Stable dynamic states at the nematic liquid crystals in weak shear flow

We study stable equilibria of liquid crystals in the flow being at rest and the stable dynamic states for nematic liquid crystals under weak shear flow for the Doi model M. Doi, S.F. Edwards, The Theory of Polymer Dynamics, Oxford University Press, 1986]. It is first theoretically proven that there is a hysteresis phenomenon in the flow being at rest when the non-dimensional potential intensity among particles changes. Furthermore, in the weak shear flow, we show that there exist many stable dynamic states: flow aligning, tumbing, log-rolling and kayaking, which depend on the initial concentrated orientation of liquid crystal particles. The results are consistent with those of numerical simulation M.G. Forest, Q. Wang, R. Zhou, The weak shear kinetic phase diagram for nematic polymers, Rheol. Acta 43 (2004) 17-37; M.G. Forest, R. Zhou, Q.Wang, Full-tensor alignment criteria for sheared nematic polymers, J. Rheol. 47 (2003) 105-127] and experimental discoveries W.R. Burghardt, Molecular orientation and rheology in sheared lyotropic liquid crystalline polymers, Macromol. Chem. Phys. 199 (1998) 471-488; Ch. Gähwiller, Temperature dependence of flow alignment in nematic liquid crystals, Phys. Rev. Lett. 28 (1972) 1554-1556]. 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.