Anisotropic viscosity model

This expression has been used to describe uniaxial restricted molecular reorientation in liquid crystals [7.11, 7.41]. The above series expansion converges rapidly generally, only the first five terms are required [7.43]. Using the anisotropic viscosity model to describe the a— and / — motion, in the single exponential approximation it is found that... 

Finally, restricted 7-motion within an apex angle 0o [Eq. (7.80)] has been employed to interpret [7.11] spectral densities of aromatic deuterons in the nematic and smectic A phases of 50.7-d4. Assuming that a-, and 7-motion are completely uncorrelated, and neglecting the -motion [i.e., set / (0) = /3 t)], the spectral densities were evaluated by describing the a-motion by a simplified model of uniaxial free rotational diffusion about the director and the 7-motion by the reduced correlated functions given in Eq. (7.80). It remains to be examined whether the anisotropic viscosity model in conjunction with restricted 7-motion [i.e., Eq. (7.82)] would be better in interpreting spectral densities of motion in various smectic phases. Also, there is still no convincing NMR evidence for biased 7-motion in nematic or smectic A phases. 

Another rotational diffusion model known as the anisotropic viscosity model156,157 is very similar to the above model, and its main feature is to diagonalize the rotational diffusion tensor in the L frame defined by the director. A similar (but not the same) expression as Eq. (71) is J R(r)co)... 

From now on, the permeation in (16) is neglected as it is several orders of magnitude smaller than the advection due to the radial component of the velocity vr (now playing the role of vz in the planar case). As far as the velocity perturbation is concerned, our aim is to describe its principal effect-the radial motion of smectic layers, i.e., instead of diffusion (permeation) we now have advective transport. In this spirit we make several simplifications to keep the model tractable. The backflow-flow generation due to director reorientation-is neglected, as well as the effect of anisotropic viscosity (third and fourth line of (19)). Thereby (19) is reduced to the Navier-Stokes equation for the velocity perturbation, which upon linearization takes the form... 

However, apparently the value of Cs is not universal in LES either. In practice, Cs is adjusted to optimize the model results. Deardorff [28] [29] quoted several values of Cg based on Lilly s estimates. The exact value chosen depends on various factors like the filter used, the numerical method used, resolution, and so forth, but they are generally of the order of Cg = 0.2. However, from comparison with experimental results, Deardorff concluded that the constant in the Smagorinsky effective viscosity model should be smaller than this, and a value of about Cg = 0.10 was used. In addition, for the case of an anisotropic resolution (i.e., having different grid width Ax, Ay and Az in the different co-ordinate directions), the geometry of the resolution has to be accounted for. 

A modest extension [7.10] to this model is to include rotational motion (7-motion) about a molecular z axis. This has been called the third-rate anisotropic viscosity, or simply the third-rate model [7.37-7.39]. The correlation functions can be written as... 

The models of liquids with anisotropic viscosity are basically used for establishing the features of the rheological behavior of low-molecular-weight liquid crystals. Their anisotropic viscoelastic behavior is the most significant distinctive feature of polymer LC. LC polymers should be considered nonlinear anisotropic viscoelastic liquids. Their anisotropic properties are inadequately characterized by only one viscosity tensor. It is necessary to introduce another relaxation time tensor which will describe the anisotropy of the relaxation properties of LC polymers. The work in this direction has only just begun, and only the basic approaches to the study of the anisotropic viscoelasticity of LC polymers will be reported here. 

More advanced models, for example the algebraic stress model (ASM) and the Reynolds stress model (RSM), are not based on the eddy-viscosity concept and can thus account for anisotropic turbulence thereby giving still better predictions of flows. In addition to the transport equations, however, the algebraic equations for the Reynolds stress tensor also have to be solved. These models are therefore computationally far more complex than simple closure models (Kuipers and van Swaaij, 1997). 

In a simple system, when a solute is placed in water, it perturbs the water molecules creating anisotropic conditions on a long time scale compared with the reorientation time of bulk water. Traditionally, this was attributed to fast exchange between the "free" and "boimd" water. Unfortimately, such a simple model has been shown to fail at fitting a number of experimental data. Further consideration is needed according to the following observations. In the slow process, some of the water molecules are strictly related to the concentration of solute and modulated by the solution viscosity (Halle et al., 1982 Hills and Multinuclear, 1991). The water molecules in the fast... 

Figure 3 illustrates the relationship between steady shear viscosity and shear rate for PBTA homopolymer solutions in NMP/4% LiCl with various concentrations. This figure clearly revels the shear-thinning effect for isotropic (C C r) solutions and anisotropic (C > Ccj-) solutions with the most shear rate region. Meanwhile, a Newtonian plateau appears in a low shear rate region for anisotropic solutions, especially for C = 6 wt% and C = 6.5 wt%. Furthermore, the experimental data could be fitted with theoretical non-Newtonian fluid model. Among which, power-law model was applied for isotropic solutions and Carreau model (22) for anisotropic solutions, as shown below ... 

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