Talks

We will show the new rigid rod-like model in a polymeric fluid. The constitutive relations considered are motivated by the kinetic theory. The micro equation has five spatial freedom variables, two of them are in the configuration domain and the others are in the macro flow domain. It is obtained the local existence of the solution with large initial data and global existence of the solution with small Deborah and Reynolds constants in periodic domains. For the case of no flow we will give the structure of stationary solutions to the micro equation with Maier-Saupe potential on the sphere. The stationary solutions are shown to be necessarily a set of axially symmetric functions, and a complete classification of parameters for phase transitions to these stationary solutions is obtained. It is shown that the number of stationary solutions hinges on whether the potential intensity crosses two critical values 6.731393 and 7.5. Furthermore, we present explicit formulas for all stationary solutions. It is first theoretically proven that there is a hysteresis phenomenon when the non-dimensional potential intensity among particles changes. In the weak shear flow, we show that there exist many stable dynamic states: flow-aligning, tumbling, log-rolling and kayaking, which depend on the initial concentrated orientation of liquid crystal particles. Theoretical analysis is reported the first time that the Kayaking state does not circulate around a fixed direction but the asymmetric axis will periodically change.