Vorticity direction

Smectic A liquid crystals are known to be rather sensitive to dilatations of the layers. As shown in [34, 35], a relative dilatation of less than 10-4 parallel to the layer normal suffices to cause an undulation instability of the smectic layers. Above this very small, but finite, critical dilatation the liquid crystal develops undulations of the layers to reduce the strain locally. Later on Oswald and Ben-Abraham considered dilated smectic A under shear [36], When a shear flow is applied (with a parallel orientation of the layers), the onset for undulations is unchanged only if the wave vector of the undulations points in the vorticity direction (a similar situation was later considered in [37]). Whenever this wave vector has a component in the flow direction, the onset of the undulation instability is increased by a portion proportional to the applied shear rate. 

Clearly flow aligning behavior of the director is present and do increases linearly with the tilt angle, do. Above a threshold in the Spain rate, y 0.011, undulations in vorticity direction set in. In Fig. 14 the results of simulations for y 0.015 are shown. In Fig. 15 we have plotted the undulation amplitude obtained as a function of the shear rate. The dashed line indicates a square root behavior corresponding to a forward bifurcation near the onset of undulations. This is, indeed, what is expected, when a weakly nonlinear analysis based on the underlying macroscopic equations is performed [54], In Fig. 16 we have plotted an example for the dynamic behavior obtained from molecular dynamics simulations. It shows the time evolution after a step-type start for two shear rates below the onset of undulations. The two solid lines correspond to a fit to the data using the solutions of the averaged linearized form of (27). The shear approaches its stationary value for small tilt angle (implied by the use of the linearized equation) with a characteristic time scale t = fi/Bi. 

Fig. 14 Undulations in a simulated model system. At a strain rate of y = 0.015 clearly undulations have developed. As considered in the theory, undulations in the vorticity direction are present. Note that the undulation amplitude does not change along the z- direction. Extracted from Fig. 5.5 of [54]...

Fig. 18 Ground state flow alignment director tilt in the Z direction at r = 0.1SR as a function of the polar angle 0. The tilt is zero at 0 = 0 and

If X, y and z are respectively the flow, gradient and vorticity directions, birefringence measurements in the y-z and x-z planes also lead to the second and third normal stress differences, providing the material verifies the linear... 

Figure 6.35 Neutron scattering pat-tems with the neutron beam directed in the gradient direction for various shear rates y of suspension A4G described in the caption to Fig. 6-34. The horizontal direction is the flow direction and the vertical direction is the vorticity direction. (From Laun et al. 1992, with permission from the Journal of Rheology.)...

Kamis and Mason 1966 Iso et al. 1996a) and analysis (Leal 1975) show that the Jeffery orbit of a single isolated fiber is modified by a spiraling drift of the fiber axis towards the vorticity direction, which is the stable orientation in this case. This drift can be accounted... 

Figure 10.7 (a-c) The Leslie viscosities 0 3 and o 2 determine the direction and rate of rotation of the director (represented by the cylinders) in the orientations shown, For negative values of 3 and 2 (the usual signs for rod-like nematics), the rotation directions are shown by the arrows. The viscosity ua, determines the viscosity of the liquid when the director is in the vorticity direction. (Adapted from Skarp et al., reprinted with permission from Mol. Cryst. Liq. Cryst. 60 215, Copyright 1980, Gordon and Breach Publishers.)... 

As Er is increased further to around 10 in 8CB (at 37°C), there is a roll-cell instability involving (a) a periodic modulation of the director field in the vorticity direction and (b) a cellular flow. The rolls cells are parallel to the primary flow direction (Pieranski and Guyon 1974) (see Fig. 10-19). These transitions in the director field have been both predicted from the Leslie-Ericksen theory (Manneville and Dubois-Violette 1976 Larson 1993) and... 

Figure 10.19 Velocity field (a) and director pattern (b) in roll cells that form in a tumbling nematic initially oriented in the vorticity direction of a shearing flow. (From Larson 1993, with permission from the Journal of Rheology.)...

Fig. 1 Geometry for shear flow, with three possible anchoring conditions on the top and bottom planes (a) n fixed along the flow direction (planar anchoring) (b) n fixed along the velocity gradient (homeotropic anchoring) and (c) n fixed along the vorticity direction (log rolling).

Fig. 22 Steady state incoherent intermediate scattering functions (z) measured in the vorticity direction as functions of accumulated strain jf for various shear rates y data from molecular dynamics simulations of a supercooled binary Lenard-Jones mixture below the glass transition ate taken from [91]. These collapse onto a yield scaling function at long times. The wavevector is q = 3.55/R (at the peak of Sq). The quiescent curve, shifted to agree with that at the highest y, shows ageing dynamics at longer times outside the plotted window. The apparent yielding master function from simulation is compared to those calculated in ISHSM for glassy states at or close to the transition (separation parameters s as labeled) and at nearby wave vectors (as labeled). ISHSM curves were chosen to match the plateau value fq, while strain parameters yc = 0.083 at = 0 solid line) and y, = 0.116 at e = 10 dashed line) were used from [45]...

A Coueite cell [Hanl] suitable for use in an electromagnet is depicted in Fig, 10.1.6. left [RoflT Shear stress, velocity gradient, and the vorticity axis are orthogonal to each other in each volume element. The vorticity direction is along the rotation axis of the cell. The molecular deformation vanishes to first order in that direction, so that for... 

Hence, physically, it represents the local rotational motion of the fluid. We obtain an equation for transport of vorticity directly by taking the curl of all terms in Eq. (10-1). Because... 

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