Commun. Comput. Phys., 7 (2010), pp. 317-332. Numerical Investigation of Tumbling Phenomena Based on a Macroscopic Model for Hydrodynamic Nematic Liquid Crystals Hui Zhang 1, Qichuan Bai 1*1 School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing, 100875, China. Received 28 September 2008; Accepted (in revised version) 13 February 2009 Available online 24 August 2009 doi:10.4208/cicp.2009.08.206 Abstract This paper is concerned with the numerical investigation of a macroscopic model for complex fluids in "1+2" dimension case. We consider the planar pressure driven flow where the direction of the molecules is constrained in the shear plane. The modified Crank-Nicolson finite difference scheme satisfying a discrete energy law will be developed. By using this scheme, it is observed numerically that the direction of the molecules will tumble from the boundary layer and later on the inner layer with a much longer time period. This is consistent with the theoretical prediction. Moreover, we find some complex phenomena, where the tumbling rises from boundary layer and is then embedded into the interior area more clearly when the viscosity coefficient $\mu$ of the macro flow has a larger value. The norm of the molecular director \textbf{d} will endure greater change as well. This implies that the viscosity of flow plays the role of an accelerator in the whole complex fluids. Comparing these results with the theoretical analysis, we can find that the gradient of the velocity has direct impact on the tumbling phenomena. These results show that the proposed scheme is capable of exploring some physical phenomena embedded in the macro-micro model. AMS subject classifications: 76T99, 76D05, 65M06 Notice: Undefined variable: pac in /var/www/html/issue/abstract/readabs.php on line 164 Key words: Complex fluids, energy law, anchoring condition, modified Crank-Nicolson scheme, tumbling. *Corresponding author. Email: hzhang@bnu.edu.cn (H. Zhang), qichuanbai@gmail.com (Q. C. Bai)