Stokes流问题中的辛体系方法
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摘要
在化学工程、环境工程、物理化学、生物力学、地球物理、气象等领域中大量的问题归结为粘性流体问题。因而近百年来对这个问题的研究一直没有间断,并且随着其广泛应用,近几十年来发展成为了一个比较活跃的领域和方向。Stokes流是粘性流体问题的主要的和经典的流动模型。在该问题的研究中,传统的方法是在拉格朗日体系下欧几里德空间中进行问题的求解,不可避免地带来了高阶偏微分方程的求解和边界条件处理难等问题。因此针对此问题探讨一种新的和有效的求解方法是必要的。
本文从不可压缩牛顿流体的本构方程和Stokes方程出发,借助于耗散能导出问题的哈密顿作用量,即拉格朗日函数,从而获得哈密顿密度函数。进一步利用变分方程,建立了正则(对偶)方程组。这样将辛体系(哈密顿体系)的理论引入到了平面和空间Stokes流问题中。在辛空间中,问题的求解归结为正则方程的本征值与本征解的问题。利用本征解之间的辛共轭正交归一关系和其完备性,原问题的解可以由本征解的线性组合表示。结合边界条件可确定解析解和半解析解,从而建立了一套辛体系求解方法。
本文以二维和三维问题的Stokes流问题作为首要研究对象,并对非定长小雷诺数流问题进行了探讨。主要研究工作如下:首先,将辛体系引入到二维Stokes流问题中,并研究了板驱动流动、剪切流动、入口流动及二维管道流动等问题,得到了问题解的解析表达式,并进行了数值计算,给出了流场的速度、应力、压强等的物理特性,同时还给出了流线图、矢量图。在分析了数值结果的基础上,揭示了Stokes流的流动机理,特点及流动的端部效应等现象。其次,将辛体系方法推广到空间Stokes流问题中,并求解出零本征值本征解和非零本征值本征解。研究了空间问题中本征解之间特殊的辛正交共轭关系,给出了各阶非零本征解的模态。研究了空间入口流问题中入口长度和入口半径之间的关系曲线,得到一些规律。对端部旋转板引起的空间流动问题也进行了研究和讨论。最后,对非定常小雷诺数流问题进行了探讨。
研究结果表明,对偶方程(正则方程)的本征解具有明确的物理意义:零本征值本征解描述了基本的流动,而非零本征值本征解则显示着局部效应的特点,它可直接描述边界效应及其衰减过程。辛体系方法是一种简单、直接、高效的求解方法。它同时也为其它问题的研究提供了一种思路。
There are many problems that can be reduced to viscous fluid problem in the chemical industry, environment engineering, physical chemistry, biomechanics, geophysics as well as meteorology etc. The research on this subject has been ongoing for almost a century. And because of its extensive applications, it has developed into an active research field. Stokes flow is the main and typical flow model of viscous fluid. The traditional method solves this problem in Euclidean space under the Lagrange system, which involves solving higher orders of partial differential equations and faces the difficulty of handling the boundary conditions. So it is necessary to investigate a new and efficient solving method.
Based on the principle of energy dissipation, The Lagrange function, Hamiltonian action, is derived from the constitutive equations of incompressible Newtonian fluid and Stokes equations. Then the Hamiltonian function can be obtained. Finally, the canonical (dual) equations are obtained by appling the variation principle, and the symplectic system (Hamiltonian system) is introduced into plane and space Stokes flow problems. In the system, the fundamental problem is reduced to eigenvalue and eigensolution problem. Because of the completeness of eigensolution space and adjoint relationships of the symplectic ortho-normalization for eigensolutions, the solution can be can be expanded by a linear combination of the eigensolutions. Under the given boundary conditions, the expansion coefficients can be obtained, and then the semi-analytical solution of the problem. The close method of the symplectic system is presented.
Two and three dimensional problems are investigated, and so is non-steady low Reynolds number flow. The main research work is as follows: first, the symplectic system is introduced into two-dimensional Stokes flow problem. The lid-driven flow, shear flow as well as channel flow is studied, and the analytical expressions of the solutions are also given. After the numerical computation, the velocities, stresses and pressures of the flow are obtained, meanwhile, the streamline patterns and velocity vectors are also plotted. Based on the numerical analysis, the flow mechanism, flow characteristics and the end effects etc. are also revealed. Second, the symplectic system is introduced into three-dimensional Stokes flow problem, and the zero eigenvalue solutions and nonzero eigenvalue solutions are given. The special adjoint relationships of the symplectic ortho-normalization is investigated, and the mode of every nonzero eigensolution are given. As an example, the influence of inlet radius on Stokes flow in a circular tube is studied, and some laws are derived. The spatial flow driven by two end lids is also the example discussed. At the end of the dissertation, non-steady low Reynolds number flow problem is studied preliminarily.
The research results show that the eigensolutions of the canonical equations have their definite physical meanings: zero eigenvalue solutions are the fundamental flows; and the nonzero eigenvalue solutions reveal the local effects, they can describe end effect and its decay process. The symplectic system method is simple, direct and efficient. It also provides a path for solving other problems.
本文从不可压缩牛顿流体的本构方程和Stokes方程出发,借助于耗散能导出问题的哈密顿作用量,即拉格朗日函数,从而获得哈密顿密度函数。进一步利用变分方程,建立了正则(对偶)方程组。这样将辛体系(哈密顿体系)的理论引入到了平面和空间Stokes流问题中。在辛空间中,问题的求解归结为正则方程的本征值与本征解的问题。利用本征解之间的辛共轭正交归一关系和其完备性,原问题的解可以由本征解的线性组合表示。结合边界条件可确定解析解和半解析解,从而建立了一套辛体系求解方法。
本文以二维和三维问题的Stokes流问题作为首要研究对象,并对非定长小雷诺数流问题进行了探讨。主要研究工作如下:首先,将辛体系引入到二维Stokes流问题中,并研究了板驱动流动、剪切流动、入口流动及二维管道流动等问题,得到了问题解的解析表达式,并进行了数值计算,给出了流场的速度、应力、压强等的物理特性,同时还给出了流线图、矢量图。在分析了数值结果的基础上,揭示了Stokes流的流动机理,特点及流动的端部效应等现象。其次,将辛体系方法推广到空间Stokes流问题中,并求解出零本征值本征解和非零本征值本征解。研究了空间问题中本征解之间特殊的辛正交共轭关系,给出了各阶非零本征解的模态。研究了空间入口流问题中入口长度和入口半径之间的关系曲线,得到一些规律。对端部旋转板引起的空间流动问题也进行了研究和讨论。最后,对非定常小雷诺数流问题进行了探讨。
研究结果表明,对偶方程(正则方程)的本征解具有明确的物理意义:零本征值本征解描述了基本的流动,而非零本征值本征解则显示着局部效应的特点,它可直接描述边界效应及其衰减过程。辛体系方法是一种简单、直接、高效的求解方法。它同时也为其它问题的研究提供了一种思路。
There are many problems that can be reduced to viscous fluid problem in the chemical industry, environment engineering, physical chemistry, biomechanics, geophysics as well as meteorology etc. The research on this subject has been ongoing for almost a century. And because of its extensive applications, it has developed into an active research field. Stokes flow is the main and typical flow model of viscous fluid. The traditional method solves this problem in Euclidean space under the Lagrange system, which involves solving higher orders of partial differential equations and faces the difficulty of handling the boundary conditions. So it is necessary to investigate a new and efficient solving method.
Based on the principle of energy dissipation, The Lagrange function, Hamiltonian action, is derived from the constitutive equations of incompressible Newtonian fluid and Stokes equations. Then the Hamiltonian function can be obtained. Finally, the canonical (dual) equations are obtained by appling the variation principle, and the symplectic system (Hamiltonian system) is introduced into plane and space Stokes flow problems. In the system, the fundamental problem is reduced to eigenvalue and eigensolution problem. Because of the completeness of eigensolution space and adjoint relationships of the symplectic ortho-normalization for eigensolutions, the solution can be can be expanded by a linear combination of the eigensolutions. Under the given boundary conditions, the expansion coefficients can be obtained, and then the semi-analytical solution of the problem. The close method of the symplectic system is presented.
Two and three dimensional problems are investigated, and so is non-steady low Reynolds number flow. The main research work is as follows: first, the symplectic system is introduced into two-dimensional Stokes flow problem. The lid-driven flow, shear flow as well as channel flow is studied, and the analytical expressions of the solutions are also given. After the numerical computation, the velocities, stresses and pressures of the flow are obtained, meanwhile, the streamline patterns and velocity vectors are also plotted. Based on the numerical analysis, the flow mechanism, flow characteristics and the end effects etc. are also revealed. Second, the symplectic system is introduced into three-dimensional Stokes flow problem, and the zero eigenvalue solutions and nonzero eigenvalue solutions are given. The special adjoint relationships of the symplectic ortho-normalization is investigated, and the mode of every nonzero eigensolution are given. As an example, the influence of inlet radius on Stokes flow in a circular tube is studied, and some laws are derived. The spatial flow driven by two end lids is also the example discussed. At the end of the dissertation, non-steady low Reynolds number flow problem is studied preliminarily.
The research results show that the eigensolutions of the canonical equations have their definite physical meanings: zero eigenvalue solutions are the fundamental flows; and the nonzero eigenvalue solutions reveal the local effects, they can describe end effect and its decay process. The symplectic system method is simple, direct and efficient. It also provides a path for solving other problems.
引文
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