Rheology simple shear

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. 

Most rheological data on polymer liquids of known structure has been obtained in simple shearing deformations. The velocity field for homogeneous simple shear in rectangular Cartesian coordinates may be expressed ... 

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. 

Three kinds of viscometric flows are used by rheologists to obtain rheological polymer melt functions and to study the rheological phenomena that are characteristic of these materials steady simple shear flows, dynamic (sinusoidally varying) simple shear flows, and extensional, elongational, or shear-free flows. 

The steady and dynamic drag-induced simple shear-flow rheometers, which are limited to very small shear rates for the steady flow and to very small strains for the dynamic flow, enable us to evaluate rheological properties that can be related to the macromolecular structure of polymer melts. The reason is that very small sinusoidal strains and very low shear rates do not take macromolecular polymer melt conformations far away from their equilibrium condition. Thus, whatever is measured is the result of the response of not just a portion of the macromolecule, but the contribution of the entire macromolecule. 

Rheological Response of Polymer Melts in Steady Simple Shear-Flow Rheometers... 

The two-way arrow between polymer rheology and fluid mechanics has not always been appreciated. Traditionally we look at polymer rheology as input to fluid mechanics, as a way to supply constitutive equations. Gary Leal pointed out the use of fluid mechanics to provide feedback to tell us whether the constitutive equation is satisfactory. In the past, we tested constitutive models by examining polymeric liquids with very simple kinematics, homogeneous flows as a rule, either simple shear or simple shear-free types of flows. We don t actually use polymers in such simple flows, and it s essential to understand whether or not these constitutive equations actually interpolate properly between those simple types of kinematics. So there s a two-way arrow that we have to pay more attention to in the future. 

In principle at least, Eq. (7.14) provides the basis for a complete calculation of the configuration-specific rheological properties of the suspension at each given instant of time t. In order to calculate therefrom the time-averaged properties of the suspension, consider the macroscopic simple shear flow discussed previously in connection with Fig. 4. 

A particularly useful expression for the complex dynamic moduli was introduced using operators and fractional derivatives for the viscoelastic constitutive equations ( ). For a rheologically simple system the complex shear modulus is... 

A few rheometers are available for measurement of equi-biaxial and planar extensional properties polymer melts [62,65,66]. The additional experimental challenges associated with these more complicated flows often preclude their use. In practice, these melt rheological properties are often first estimated from decomposing a shear flow curve into a relaxation spectrum and predicting the properties with a constitutive model appropriate for the extensional flow [54-57]. Predictions may be improved at higher strains with damping factors estimated from either a simple shear or uniaxial extensional flow. The limiting tensile strain or stress at the melt break point are not well predicted by this simple approach. 

If the slip parameter a is a non-zero constant, the requirement that the shear stress be a monotonically increasing function of the shear rate in simple shear flow imposes a constraint upon the viscosity ratio. Using a spectrum of relaxation times loosens this constraint and allows for more realistic fitting of the rheological data. 

Doi and Edwards (1978) and Kuzuu and Doi (1980) have solved the Smoluchowski equation (6-47)-(6-48) for simple shearing and elongational flows, and they obtained predictions of rheological behavior that are similar to those of the reptation theory for concentrated flexible polymers discussed in Section 3.7.5.1. Figure 6-19, for example, shows the shear-rate-dependence of the shear viscosity and first and second normal stress coefficients predicted by the Doi-Edwards theory for semidilute rods these results are similar to those predicted by the Doi-Edwards theory for entangled flexible molecules. At... 

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. 

In the previous sections, the non-Newtonian viscosity rj) was used to characterize the rheology of the fluid. For a viscoelastic fluid, additional coefficients are required to determine the state of stress in any flow. For steady simple shear flow, the additional coefficients are given by... 

In many of the rheological test methods, the sample is subjected to simple shear. The best example for illustration is that of a pack of cards (Fig. 3). If a shear stress T is applied to the top ( reference ) card of the pack, the displacement of each individual card of the pack is proportional to its distance from the reference card. The larger the distance is, the less is the card displaced. The relative displacement of two cards divided by their separation (51/1) is called simple shear strain. ... 

S. R. Rastogi, N. J. Wagner, and S. R. Lustig, J. Chem. Phys., 104, 9234 (1996). Rheology, Self-Diffusion, and Microstructure of Charge Colloids Under Simple Shear by Massively Parallel Nonequilibrium Brownian Dynamics. 

Shear rheology occurs when simple shearing deformation is applied to the material and shear rheological properties are measured, Figure 3.63. 

If we interpret this question as asking whether models exist for the general class of complex/non-Newtonian fluids that are known to provide accurate descriptions of material behavior under general flow conditions, the current answer is that such models do not exist. Currently successful theories are either restricted to very specific, simple flows, especially generalizations of simple shear flow, for which rheological data can be used to develop empirical models, or to very dilute solutions or suspensions for which the microscale dynamics is dominated by the motion deformation of single, isolated macromolecules or particles/drops.24... 

We can see that Eqs. (2 101) (2-104) are sufficient to calculate the continuum-level stress a given the strain-rate and vorticity tensors E and SI. As such, this is a complete constitutive model for the dilute solution/suspension. The rheological properties predicted for steady and time-dependent linear flows of the type (2-99), with T = I t), have been studied quite thoroughly (see, e g., Larson34). Of course, we should note that the contribution of the particles/macromolecules to the stress is actually quite small. Because the solution/suspension is assumed to be dilute, the volume fraction is very small, (p 1. Nevertheless, the qualitative nature of the particle contribution to the stress is found to be quite similar to that measured (at larger concentrations) for many polymeric liquids and other complex fluids. For example, the apparent viscosity in a simple shear flow is found to shear thin (i.e., to decrease with increase of shear rate). These qualitative similarities are indicative of the generic nature of viscoelasticity in a variety of complex fluids. So far as we are aware, however, the full model has not been used for flow predictions in a fluid mechanics context. This is because the model is too complex, even for this simplest of viscoelastic fluids. The primary problem is that calculation of the stress requires solution of the full two-dimensional (2D) convection-diffusion equation, (2-102), at each point in the flow domain where we want to know the stress. 

The rheological nature of a material can be defined by a relation between components of stress and deformation in simple shear, as illustrated in Figure 5.1. 

The relationship between an applied stress or strain and the response of the material, shear rate, or deformation is the aim of the rheology of suspensions. Normally, both the stress and the strain are tensors with each having nine components. In simple shear, which is the most common way of determining the rheological behavior, the shear stress oxy (some literature also uses the symbol r to stand for the shear stress) can be related to the shear rate y by... 

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