Rheological flows simple shear flow

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. 

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... 

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. 

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. 

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... 

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. 

For Newtonian lipid-based food systems, it is sufficient to measure the ratio of shearing stress to the rate of shear, from which the viscosity can be calculated. Such a simple shear flow forms the basis for many rheological measurement techniques. The rheological properties resulting from steady shear flow for variety of food systems have been studied by many laboratories (Charm, 1960 Holdsworth, 1971 Middleman, 1975 Elson, 1977 Harris, 1977 Birkett, 1983 Princen, 1983 Shoemaker and Figoni, 1984 Hermansson, 1994 Kokini et al., 1994, 1995 Morrison, 1994 Pinthus and Saguy, 1994 and Meissner, 1997). 

Brady, J. R, and Bossis, G., The rheology of concentrated suspensions of spheres in simple shear flow by numerical simulations, J. Fluid Mech., 155,105-129 (1985). 

Eberle APR, et al. Modeling the rheology and orientation distribution of short glass fibers suspended by polymeric fluids simple shear flow. ANTEC, conference proceedings. Society of Plastics Engineers 2007. 

It is well known that the rheological properties of partially hydrolyzed polyacrylamide depend on the stresses associated with a given flow field. In a simple shear flow, the apparent viscosity is constant at low shear rates (Newtonian behavior). At a critical shear rate, the apparent viscosity decreases as the shear rate is increased, i.e., a shear thinning behavior [48]. The viscosity shear-rate data of water soluble-polymers are commonly fitted using the Carreau viscosity model [49]. According to this model, the apparent viscosity, p, is a function of the shear rate, Y, as follows ... 

The steady simple shear flow in Fig. 3 is fully controllable, because the velocity profile of the flow depends only on the geometry and motion of the plates, regardless of the rheological properties of the fluid. Fully developed flow in a capillary tube is only partially controllable. The material properties can affect the velocity proflle, which must be known in order to calculate the shear rate that the sample is experiencing. If a Newtonian fluid... 

Pople JA, Mitchell GR (1997) WAXS studies of global molecular orientation induced in nematic liquid crystals by simple shear flow. Liquid Crystals 23 467 Potschke P, Abdel-Goad M, Alig I, Dudkin S, Lellingta- D (2004) Rheological and dielectrical characterization of melt mixed polycarbonate-multiwalled carbon nanotube cranposites. Polymer 45 8863... 

Although most viscometric and rheological studies are carried out in simple shear flows such as rotational viscometers, real flows experienced by real liquids are very often extensional (stretching or elongational) in nahire, and for some liquids there can be a very large difference between their shear and extensional viscosities. 

Because so much rheological work has been done with simple shear flows where 7 7 Ijo = 0 (note eq. 2.2.23), most functional forms for tj have assumed rjilho) only. 

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... 

Metzner A B, Houghten W T, Sailor R A and White J L (1961), A method for the measurement of normal stresses in simple shearing flow . Transactions of Society of Rheology, 5,133-147. 

Here the usual rheological convention is used, with x being the flow direction, y the shear and z the neutral directions this simple shear flow is the prototypical viscometric motion. ... 

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