Isotropic viscoelastic fluid

Note that the normal stress differences Ni and N2 are linear in the shear rate y, as is the shear stress. In contrast, for isotropic viscoelastic fluids, Ni shear rates (see Section 1.4.3). 

The otoliths are an overdamped second-order system whose structure is shown in Figure 64.1a. In this model the otoconial layer is assumed to be rigid and nondeformable, the gel layer is a deformable layer of isotropic viscoelastic material, and the fluid endolymph is assumed to be Newtonian fluid. A small element of the layered structure with surface area dA is cut from the surface and a vertical view of this surface element, of width Ax, is shown in Figure 64.1b. To evaluate the forces that are present, free body diagrams are constructed of each elemental layer of the small differential strip. See the nomenclature table for a description of all variables used in the following formulas (for derivation details see Grant et al., 1984 and 1991). 

The time-dependent growth of Nx after start-up of steady shearing for a polyethylene melt is shown in Fig. 1-10. Note that at steady state the first normal stress difference is larger than the shear stress at this particular shear rate. The normal stress differences usually are more shear-rate-dependent than the shear stress. In fact, if the isotropic liquid belongs to a fairly general class known as viscoelastic simple fluids with fading memory (Coleman and Noll 1961), then at low shear rates the normal stress differences depend quadratically... 

A single weightless Hookean, or ideal, elastic spring with a modulus of G and a simple Newtonian (fluid) dash pot or shock absorber having a liquid with a viscosity are convenient to use as models illustrating the deformation of an elastic solid and an ideal liquid. Because polymers are often viscoelastic solids, combinations of these models are used to demonstrate deformations resulting from the application of stress to an Isotropic solid polymer. 

The flow of a viscoelastic liquid between infinite parallel walls is a viscometric flow, or a flow with constant stretch history. The velocity profile for a fluid that is isotropic at rest is determined in such a flow only by the shear viscosity, although the stress distribution depends on the viscoelastic parameters. A nearly parallel flow for which the Deborah number is low, and stress growth and relaxation is not important, can be treated as though the local flow were that between infinite parallel walls in that case the viscoelasticity is not important for determining the flow field and the process can be analyzed with the lubrication or Hele-Shaw equations as though the poljmier were purely viscous. Effects attributable to the viscoelastic parameters (eg, interface movement in co-extrusion)... 

From the above considerations it is clear that our discussion of spinodal decomposition in terms of a generaUzed nonlinear diSusion equation (Eq. (5)) was very incomplete, since there it was tacitly assumed that the concentration field c(x, t) is the only relevant slow variable in the problems while in reality a second slow variable, the velocity field v(x, t), needs to be included, even if the fluid on average is at rest. A particularly interesting comphca-tion arises for fluids exposed to shear flow, where the direction of (relative to the flow direction) matters [11], while for fluids at rest S(k, t) is isotropic. This problem is beyond our scope here, as are the subtleties which arise when the two constituents of a binary mixture have vastly different viscosity (viscoelastic phase separation [12]). 

Abstract Phase separation in isotropic condensed matter has so far been believed to be classified into solid and fluid models. When there is a large difference in the characteristic rheological time between the components of a mixture, however, we need a model of phase separation, which we call viscoelastic model . This model is likely a general model that can describe all types of isotropic phase separation including solid and fluid model as special cases. We point out that this dynamic asymmetry between the components is quite common in complex fluids, one of whose components has large internal degrees of freedom. We also demonstrate that viscoelastic phase separation in such dynamically asymmetric mixtures can be characterized by the order-parameter switching phenomena. The primary order parameter switches from the... 

This generality of the viscoelastic model is summarized in Table 1. The viscoelastic model in the classification of isotropic phase separation corresponds to viscoelastic matter in the classification of isotropic condensed matter. Corresponding to the classification of isotropic matter into solids, viscoelastic matter, and fluids, we... 

When a viscoelastic material is sheared between two parallel surfaces at an appreciable rate of shear, in addition to the viscous shear stress T 2, there are normal stress differences Wi s Tn - 722 and N2 s 722 - T23. Here 1 is the flow direction, 2 is perpendicular to the surfaces between which the fluid is sheared, as defined by eq 1.4.8, and 3 is the neutral direction. The largest of the two normal stress differences is N, and it is responsible for the rod climbing phenomenon mentioned at the beginning of this book. For isotropic materials, Ni has always been found to be positive in sign (unless it is zero). In a cone and plate rheometer this means that the cone and plate surfaces tend to be pushed q>art. N2 is usually found to be negative and smaller in magnitude than Ni typically the ratio —N2/N1 lies between 0.05 and 0.3 (Keentok et al., 1980 Ramachandran et al., 1985). Figure 4.2.1 shows the... 

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