立了简化模型.最后,将该简化模型写成抽象形式.在第三章中,首先,我们推导了所需要的中心流形公式;再借助分离变量法计算了简化模型线性方程的特征向量与特征值,并验证了该导电流体发生动力学相变所满足的稳定性交换准则.然后,我们考虑了五种情况的相变.具体的方法为:借助中心流形公式,将该简化模型(偏微分方程)约化成为一个有限维常微分方程,进而通过分析常微分方程的稳定性,得到了决定相变类型的无量纲数;再通过分析无量纲数,得到了五种情况的相变定理.并对每个定理结论给出了相应的物理解释;最后,给出了具体的数值例子和参数区域图,并画出了模型分歧出的非平凡稳定状态的对流结构.在第四章中,我们对本文进行了总结,主要是对磁场和旋转效应对导电流体的相变影响的具体阐述以及对相变动力学理论的展望.更多还原

【Abstract】 This dissertation studies the stability and phase transition phenomenon of an electrically conducting rotating fluid with an external uniform horizontal magnetic field from the perspective of dynamic transition of nonlinear operator.By analyzing the stability,transition and bifurcation of the conducting fluid,the full mathematical analysis for the dynamic transition of the fluid is accomplished,and the influence induced by magnetic field and rotating effect are analyzed.The dissertation is divided into four chapters.In Chapter 1,we first introduce the development and application of the main method-dynamic transition theory.Then,we introduce the background of the model,the main content of the study and the innovation points.In Chapter 2,we introduced some relevant preliminaries,including the spectrum theory of linear complete continuous field,dynamic transition theory and averaging method.Then,we build simplified model via non-dimensional procedure.Finally,we rewrite the simplified model into abstract form.In Chapter 3,we first derive the required center manifold formula;The eigenvalues and eigenvectors of linear part of the simplified model are calculated through the method of separate variables and the principle of exchange of stability(PES)condition of the conducting fluid is derived.Next,phase transition in five distinct scenarios are considered.The specific method is that the simplified model(partial differential equation)is reduced into a finite dimensional ordinary differential equation by using the central manifold formula,then by analyzing the stability of the ODE,the nondimensional numbers determining the transition type are acquired.Then,by analyzing the non-dimensional numbers,five transition theorems are obtained.And the physical interpretation is offered for every conclusion.At last,several numerical examples and parameter diagrams are given and the convection structures of the bifurcated nontrivial stable state are drawn.In Chapter 4,we conclude the paper with a detailed description of the effects of magnetic fields and rotational effects on the phase transition of conducting fluids and an outlook on phase transition theory.更多还原