Transversely isotropic fluid

Comparison with Ericksen s Transversely Isotropic Fluid Theory... 

Ericksen proposed a theory for fluids such as nematic liquid crystals which could become anisotropic during flow. By assuming symmetry around the director, the expression for the stress tensor was somewhat simplified. We compare here briefly Ericksen s transversely isotropic fluid theory with the transient behavior observed for thermotropic copolyesters of PHB/PET. 

Fig. 10. Prediction of the shear stress using Ericksen s transversely isotropic fluid theory. The domains were assumed to be initially oriented along 2-direction (i.e. perpendicular to the shear surface).

The shear stress growth on the inception of shear flow may reflect the orientation of the liquid crystalline domains. Orientation seems to occur within less than 2 strain units in shear flow. This primary normal stress difference can exhibit different phenomena from the shear stress response. In particular for the 60 mole % PHB/PET system, values of N are positive and rise gradually to the equilibrium values whereas the 80 mole % PHB/PET system can exhibit negative values of N. Ericksen s transversely isotropic fluid theory can qualitatively handle some of the observed phenomena. Further studies which couple the transient flow behavior to the orientation and morphology need to be carried out. 

Ericksen (1960) proposed a model for the transversely isotropic fluid (TIF), given by ... 

The first rheological theory for Uquid crystals was developed by LesUe and Ericksen, building on Ericksen s earUer transversely isotropic fluid. The theory is formulated in terms of a director field n, and it is similar to the fiber theory in the preceding section, except that it includes a contribution to the free energy from an interactive potential that causes the molecules to align at rest. The usual form of the free energy F resulting from distortions of the director field is... 

J.L. Ericksen, Transversely Isotropic Fluids, Kolloid-Zeit, 173, 117-122 (1960). 

Abstract A poromechanics formulation for transversely isotropic chemically active poroelastic media under non-isothermal conditions is presented. The formation pore fluid is modeled as a two-species constituent comprising of the solute and the solvent. The model is applied to study the thermo-chemical effects on the stress and pore pressure distributions in the vicinity of an inclined borehole drilled in a chemically active transversely isotropic formation under non-isothermal conditions. 

The borehole is assumed to be infinitely long and inclined with respect to the in-situ three-dimensional state of stress. The axis of the borehole is assumed to be perpendicular to the plane of isotropy of the transversely isotropic formation. Details of the problem geometry, boundary conditions and solutions for the stresses, pore pressure and temperature are available in [7], The solution is applied to assess the thermo-chemical effects on stresses and pore pressures. Both the formation pore fluid and the wellbore fluid are assumed to comprise of two chemical species, i.e., a solute fraction and solvent fraction. The formation material properties are those of a Gulf of Mexico shale [7] given as E = 1853.0 MPa u = 0.22 B = 0.92 k = 10-4 md /r = 10-9 MPa.s Ch = 8.64 x 10-5 m2/day % = 0.9 = 0.14 cn = 0.13824 m2/day asm = 6.0 x 10-6 1°C otsf = 3.0 x 10-4 /°C. A simplified example is considered wherein the in-situ stress gradients are assumed to be trivial and pore pressure gradients of the formation fluid and wellbore fluid are assumed to be = 9.8 kPa/m. The difference between the formation temperature and the wellbore fluid temperature is assumed to be 50°C. The solute concentration in the pore fluid is assumed to be more than that in the wellbore fluid such that mw — mf> = —1-8 x 10-2. 

Due to the relatively high porosity and low permeability, fluid saturated pores and the existence of bedding planes in most shales, it is necessary to take into account the effects of material anisotropy and poroelastic effects (Biot, 1956) on wellbore stability in such formations. The constitutive equation for a transversely isotropic poroelastic material can be found, for example, in Cheng (1997). 

The above procedure guaranties that all coefficients are smooth functions of the wave-vectors. Since the field B satisfies an anisotropic Poisson equation, transformed to Fourier space, its long-range character is evident. In nematics its effect turns out to enhance transverse modulations in distinct contrast to isotropic fluids in most cases. As will become clear below, the mean flow contributions can be neglected in the immediate vicinity of threshold. 

The terms occurring are those allowed by symmetry. Due to the anisotropy more terms appear than in isotropic systems [94]. The clue for the characteristic appearance of the zigzag instabihty as a secondary bifurcation is that 4 is typically negative in nematics [23, 24] leading, in contrast to isotropic fluids, to amplification of transverse modulations of roll patterns. Model calculations that include this feature [25, 26] were quite successful in describing quahtatively the secondary instability and the behaviour beyond. 

In more recent years. White and Suh [53] and White et al. [54] have developed three-dimensional models of compoimds with anisotropic disldike particles, which exhibit yield values. These produce direction dependent flow characteristics. Later Robinson et al. [55] described a transversely isotropic Bingham fluid model. 

Consider first an anisometric molecule with the longitudinal p, and transversal p, permanent dipole moments in an isotropic phase. There are two relaxation modes mode 1, rotations of p, around the long axis, and mode 2, reorientation of p,. Figure 10-1. The mode 1 has a smaller relaxation time, Tj < Tj, because of the smaller moments of inertia involved. When this isotropic fluid is cooled down into the NEC phase, the dynamics is affected by the appearance of the nematic potential associated with the orientational order along the director n. The mode 1 remains almost the same as in the isotropic phase, and contributes to both the parallel and perpendicular components of dielectric polarization (determined with respect to n). Mode 2 is associated with small changes of the angle between p, and n it contributes to the parallel component of dielectric polarization. Mode 3 is associated with conical rotations of p, around the director (as the axis of the cone) it is effective when the applied electric field is perpendicular to n and contributes... 

The simplest situation is one in which a planar substrate lacks any crystallographic structure. Then the confined fluid is homogeneous and isotropic in transverse (x,y) directions. All off-diagonal elements of T vanish, Txx = Tyy = T, and Eq. (5) simplifies to... 

Because of the quadratic dependence on g, the echo attenuation for an isotropically diffusing fluid in the vicinity of the wire depends exponentially on the fourth power of the distance r from the wire center. This results in an enormous dynamic range across the image plane transverse to the wire so that it is possible to measure molecular diffusion coefficients that differ by many orders of magnitude without the need to greatly vary the amplitude or duration of the current pulses. 

In the simplest liquid-crystalline phase, namely the uniaxial nematic, there is at rest a special direction designated by a unit vector n called the director (see Fig. 10-2). In the plane transverse to the director, the fluid is isotropic. The most common nematics are composed of oblong molecules that tend to point in a common direction, which defines the director orientation. Oblate, or disc-like, molecules can also form uniaxial nematics for these discotic nematics, the director is defined by the average orientation of the short axis of the molecule. Lath-like molecules or micelles (shaped like rectangular slabs), in which all three dimensions of the molecule are significantly different from each other, can form biaxial nematics (Praefcke et al. 1991 Chandrasekhar 1992 Fialtkowski 1997). A biaxial... 

They required that the turbulence should be locally isotropic and steady, that the particle Reynolds number should be small, that the particles concentration was small, and that the particle diameter should be much smaller than the length scale of the energy containing eddies for the diffusion controlled range of the model. The model is based on the ability of the particle to respond to the motion of the surrounding fluid. It depends on particle size and density, turbulence structure of the fluid, and transversal particle concentration differences. 

The presence of an isotropic ferroelectric transition is also reflected by the total configm-ational potential energy per particle U /N plotted in the inset of Fig. 6.6. Increasing P from the initial smaller values, U) /N first increases, but then begins to decrease at a transverse pressure of about 7 11 2.0 where Pi begins to rise rather sharply. Clearly, the decrease of (/) /N can only be caused by the dipolar interactions, because the short-range fluid fluid and the fluid substrate potentials are purely repulsive. 

Let the relative viscosity (normalized by the suspending fluid viscosity) as measured in the direction of the cylinder axis (longitudinal direction) as prL and the relative transverse viscosity be / rT. At the low shear limit and in a dilute system, the viscosity is expected to be isotropic. Eshelby (100) obtained... 

In our case of two infinite surfaces which are separated by a medium that transmits stress (e.g., the charged fluid discussed earlier), the force per unit area in the z direction between two rigid surfaces of area A whose normals are in the z and —z directions, is therefore given by /7zz(z = D) = ITc(D) where the subscript i denotes that this is the longitudinal component of the stress tensor. The transverse component of the stress tensor, /7,(z) = TIxx(z) = Hyy(z) (for a system that is isotropic in the plane perpendicular to the z direction), is related to the force in the Jc, y direction for displacements that increase the cross-sectional area of the system. The relevant normal vectors are also in... 

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