Viscosity rheological models

One simple rheological model that is often used to describe the behavior of foams is that of a Bingham plastic. This appHes for flows over length scales sufficiently large that the foam can be reasonably considered as a continuous medium. The Bingham plastic model combines the properties of a yield stress like that of a soHd with the viscous flow of a Hquid. In simple Newtonian fluids, the shear stress T is proportional to the strain rate y, with the constant of proportionaHty being the fluid viscosity. In Bingham plastics, by contrast, the relation between stress and strain rate is r = where is... 

An effective viscosity rp has been introduced in the Reynolds equation to describe the non-Newtonian lubricant properties. Ignoring the variation of viscosity across the film thickness, one may evaluate the effective viscosity via the following rheological model that considers a possible shearthinning effect [19],... 

Other schemes have been proposed in which data are fit to a lower, even order polynomial [19] or to specific rheological models and the parameters in those models calculated [29]. This second approach can be justified in those cases when the range of behavior expected for the shear viscosity is limited. For example, if it is clear that power-law fluid behavior is expected over the shear rate range of interest, then it would be possible to calculate the power-law parameters directly from the velocity profile and pressure drop measurement using the theoretical velocity profile... 

The sample is subjected to compression by moving the crosshead downwards at a constant speed. The sample is extruded from between the two discs, undergoing elongational or biaxial flow the sample is stretched radially and azimuthally as it flows outwards between the approaching discs. Lubrication ensures that shear flow cannot occur. Elongational viscosity is calculated directly from the measured force-distance data, disc radius and crosshead speed no rheological model is required (Campanella and Peleg, 2002). 

Estimation of Parameters. The resin viscosity, tj, as a function of time and/or temperature can be obtained using either a generalized dual-Arrhenius rheology model (Equation 5) or the thickness - time relationship for the neat resin from a separate squeeze-flow experiment (7). 

Rheological Models for Apparent Viscosity-Shear Rate Data... 

Lepez et al. (1990) examined the rheology of glass-bead-filled HOPE and PS. They found that a Cross model describes the viscosity-shear-rate relationship, a Quemada model describes the concentration dependence of the viscosity, and a compensation model applies for the tempemture dependence of the viscosity. This model is expressed as... 

In the following we often consider a rheological model more general than (6.1.4). In the three-dimensional case, this model is described by Eq. (6.1.1), where the apparent viscosity fi arbitrarily depends on the quadratic invariant of the shear rate tensor,... 

What we have done in Eqn 4-1 is to define viscosity by a rheological model applicable to any material which flows. Note that Eqn 4-1 does not imply that the ratio 6t /6y is necessarily constant volume of the fluid or during all the time of flow, many different modes of flow in nature, each with a characteristic relation for 5t. . /6y as illustrated in Fig. 4-2. if St - -/Sy is constant and -c i -t I... 

In other cases, including those involving considerable deformation of macromolecules (conformational changes) in the flow, an inverse phenomenon constituting an increase in viscosity with increasing flow rate may also take place. This phenomenon can not be described in terms of the simplest rheological models with constant parameters. Systems in which viscosity is dependent on the strain rate are referred to as anomalous, or non-Newtonian fluids. In sufficiently dilute systems, i.e. in the absence of interactions between particles, changes in viscosity due to the orientation and deformation of dispersed particles are usually rather small. 

We will now examine the simplest rheological model, the Hooke law for elasticity and Newton law for viscosity. In Hooke s law the tensor of strain Un is a linear fimction of the stress tensor of S, i.e. the deformation is proportional the acting forces. If the inertial stress, the elastic or Hooke s stress and the viscous stress are additive, we can write... 

A number of viscoelastic (i.e., rheological) models have been proposed to model steady-state creep in soils. A selection of fom of these models is presented in Figure 8.47. A model incorporating spring constants, and E2 a slider element of resistance x and a dash-pot with viscosity, v, was proposed by Murayama and Shibata (1964), which is shown in Figure 8.47a. The time-dependent deformation is controlled by the slider element Xq. Deformation will only occur for applied stresses in excess of Xq. 

Since the power-law and the Bingham plastic fluid models are usually adequate for modelling the shear dependence of viscosity in most engineering design calculations, the following discussion will therefore be restricted to cover just these two models where appropriate, reference, however, will also be made to the applications of other rheological models. Theoretical and experimental results will be presented separately. For more detailed accounts of work on heat transfer in non-Newtonian fluids in both circular and non-circular ducts, reference should be made to one of the detailed surveys [Cho and Hartnett, 1982 Irvine, Jr. and Kami, 1987 Shah and Joshi, 1987 Hartnett and Kostic, 1989 Hartnett and Cho, 1998]. 

Abstract. The viscosity is a physical parameter which controls not only the melting and fining of melts, but also the stress relaxation and the nucleation and crystallization phenomena. Here the basis of viscous flow is presented and discussed. Rheological models and some measurement methods fiber extension, beam bending and indentation are described. 

FIGURE 4 Comparison of rheological model of Eqs. (47)-(49) with experiment for natural rubber, (a) Steady-state shear viscosity, (b) Transient shear viscosity at beginning of flow, (c) Stress, relaxation following now. 

The formulations of (47) and (55) have been criticized by Leonov [L9, Lll], among others, as not being tested for consistency with the second law of thermodynamics. For Newtonian fluids, such testing requires a positive shear viscosity. For a linear viscoelastic material, one may show that the relaxation modulus function must be always positive to satisfy the second law. The requirements for Eqs. (47) and (55) are not so clear. Leonov has sought to develop nonlinear viscoelastic rheological models based on thermodynamic arguments. 

Rheological models have also been developed to describe fluid behavior over the shear rate range which include Newtonian behavior at low and high shear rates. The Carreau Model Pi has been found to fit polymer data satisfactorily. Equation 2 is the Carreau Model A. In Equation 2, i is the Newtonian viscosity in the low shear region, x is the Newtonian viscosity in the high shear regions, and i is the shear rate. The parameter n is the power-law exponent and Tj- is a characteristic time constant. All parameters are determined by fitting experimental data. 

The Gogarty correlation was developed because it was not possible to predict the apparent viscosity of polyacrylamide solutions in situ. Gogarty demonstrated that it is possible to correlate data using rheological models. However, it is necessary to have good experimental data for the polymer/rock system of interest to develop correlations needed for design and simulation. 

In the very important case of non-Newtonian fluid flow, the viscosity p, which is defined in this paper, has to be replaced by the apparent viscosity of the generalized Newtonian fluid when it is possible (pseudoplastic, dilatant, or plastic fluids). This apparent viscosity is defined from the flow rheological model representing the fluid by = f(y). 

Most polymer processes are dominated by the shear strain rate. Consequently, the viscosity used to characterize the fluid is based on shear deformation measurement devices. The rheological models that are used for these types of flows are usually termed Generalized Newtonian Fluids (GNF). In a GNF model, the stress in a fluid is dependent on the second invariant of the stain rate tensor, which is approximated by the shear rate in most shear dominated flows. The temperature dependence of GNF fluids is generally included in the coefficients of the viscosity model. Various models are currently being used to represent the temperature and strain rate dependence of the viscosity. 

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