Poiseuille ¯ow of nematic liquid crystals M. Carme Calderer *,1 , Chun Liu Department of Mathematics and Center for Materials Research, Penn State University, 405 McAllister Building, University Park, PA 16802, USA Received 12 April 1999; accepted 14 April 1999 Abstract We study ¯ow phenomena of nematic liquid crystals with variable degree of orientation. There are three non-dimensional parameters, the Ericksen number, the Reynolds number and the Interface number that are very relevant in determining the ¯ow behavior. We establish a dissipative relation for the governing system that holds for general ¯ow domains. In the case of plane Poiseuille ¯ow, we study the well-posedness of the governing system of dierential equations. We discuss stationary con®gurations with many defects that are due to the large Ericksen number of the ¯ow. We also present numerical simulations of the boundary value problem for the stationary system. Ó 2000 Published by Elsevier Science Ltd. All rights reserved. Keywords: Liquid crystals; Poiseuille ¯ow; Defect structures 1. Introduction We study time dependent and stationary ¯ow of nematic liquid crystals with variable degree of orientation. We show that the governing system possess a dissipative property. In the case of planar Poiseuille ¯ow we establish well-posedness of the time dependent problem. In the case of ¯ow with large Ericksen number, we ®nd stationary con®gurations that present a rich structure of defects and texture. We also carry out numerical simulations of such a ¯ow. The model that we analyze was developed by Ericksen [8] and emphasizes the role of the order parameter in describing defects and non-Newtonian properties of the ¯ow. The variables to de- scribe such ¯ows include the velocity ®eld v, the pressure p, the director, n, and the order pa- rameter s that corresponds to the variable degree of orientation. The presence of n and s in the constitutive equations contributes to the modeling of the anisotropic behavior of the ¯ow. www.elsevier.com/locate/ijengsci International Journal of Engineering Science 38 (2000) 1007±1022 * Corresponding author. Tel.: +1-814-865-3611. E-mail address: mcc@math.psu.edu (M.C. Calderer). 1 Supported by NSF Grant no. DMS-9704714, 1997±1999. 0020-7225/00/$ - see front matter Ó 2000 Published by Elsevier Science Ltd. All rights reserved. PII:S0020-7225(99)00099-3