Torsional shear flow, nematics

So far we have introduced four Miesowicz viscosities. Two other viscosities can be proposed by considering the following. The director n in Fig. 4.1(a), if free to move, will rotate due to a viscous torque the viscosity coefficient 71 is introduced to describe this situation and characterises the torque associated with a rotation of n. For this reason 71 is often called the rotational viscosity or twist viscosity. The coefficient 71 generally determines the rate of relaxation of the director. Also, a rotation of n due to body forces will induce a flow. The viscosity coefficient 72 characterises the contribution to the torque due to a shear velocity gradient in the nematic and is sometimes referred to as the torsion coefficient in the velocity gradient it leads to a coupling between the orientation of the director and shear flow. The two viscosities 71 and 72 have no counterpart in isotropic fluids. We therefore have a total of six viscosities four Miesowicz viscosities plus 71 and 72. It turns out, as will be seen in the problems to be discussed in later Sections, that 7i and 72 are precisely the viscosities introduced in the constitutive theory at equations (4.78) and (4.79), namely. 

An important aspect of the macroscopic structure of liquid crystals is their mechanical stability, which is described in terms of elastic properties. In the absence of flow, ordinary liquids cannot support a shear stress, while solids will support compressional, shear and torsional stresses. As might be expected the elastic properties of liquid crystals are intermediate between those of liquids and solids, and depend on the symmetry and phase type. Thus smectic phases with translational order in one direction will have elastic properties similar to those of a solid along that direction, and as the translational order of mesophases increases, so their mechanical properties become more solid-like. The development of the so-called continuum theory for nematic liquid crystals is recorded in a number of publications by Oseen [ 1 ], Frank [2], de Gennes and Frost [3] and Vertogen and de Jeu [4] extensions of the theory to smectic [5] and columnar phases [6] have also been developed. In this section it is intended to give an introduction to elasticity that we hope will make more detailed accounts accessible the importance of elastic properties in determining the... 

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