Unit-sphere description of nematic flows

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Abstract
The unit-sphere description of nematic liquid-crystal configurations first introduced by Thurston [J. Appl. Phys. 52, 3040 (1981)] and dealing with elastic properties of nematic liquid crystals only is extended to include also the study of nematic flows. It is shown how the general features of the flow properties of nematic liquid crystals can be studied by the mapping of the hydrodynamic torque on the unit sphere. The effects of the application of electric or magnetic fields to a nematic liquid crystal under shear are discussed by adding the corresponding torque to the maps. Introducing a dimensionless quantity ɛ, which is the ratio of the hydrodynamic and the electric (magnetic) torques, one can deduce whether the hydrodynamic effects are dominating over the field effects or vice versa. By drawing the sequence of torque maps which appears as ɛ goes from minus to plus infinity [ɛ having the same sign as the dielectric (magnetic) anisotropy] it is shown how one can get a good qualitative understanding of the flow behavior just by the inspection of these maps. By calculating the eigenvalues of the singular points of the torque maps, their stability is determined. The stable ones correspond to flow alignment of the director, while the unstable ones can lend themselves to the study of hydrodynamic instabilities. It is also shown how one in a simple manner can derive expressions of relaxation times, boundary layers, and thresholds of hydrodynamic instabilities by knowing the eigenvalues of the singular points. Finally, it is proven that the equations governing the director profile are formally equivalent to those of a particle moving on a smooth sphere upon which the hydrodynamic torque pattern is imposed, the elastic constant (in the one-constant approximation) playing the same role as the mass of the particle. Using this equivalence it is discussed how one can approach the shear-flow problem of nematic liquid crystals by using the powerful tools of analytical mechanics.