Equations of motion of nematic liquid-crystal media: PMM vol. 35, n≗ 5, 1971, pp. 879–891

Abstract

Equations of motion for nematic liquid-crystal media in a magnetic field and also the equations of thermal conductivity are obtained. Together with the condition of continuity and the equation of state, these relationships determine the fields of nine quantities which characterize the nematic fluid: densities, pressures, temperatures, basis vectors of the local axis of anisotropy, rates of collective rotations of molecules near their “long” axes, and vectors of translational velocity. Initial conditions and boundary conditions are formulated. Special cases are examined: equilibrium of the medium in a homogeneous magnetic and temperature field, disinclinations, orientational boundary layer, and also the flow in a flat capillary in a magnetic field and the drag of fluid by a rotating magnetic field. Based on obtained results, an explanation is given for a number of effects which have been discovered experimentally earlier. Liquid crystals occupy on the thermodynamic scale of states an intermediate (mesophase) position between anisotropic crystals and isotropie liquids. Two fundamental varieties of mesophases exist: the smectic and the nematic. In the liquid crystal medium of the smectic type the one-dimensional long-range coordination structure is preserved. The molecules are organized in regularly spaced parallel monolayers. In the medium of the nematic type the long-range order is completely absent in the spatial arrangement of molecules, just as in the ordinary liquid. However, in contrast to a liquid and in similarity to a solid crystal the long-range order of orientation for the “long” molecular axes is preserved. The orientational order is characterized in each point of the medium by the axis of mean molecular orientation. This axis is simultaneously the local axis of symmetry of the medium. In their mechanical properties the nematic media are quite close to liquids. Experiments show [1, 2] that the behavior of nematic liquids in a force field, a temperature field, a magnetic field, and an electrical field has a number of anomalies (anisotropy of viscosity, scale effect, orientation in hydrodynamic flow, drag of the medium by a rotating magnetic field, and others.) The peculiar combination of mechanical properties makes liquid crystal media interesting objects for investigation from the point of view of continuum mechanics. At the present time the hydrostatic theory [3 – 9] is the most developed. In papers [10 – 12] linear hydrostatics is examined with consideration of thermal conductivity and effects of rotational viscosity. The hydrodynamic theory which takes into account elastic and thermal effects in a magnetic field is just being developed [13, 14]. Some results are available in paper [9]. Liquid crystals belong to so-called media with moments or media with rotational degrees of freedom [15]. At the phenomenological level these degress of freedom are taken into account in asymmetric mechanics of continuous media. On the basis of ideas of asymmetric mechanics [16, 17] the general hydrodynamic theory for nematic liquids is developed in [13, 14] taking into account elastic, thermal and magnetic effects. Based on this development, the goal of the present paper is to obtain a closed system of equations of motion, to formulate boundary conditions and initial conditions, to clarify the most essential characteristics of the equations, and to examine the simplest cases of motion of nematic liquids.