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- Kickback in nematic liquid crystalsPublication . Costa, Fernando Pestana da; Grinfeld, Michael; Langer, Mathias; Mottram, Nigel J.; Pinto, João TeixeiraWe describe a nonlocal linear partial differential equation arising in the analysis of dynamics of a nematic liquid crystal. We confirm that it accounts for the kickback phenomenon by decoupling the director dynamics from the flow. We also analyse some of the mathematical properties of the decoupled director equation.
- Uniqueness in the Freedericksz transition with weak anchoringPublication . Costa, Fernando Pestana da; Grinfeld, Michael; Mottram, Nigel J.; Pinto, João TeixeiraIn this paper we consider a boundary value problem for a quasilinear pendulum equation with non-linear boundary conditions that arises in a classical liquid crystals setup, the Freedericksz transition, which is the simplest opto-electronic switch, the result of competition between reorienting effects of an applied electric field and the anchoring to the bounding surfaces. A change of variables transforms the problem into the equation xττ = −f (x) for τ ∈ (−T , T ), with boundary conditions xτ = ±βT f (x) at τ = ∓T , for a convex non-linearity f . By analysing an associated inviscid Burgers’ equation, we prove uniqueness of monotone solutions in the original non-linear boundary value problem. This result has been for many years conjectured in the liquid crystals literature, e.g. in [E.G. Virga, Variational Theories for Liquid Crystals, Appl. Math. Math. Comput., vol. 8, Chapman & Hall, London, 1994] and in [I.W. Stewart, The Static and Dynamic Continuum Theory of Liquid Crystals: A Mathematical Introduction, Taylor & Francis, London, 2003].
- A mathematical study of a bistable nematic liquid crystal devicePublication . Costa, Fernando Pestana da; Grinfeld, Michael; Mottram, Nigel J.; Pinto, João TeixeiraWe consider a model of a bistable nematic liquid crystal device based on the Ericksen– Leslie theory. The resulting mathematical object is a parabolic PDE with nonlinear dynamic boundary conditions. We analyze well-posedness of the problem and global existence of solutions using the theory developed by Amann. Furthermore, using phaseplane methods, we give an exhaustive description of the steady state solutions and hence of the switching capabilities of the device.