Nonlinear reversible hydrodynamics of liquid crystals and crystals

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Abstract
The unified nonlinear reversible hydrodynamics of nematics, crystals, smectics A, cholesterics and smectics C is presented. Besides the expected but nonetheless new convective terms, additional nonlinear contributions are obtained in the stress tensor and in the equation for the hydrodynamic variables characterizing the broken symmetries. A brief section treats nematic liquid crystals. The results found are in agreement with previously derived theories. Then we discuss crystals. Next smectics A are considered including the examination of terms which describe rotations of the director. The influence of an external magnetic field is discussed as well. The hydrodynamic treatment of cholesterics is described, where special emphasis is laid on the investigation of the differences between smectics A and cholesterics. Finally the nonlinear hydrodynamics of smectics C is given. The equations show numerous nonlinear contributions reflecting the different broken symmetries of smectics C and the interplay between them. As a quite remarkable fact the dependence of these nonlinear coefficients on all scalar quantities present in smectics C is established. This has been done for the other presented systems as well but those have less scalars illustrating once more the special role played by smectics C as a biaxial system. In order to apply the presented theory to specific problems it is, of course, necessary to supplement the nonlinear reversible hydrodynamic equations by irreversible contributions derived e.g. by Martin, Parodi and Pershan ; de Gennes; Moritz and Franklin