Velocity anisotropy Viscosity

The flow velocity, pressure and dynamic viscosity are denoted u, p and fj and the symbol (...) represents an average over the fluid phase. Kim et al. used an extended Darcy equation to model the flow distribution in a micro channel cooling device [118]. In general, the permeability K has to be regarded as a tensor quantity accounting for the anisotropy of the medium. Furthermore, the description can be generalized to include heat transfer effects in porous media. More details on transport processes in porous media will be presented in Section 2.9. 

The two-equation models (especially, the k-s model) discussed above have been used to simulate a wide range of complex turbulent flows with adequate accuracy, for many engineering applications. However, the k-s model employs an isotropic description of turbulence and therefore may not be well suited to flows in which the anisotropy of turbulence significantly affects the mean flow. It is possible to encounter a boundary layer flow in which shear stress may vanish where the mean velocity gradient is nonzero and vice versa. This phenomenon cannot be predicted by the turbulent viscosity concept employed by the k-s model. In order to rectify this and some other limitations of eddy viscosity models, several models have been proposed to predict the turbulent or Reynolds stresses directly from their governing equations, without using the eddy viscosity concept. 

In equations (5)-(8), i is the molecule s moment of Inertia, v the flow velocity, K is the appropriate elastic constant, e the dielectric anisotropy, 8 is the angle between the optical field and the nematic liquid crystal director axis y the viscosity coefficient, the tensorial order parameter (for isotropic phase), the optical electric field, T the nematic-isotropic phase transition temperature, S the order parameter (for liquid-crystal phase), the thermal conductivity, a the absorption constant, pj the density, C the specific heat, B the bulk modulus, v, the velocity of sound, y the electrostrictive coefficient. Table 1 summarizes these optical nonlinearities, their magnitudes and typical relaxation time constants. Also included in Table 1 is the extraordinary large optical nonlinearity we recently observed in excited dye-molecules doped liquid... 

Because liquid crystals are fluids, they also show anisotropy in their flow behaviom. This can easily be imderstood by imagining measuring the viscosity of a liquid crystal by placing it between two flat plates and measuring the force necessary to move one plate past the other at a certain velocity. In Figure 2.1, the plates lie in the xy plane and are separated by a distance d. The bottom plate is fixed and the force acting on the top plate is in the x-direction, F. The velocity of the top plate is also in the x-direction, v. ... 

Measuring the torque on a sample of a nematic liquid crystal in a magnetic field rotating with an angular velocity smaller than the critical one represents a relatively simple method for the determination of the rotational viscosity coefficient. Below the critical angular velocity Eq. (24) is valid with 0 = F. Neither the phase lag F-0 nor the anisotropy of the magnetic susceptibility have to be known. This method will be thoroughly discussed in Chap. Ill, Sect. 2.6 of Vol. 2A of this Handbook. 

The anisotropy of the viscosity is an important feature of the rheological properties of nematics the viscosity of the solution has different values as a function of the mutual orientation of the director and the direction and gradient of the velocity (in simple shear flow). It is known (cf. [1]) that the anisotropy of the viscosity is described by the Leslie coefficients aj-Og. In the simplest... 

This kind of rheological behavior of LC solutions and melts of polym at different temperatures or in different concentrations permits confidently postulating a common cause which determines their viscous prqwrties. This common cause is the appearance of LC order at the transitioi point and the appearance of anisotropy of the viscosity, which is the dependence of the internal friction in flow on the mutual position of the directions of orientation of the macromolecules, velocity, and velocity gradient and on the presence of disclinations and the degree of disposion of the system at rest and in the flow state. 

This result allows the rotational viscosity 71 to be determined by measurement of the torsion in the wire for uj < oJc- In this situation neither the phase lag nor the magnetic anisotropy need to be known for the measurement of 71 to be made via small angular velocities. Frost and Gasparoux [226] used the result (5.86) to measure the rotational viscosity 71 for the nematics MBBA and PAA. 

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