Simple shear flow normal stresses

Viscoelastic Fluids. These materials exhibit elastic and viscous properties simultaneously. Under simple shear flow, normal stresses are generated as shown in Figure 4-1. The magnitude of these normal forces varies with the shear rate. It is usual to work in terms of normal stress differences rather than normal stresses themselves ... 

The theoretical basis for spatially resolved rheological measurements rests with the traditional theory of viscometric flows [2, 5, 6]. Such flows are kinematically equivalent to unidirectional steady simple shearing flow between two parallel plates. For a general complex liquid, three functions are necessary to describe the properties of the material fully two normal stress functions, Nj and N2 and one shear stress function, a. All three of these depend upon the shear rate. In general, the functional form of this dependency is not known a priori. However, there are many accepted models that can be used to approximate the behavior, one of which is the power-law model described above. 

Thus, only the normal Reynolds stresses (i = j) are directly dissipated in a high-Reynolds-number turbulent flow. The shear stresses (i / j), on the other hand, are dissipated indirectly, i.e., the pressure-rate-of-strain tensor first transfers their energy to the normal stresses, where it can be dissipated directly. Without this redistribution of energy, the shear stresses would grow unbounded in a simple shear flow due to the unbalanced production term Vu given by (2.108). This fact is just one illustration of the key role played by 7 ., -in the Reynolds stress balance equation. 

Difference between the first two normal stresses cti 1 and 022 in simple shear flow... 

Metzner AB, Houghton WT, Salior RA, White JL. A method for measurement of normal stresses in simple shearing flow. Trans Soc Rheol 1961 133-147. 

First normal stress function, pt t — p22 at steady state in steady simple shear flow. 

Williams,M.C. Concentrated polymer solutions. Part III. Normal stresses in simple shear flow. A.I.Ch.E. J. 13,955-961 (1967). 

In a complex, polymeric liquid, normal stresses as well as the shear stress can be present, and these contributions will influence the shape of the structure factor. The simplest rheological constitutive model that can account for normal stresses is the second-order fluid model [64], where the first and second normal stress differences are quadratic functions of the shear rate. Calculations using this model [92,93,94,90,60], indicate that the appearance of normal stresses can rotate the structure factor towards the direction of flow in the case of simple shear flow and can induce a four-fold symmetry in the case of exten-sional flow. 

As the flow accelerates into the gaps around the cylinder, it possesses a greater relative amount of extension. Ultimately, at distances far downstream from the cylinder, the flow is expected to relax back toward a parabolic profile. In these plots, the symbols represent the measured velocities and the solid curves are the results of a finite element, numerical simulation. The constitutive equation used was a four constant, Phan-Thien-Tanner mod-el[193], which was adjusted to fit steady, simple shear flow shear and first normal stress difference measurements. The fit to the velocity data is very satisfactory. 

A graphic example of the consequences of the existence of in stress in simple steady shear flows is demonstrated by the well-known Weissenberg rod-climbing effect (5). As shown in Fig. 3.3, it involves another simple shear flow, the Couette (6) torsional concentric cylinder flow,3 where x = 6, x2 = r, x3 = z. The flow creates a shear rate y12 y, which in Newtonian fluids generates only one stress component 112-Polyisobutelene molecules in solution used in Fig. 3.3(b) become oriented in the 1 direction, giving rise to the shear stress component in addition to the normal stress component in. 

Next we define the two normal stress difference functions that arise in simple shear flows... 

For the simple shear flow, the only one component of the velocity gradient tensor differs from zero, namely, v 2 0. The shear stress and the differences of the normal stresses are defined by equation (9.61) as... 

In the case of elastic fluids and for simple shear flow, the first normal stress difference is N[ =on — o22- When shearing a fluid between two plates (x, direction), the first normal stress difference N( forces the plates apart (x2 direction). The first normal stress difference N i is shown together with the measured shear stress x as a function of the shear rate in Fig. 3.9. In the range of shear rates investigated, the shear stress in the case of silicone oil is substantially greater than the normal stress difference and we see substantially greater normal stress differences for viscoelastic PEO solution than for viscous silicone oil. 

The stress in viscoelastic liquids at steady-state conditions is defined, in simple shear flow, by the shear rate and two normal stress differences. Chapter 13 reviews the evolution of both the normal stress differences and the viscosity with increasing shear rate for different geometries. Semiquantitative approaches are used in which the critical shear rate at which the viscosity starts to drop in non-Newtonian fluids is estimated. The effects of shear rate, concentration, and temperature on die swell are qualitatively analyzed, and some basic aspects of the elongational flow are discussed. This process is useful to understand, at least qualitatively, the rheological fundamentals of polymer processing. 

It is known that incompressible newtonian fluids at constant temperature can be characterized by two material constants the density p and the viscosity T. The characterization of a purely viscous nonnewtonian fluid using the power law model (or any of the so-called generalized newtonian models) is relatively straightforward. However, the experimental description of an incompressible viscoelastic nonnewtonian fluid is more complicated. Although the density can be measured, the appropriate expression for r poses considerable difficulty. Furthermore there is some uncertainty as to what other properties need to be measured. In general, for viscoelastic fluids it is known that the viscosity is not constant but depends on shear rate, that the normal stress differences are finite and depend on shear rate, and that the stress may also depend on the preshear history. To characterize a nonnewtonian fluid, it is necessary to measure the material functions (apparent viscosity, normal stress differences, etc.) in a relatively simple or standard flow. Standard flow patterns used in characterizing nonnewtonian fluids are the simple shear flow and shear-free flow. 

FIGURE 4.2 Simple shear flow deflnition of shear stress, strain, and shear rate Xyy, and are the Cartesian coordinates and the direction of normal stresses in shear flow. 

The PTT model predicts shear thinning and first normal stress difference for both steady and transient shearing. It also predicts a non-zero (negative) second normal stress difference in simple shear flow N2 — — Nil2. The main advantage of the PPT model is that it predicts reasonable extensional flow behavior at all extensional rates. 

Housiadas and Tanner (2009), following the approach of Greco et al. (2005), have used a perturbation analysis to obtain the analytical solution for the pressure and the velocity field up to 0 (pDe) of a dilute suspension of rigid spheres in a weakly viscoelastic fluid, where

volume fraction of the spheres and De is the Deborah number of the viscoelastic fluid. The analytical solution was used to calculate the bulk first and second normal stress in simple shear flows and the elongational viscosity. The main results are... 

For a one-dimensional steady shear flow of a fluid between two planes, the velocities of an inflnitesimal element of fluid in the y- and z-directions are zero. The velocity in the x-direction is a function of y only. Note that in addition to the shear stress Tyx (refer to r subsequently) there are three normal stresses denoted by Txx, tyy, Tzz within the sheared fluid. Weissenberg in 1947 [6] was the first to observe that the shearing motion of a viscoelastic fluid gives rise to unequal normal stresses, known as Weissenberg effects. Since the pressure in a non-Newtonian fluid cannot be defined, and as the normal stress differences [2, 3], Txx — Tyy = Vi and Tyy — Tzz = V2, are more readily measured than the individual stresses, it is therefore customary to express N and N2 together with the shear stress t as functions of the shear rate /yx to describe the viscoelastic behavior of a material in a simple shear flow. 

This section considers the behavior of polymeric liquids in steady, simple shear flows - the shear-rate dependence of viscosity and the development of differences in normal stress. Also considered in this section is an elastic-recoil phenomenon, called die swell, that is important in melt processing. These properties belong to the realm of nonlinear viscoelastic behavior. In contrast to linear viscoelasticity, neither strain nor strain rate is always small, Boltzmann superposition no longer applies, and, as illustrated in Fig. 3.16, the chains are displaced significantly from their equilibrium conformations. The large-scale organization of the chains (i.e. the physical structure of the liquid, so to speak) is altered by the flow. The effects of finite strain appear, much as they do when a polymer network is deformed appreciably. 

The theory has only a single adjustable parameter, which corresponds to the Rouse time (the characteristic relaxation time for an unconfined chain) of the polymer, and it does a quite reasonable job of predicting the hnear viscoelastic response and the transient and steady-state shear and normal stresses in simple shear, ft is not as good as more complex tube-based models hke the pom-pom model, and it cannot be used for nonviscometric flows because of the absence of a continuum representation, but it contains structural details and is very useful for providing insight into the mechanics of slip. 

A consequence of finite deformations is the appearance of normal stresses in simple shearing deformations. Thus, even in steady-state simple shear flow (Fig. 1-16) where the rate of strain tensor (c/. equations 3 and 5) is... 

We now introduce the major rheological material functions, with illustrations provided by typical experimental results. Figure 7.15 depicts data obtained for low density polyethylene under steady shear flow conditions, employing a cone-and-plate rheometer. Curves display both the shear rate dependence of the viscosity, with similar results as in Fig. 7.1, and the shear rate dependence of the first normal stress difference. The stresses arising for simple shear flows may be generally expressed by the following set of equations... 

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