纤维悬浮流中纤维取向的理论研究及数值模拟
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摘要
悬浮流动在自然界及工业生产和日常生活中十分普遍。而纤维悬浮流作为一种理论模型，可以用来模拟多种包含细长结构悬浮颗粒的流动，如造纸工业的纸浆流动，向列型的液晶态物质流动，以及一大类的具有细长分子链结构的聚合物的溶液流动。由于纤维(特指细长结构的悬浮颗粒)不同于球状颗粒，其结构具有很强的空间方向性，因而当其悬浮在流场中时将使悬浮流场表现出多种非各向同性效应，如很强的拉伸粘性，第一和第二法向应力差等非牛顿流体特性。对这些性质的研究有着非常重大的科学和生产意义。从纤维悬浮流的微观流动结构出发来研究悬浮流的宏观平均性质是一种最根本的方法。而查明纤维在各种悬浮流场中的运动方式，便是最基本的问题之一。本文便着重于研究纤维在悬浮流中的转动取向问题。
纤维在均匀Stokes流场中的转动满足著名的Jeffery方程。在简单剪切流动情形下，Jeffery(1922)给出了此方程的解，发现纤维在流场中作周期性转动。而本文通过扩充了Jeffery的结果，给出了纤维在一般二维流场中转动的解析表达式。发现纤维的转动有两种模式，一种是如在简单剪切流场中作周期性转动，而另一种则是渐近趋向于特定角度。文中给出了相应的周期与渐近角度方向，同时，给出了一个判别式来判定纤维的转动模式。只要给定流场的局部应变率，便可立刻判定纤维的转动状态，周期或渐近方向。
纤维悬浮流的各种流变性质与纤维的取向分布函数直接相关。取向分布函数是为了在系综平均过程中给出悬浮流的宏观性质而引入的，其表示纤维在转动过程中在各空间方向出现的可能性。对于取向分布的三种情形：无扩散效应，弱扩散效应以及强扩散效应，文中三种不同的方法进行了研究。在无扩散情形下，首先证明了纤维的转动方程与控制取向分布的Fokker-Planck在弱解条件下等价，然后通过分析变换，直接用前面求得的纤维转动解析表达式构建了取向分布函数。在强扩散情形下，巧妙结合纤维转动解析表达式，用谱方法求解了Fokker-Planck方程。因为采用球面调和函数作为基，有效避免了在球坐标中用其他方法求解(差分法或有限元方法)时所遇到的奇点问题。与差分法的比较求解证明，所提出的谱方法具有高精度，高效率等诸多优点。由于谱方法在弱扩散情形下收敛性变差，为解决此情形下纤维取向分布的高效求解问题，而采用正则摄动法，给出了在弱扩散条件下取向分布的一阶近似解。所提出的三种方法，系统完整地解决了纤维取向分布的高效求解问题。
对在纤维悬浮流中若干相关的重要问题，扩散系数，附加应力以及取向分布与流场的耦合求解等，文中也给出了相应的研究成果。
本文的研究成果，对于更清楚了解纤维在各种流动中的转动状态，提示相关的流变学机理，以及结合相应的本构关系来数值求解纤维悬浮流等方面，都有着重要贡献。
Suspension flows are very popular in industries and daily life. And fiber suspen-sion flow is a model for the flow in which slender bodies suspend, such as the pulpflow, nematic liquid crystal flow, and abundant polymer flows. Fiber which standsfor slender particulate is different from spherical particulate, since fiber is orientablewhile the latter is isotropic. Fiber suspension flow always show non-isotropic proper-ties, such as huge extensional viscosity, the first normal stress difference and the secondnormal stress difference, which are among the various peculiar properties of the Non-Newtonian fluid. It is significant to research such rheological properties. Finding outthe micro flow structure of the suspensions is rudimental for the study of the macroproperties of the suspensions. The subject of this thesis focuses on the fiber orientationin the suspension flows.
The rotation of a fiber is depicted by the well-known Jeffery equation. Jeffery (1922)have solved the equation in a simple shear flow. He found that the fiber rotates period-ically in such flow. In the thesis, the solution of Jeffery equation in general 2D flows isobtained, which covers Jeffery's result for the simple shear flow. It is found that thereare two modes for the fiber's rotation, periodical and asymptotical. A discriminant ispresented in the thesis to judge whether a fiber will rotate periodically or asymptoti-cally run to a specific direction in a flow. The expressions for the period and for theasymptotical direction have been given also.
The apparent properties of fiber suspension flow can be derived from the contribu-tion of all the fiber orientation through the ensemble average method. In the method,the fiber orientation distribution is introduced, which describes the probability of a fiberorientates in certain direction. According to the different effects of diffusion, three meth-ods have been developed to deal with the orientation distribution. Under no diffusioncircumstance, the solution of the orientation distribution is derived from the analyti-cal solution of the Jeffery equation. Under strong diffusion circumstance, a spectralmethod which ingeniously utilizes the analytical solution of the Jeffery equation is de-veloped. The spectral method has high computational efficiency and high precision, in the same time, it avoids the singularity problem which is inevitable for the finitedifference method or the finite element method to solve the Fokker-Planck equationin spherical coordinates. Under weak diffusion circumstance, the spectral method isnot feasible because of the increasing of computational errors. A regular perturbationmethod is developed under such circumstance, which is suitable for weak diffusionproblem. The three methods, as a whole, have efficiently and completely solved theorientation distribution problem.
Several important topics in the research of fiber suspension flow, i.e., diffusivity, ad-ditional stress, and the coupling solution of the orientation distribution and flow field,are discussed in the thesis also.
The achievement of the thesis contributes the understanding of the movement offiber in various flows, it gives clues to reveal the complicated rheology of suspensionflow, it also can be used to numerical simulation when combined with suitable consti-tutive equations about fiber suspension flow.
纤维在均匀Stokes流场中的转动满足著名的Jeffery方程。在简单剪切流动情形下，Jeffery(1922)给出了此方程的解，发现纤维在流场中作周期性转动。而本文通过扩充了Jeffery的结果，给出了纤维在一般二维流场中转动的解析表达式。发现纤维的转动有两种模式，一种是如在简单剪切流场中作周期性转动，而另一种则是渐近趋向于特定角度。文中给出了相应的周期与渐近角度方向，同时，给出了一个判别式来判定纤维的转动模式。只要给定流场的局部应变率，便可立刻判定纤维的转动状态，周期或渐近方向。
纤维悬浮流的各种流变性质与纤维的取向分布函数直接相关。取向分布函数是为了在系综平均过程中给出悬浮流的宏观性质而引入的，其表示纤维在转动过程中在各空间方向出现的可能性。对于取向分布的三种情形：无扩散效应，弱扩散效应以及强扩散效应，文中三种不同的方法进行了研究。在无扩散情形下，首先证明了纤维的转动方程与控制取向分布的Fokker-Planck在弱解条件下等价，然后通过分析变换，直接用前面求得的纤维转动解析表达式构建了取向分布函数。在强扩散情形下，巧妙结合纤维转动解析表达式，用谱方法求解了Fokker-Planck方程。因为采用球面调和函数作为基，有效避免了在球坐标中用其他方法求解(差分法或有限元方法)时所遇到的奇点问题。与差分法的比较求解证明，所提出的谱方法具有高精度，高效率等诸多优点。由于谱方法在弱扩散情形下收敛性变差，为解决此情形下纤维取向分布的高效求解问题，而采用正则摄动法，给出了在弱扩散条件下取向分布的一阶近似解。所提出的三种方法，系统完整地解决了纤维取向分布的高效求解问题。
对在纤维悬浮流中若干相关的重要问题，扩散系数，附加应力以及取向分布与流场的耦合求解等，文中也给出了相应的研究成果。
本文的研究成果，对于更清楚了解纤维在各种流动中的转动状态，提示相关的流变学机理，以及结合相应的本构关系来数值求解纤维悬浮流等方面，都有着重要贡献。
Suspension flows are very popular in industries and daily life. And fiber suspen-sion flow is a model for the flow in which slender bodies suspend, such as the pulpflow, nematic liquid crystal flow, and abundant polymer flows. Fiber which standsfor slender particulate is different from spherical particulate, since fiber is orientablewhile the latter is isotropic. Fiber suspension flow always show non-isotropic proper-ties, such as huge extensional viscosity, the first normal stress difference and the secondnormal stress difference, which are among the various peculiar properties of the Non-Newtonian fluid. It is significant to research such rheological properties. Finding outthe micro flow structure of the suspensions is rudimental for the study of the macroproperties of the suspensions. The subject of this thesis focuses on the fiber orientationin the suspension flows.
The rotation of a fiber is depicted by the well-known Jeffery equation. Jeffery (1922)have solved the equation in a simple shear flow. He found that the fiber rotates period-ically in such flow. In the thesis, the solution of Jeffery equation in general 2D flows isobtained, which covers Jeffery's result for the simple shear flow. It is found that thereare two modes for the fiber's rotation, periodical and asymptotical. A discriminant ispresented in the thesis to judge whether a fiber will rotate periodically or asymptoti-cally run to a specific direction in a flow. The expressions for the period and for theasymptotical direction have been given also.
The apparent properties of fiber suspension flow can be derived from the contribu-tion of all the fiber orientation through the ensemble average method. In the method,the fiber orientation distribution is introduced, which describes the probability of a fiberorientates in certain direction. According to the different effects of diffusion, three meth-ods have been developed to deal with the orientation distribution. Under no diffusioncircumstance, the solution of the orientation distribution is derived from the analyti-cal solution of the Jeffery equation. Under strong diffusion circumstance, a spectralmethod which ingeniously utilizes the analytical solution of the Jeffery equation is de-veloped. The spectral method has high computational efficiency and high precision, in the same time, it avoids the singularity problem which is inevitable for the finitedifference method or the finite element method to solve the Fokker-Planck equationin spherical coordinates. Under weak diffusion circumstance, the spectral method isnot feasible because of the increasing of computational errors. A regular perturbationmethod is developed under such circumstance, which is suitable for weak diffusionproblem. The three methods, as a whole, have efficiently and completely solved theorientation distribution problem.
Several important topics in the research of fiber suspension flow, i.e., diffusivity, ad-ditional stress, and the coupling solution of the orientation distribution and flow field,are discussed in the thesis also.
The achievement of the thesis contributes the understanding of the movement offiber in various flows, it gives clues to reveal the complicated rheology of suspensionflow, it also can be used to numerical simulation when combined with suitable consti-tutive equations about fiber suspension flow.
引文
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