Euro. Jnl of Applied Mathematics (2014), vol. 25, pp. 397–423. c  Cambridge University Press 2013 doi:10.1017/S095679251300034X 397 Extensional flow of nematic liquid crystal with an applied electric field L. J. CUMMINGS 1 , J. LOW 2 and T. G. MYERS 1 1 Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102-1982, USA emails : linda.j.cummings@njit.edu, tmyers@crm.cat 2 Centre de Recerca Matem` atica, Campus de Bellaterra, Edifici C, 08193 Bellaterra, Barcelona, Spain email : jlow@crm.cat (Received 18 May 2012; revised 1 September 2013; accepted 10 September 2013; first published online 17 October 2013) Systematic asymptotic methods are used to formulate a model for the extensional flow of a thin sheet of nematic liquid crystal. With no external body forces applied, the model is found to be equivalent to the so-called Trouton model for Newtonian sheets (and fibres), albeit with a modified ‘Trouton ratio’. However, with a symmetry-breaking electric field gradient applied, behaviour deviates from the Newtonian case, and the sheet can undergo finite-time breakup if a suitable destabilizing field is applied. Some simple exact solutions are presented to illustrate the results in certain idealized limits, as well as sample numerical results to the full model equations. Key words: Nematic liquid crystal; Thin film; Viscous sheet; Electric field 1 Introduction Nematic liquid crystals (NLCs) are ubiquitous in nature, and find wide applications in many industrial processes. For example, many modern display devices, certain thermo- meters and some biopathogen detection methods exploit the liquid crystalline nature of chemicals. Contemporary makeup products also often rely on various liquid crystal com- pounds for their iridescent optical qualities [12]. An understanding of how liquid crystals behave under a wide variety of conditions is thus commercially important, but due to the highly complex nature of the governing dynamic equations it can be challenging to investigate the behaviour theoretically from a mechanistic viewpoint. Simple experimental setups can be very valuable as an investigative tool to reveal novel behaviour and new regimes not exhibited by Newtonian fluids. For example, a system as simple as a spreading nematic droplet can exhibit highly complex fingering instabilities [13]. The mathematical models described recently in [9, 10] reveal that these arise due to a complex interplay between fluid flow, internal elasticity and surface (anchoring) energy: strong anchoring will stabilize a film, but with weak anchoring the free surface can destabilize. In this paper we investigate another simple experimental configuration: a thin nematic sheet with one end clamped and the other pulled, subject to a constant force or prescribed speed. This simple setup allows us to make analytical progress, which can aid our overall understanding of free-surface liquid crystal dynamics. We note that the analysis represents available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S095679251300034X Downloaded from https:/www.cambridge.org/core. New Jersey Institute of Technology, on 10 May 2017 at 15:47:48, subject to the Cambridge Core terms of use,