Constitutive models, rheology

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. 

The parison is inflated fast, within seconds or less, at a predetermined rate such that it does not burst while expanding. It is a complex process that involves expansion of a nonuniform membrane-like element. Because the extension ratio is high (above 10), it is difficult to calculate the final thickness distribution. Naturally, much of the recent theoretical research on parison stretching and inflation (as in the case with thermoforming) focuses on FEM methods and the selection of the appropriate rheological constitutive models to predict parison shape, thickness, and temperature distribution during the inflation. 

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. 

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. 

Solovyov SE, Virkler TL and Scott CE (1999) Rheology of acrylonitrile-butadiene-styrene polymer melts and viscoelastic constitutive models. J Rheol 43 977-90. 

From a numerical viev point, rapid progress has been made in the last few years in studies generally devoted to the entry flow problem, together with the use of more and more realistic constitutive equations for the fluids. Consequently, more complexity was involved for the munerical problem, in relation to the nonlinearity induced by the rheological model in the governing equations. The use of nonlinear constitutive models required approximate methods for solving the equations, such as finite element techniques, even for isothermal and steady-state conditions related to a simple flow geometry. 

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. 

Figure 11.6 (a) Rheological representation of the constitutive model (b) kinematios of deformation. 

Kokini, J. L., and Plutchok, G. J. (1987a). Predicting steady and oscillatory shear rheological properties of cmc/guar blends using the Bird-Carreau constitutive model. J. Text Stud. 18, 31-42. 

By then, to be sure, we had obtained the rheological tools of the trade we needed for finding our constitutive models, but our simulation methods were still immature the contribution by Bechtel and Lang outlines the much-improved methods that have since become available. 

Y arin et al. [29, 111] gave a theory of the capillary breakup of thin jets of dilute polymer solutions and formation of the bead-OTi-the-string structure (some additional later results can be foimd in [90]). The basic quasi-one-dimensional equations of capillary jets (1.49) and (1.50) are supplemented with an appropriate viscoelastic model for the longitudinal stress. Yarin et al. [29, 111] used the Hinch rheological constitutive model, which yields the following expression... 

Non-Newtonian behavior of complex fluids is usually governed by various constitutive laws which relate the viscosity of liquids to the rate of shear. The power-law constitutive model is used in most instances due to its ability to predict rheological behaviors of a wide range of non-Newtonian liquids. The power-law model is characterized with a flow behavior index, n, and a flow consistency index, m. Specifically, n=l corresponds to Newtonian fluids whose viscosity is constant, n< corresponds to shear-thinning fluids whose viscosity decreases with increasing the rate of shear, and n>l... 

Both methods 1 and 3 involve using of constitutive model for viscosity prediction. Care should be taken to ensure the realistic behavior of the viscosity model chosen. Kennedy (1995) has shown that, while some existing models can deal with fluid behavior above the melting point, the extrapolation to low temperatures does not predict the sharp increase in viscosity for the semicrystaUine polymer. To produce a sudden rise in viscosity, we may need to incorporate the crystallization kinetics in rheology. This will be discussed in more detail in the next chapter. 

Tanner RI (2009) The changing face of rheology. J Non-Newtonian Fluid Mech 157 131-141 Tanner RI, Nasseri S (2003) Simple constitutive models for linear and branched polymers. J Non-Newtonian Fluid Mech 116 1-17... 

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