活性液晶聚合物流体的建模与数值分析

我们应用广义流体力学理论对活性液晶聚合物流体进行数学建模，应用系统自由能的平移和旋转不变性系统地推导出Ericksen应力张量和反对称张量，根据Onsager原理得到应力和应变的本构关系。加入活性应力，确保模型满足热力学原理。同时，通过研究粒子和边界相互作用的物理性质，推导出合理的边界条件。由于得到的数学模型是高度耦合、强非线性、包含高阶导数的偏微分方程组。我们采用增加中间变量来降低导数的阶数、解除方程间的耦合关系，通过引入稳定化因子和凸分裂的思想设计稳定、解耦、高精度、快速有效的数值格式对模型进行数值模拟。通过对模型进行线性稳定性分析和数值结果的全面分析，理解活性液晶系统的流变现象，相变过程，自发流动，defects和pattern等形成的原因，以及分子极性和作用力的类型对系统结果的影响等。同时验证模型的合理性。

We systematically derive a continuum model for the active liquid crystalline polymers using general hydrodynamic theories. For the translational and rotational invariance of the free energy, we get the Ericksen stress tensor and the antisymmetric tensor. We also get the relationship between strain and stress by Onsager principle. With active stress term, our model satisfies thermodynamical theories. We also study the boundary conditions of the system. We derive a stable, decouple, high precision and faster scheme to solve the systems. Using the result of theory and simulations, we analysis the rheology and instabilities of the system, study how to driven the spontaneous flow, defects and pattern formations, and verify the efficiency of the model.