Inertial term director

Although, as mentioned earlier, we have not considered the director inertial term in the derivation of equation (4.102), it is worth noting that when it is included this equation should be replaced by [168, Eqn.(53)2]... 

It is convenient at this point to summarise the Ericksen-Leslie dynamic equations for nematics in the incompressible isothermal theory when the director inertial term is neglected. These are the most frequently used forms of the equations and we state them in the notation introduced in the previous Sections. They consist of the constraints... 

To describe this effect we should write Eq. (9.21) for the director motion, i.e., the balance of torques. However, up to now, nobody has observed any effect related to the inertia of the director. Such effects would result in oscillatory character of the director relaxation. The inertial term for the director in a unit volume can be estimated as a sum of the inertia moments of the molecules in this volume. Let... 

In the twist geometry, the rotation of the liquid crystal director is not coupled to the translational motion of the molecules. The rotation of the liquid crystal director is governed by the over-damped dynamics the elastic and electric torques are balanced by the rotational viseosity torque and the inertial term can be neglected [4]. Mathematically we have... 

We study a nematic layer (Fig. 11) confined between two parallel plates of infinite dimension. The director is assumed to lie in the x,y plane and the pressure is constant. The inertial terms in Eq. (51) can be neglected under usual conditions and the x compo-... 

This equation describes director rotation in the magnetic field with the inertial term being disregarded, /i=a3-a2 is the rotational viscosity, are Leslie s coefficients. In the limit of small 0 angles, 0-< 1, Eq. (30) reduces to a linear form ... 

The other two instabilities shown in Fig. 28 may be observed only in liquid crystals (nematic, cholesteric, and smectic C). The first is the Carr-Helfrich instability, which is caused by a low-frequency electric field and occurs in the form of elongated vortices with their axis perpendicular to the original director alignment. The vortices cause a distortion of the director orientation, which is observed optically as a one-dimensional periodic pattern (Kapustin-Williams domains). The other anisotropic mode is observed only in highly conductive liquid crystals. For its interpretation the inertial term dvidt for the fluid velocity must be taken into account, which is why this mode may be called inertial mode. 

In order to discuss the behavior of the instability threshold as a function of the applied field frequency, the time dependent terms /i (dd/dt) and dQ/dt must be added to the equations for the director and charge continuity. Estimates show that the inertial term for the velocity dv/dt in the Navier-Stokes equation may be disregarded even in... 

In a liquid crystal most properties are best expressed relative to a director based coordinate system. This is not a problem in a macroscopic system where the director is virtually fixed. However, it can be a problem in a small system such as a simulation cell where the director is constantly diffusing on the unit sphere. Thus a director based frame is not an inertial frame. Correction terms should therefore be added to time dependent properties. Time correlation functions with slowly decaying tails might also be affected by the director reorientation. Transport coefficient obtained from them will consequently be incorrect. When NEMD-simulation algorithms are applied, the fictitious external field exerts a torque that constantly twists the director, which could make it impossible to reach a steady state. 

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