Rheological models normal stress difference

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 reality, however, is not as simple as that. There are several possibilities to describe viscosity, 77, and first normal stress difference coefficient, P1. The first one originates from Lodge s rheological constitutive equation (Lodge 1964) for polymer melts and the second one from substitution of a sum of N Maxwell elements, the so-called Maxwell-Wiechert model (see Chap. 13), in this equation (see General references Te Nijenhuis, 2005). 

In the next paper, Vinckier et aL [279] fitted the viscosity and the first normal stress difference of the model PIB/PDMS emulsions to Eq. (2.18). A reasonable description of the rheological behavior was obtained for the diluted and semi-diluted concentrations with the viscosity ratios A = 1.5—4. The experiments were carefiilly conducted within the range of the capillarity numbers (k < Kcr) and ... 

The type of orientational behavior strongly aflfects the rheological behavior of the fluid. In the following, rheological properties like the shear stress, the non-newtonian viscosity and the normal stress differences are presented as functions of the shear rate for a few selected vaJues for the temperature and for the other model parameters Ak and k. The underlying orientational behavior is discussed for a few representative cases and we address the question which rheologicaJ properties indicate a chaotic behavior. 

Rheological models have been described for steady shear viscosity function, normal stress difference function, complex viscosity function, dynamic modulus function and the extensional viscosity function. The variation of viscosity with temperature and pressure is also discussed. 

Leonov viscoelastic nematodynamics model was chosen for rheological simulation of TLCPs. The simulated viscosities have been compared with experimentally measured viscosity data, indicating a good fit in the shear rate range of 10-2000 s In the low shear rate region, the model did not predict negative normal stress difference for filled TLCPs. For unfilled TLCPs, good comparison has been observed between the simulated values of Ni and the experimental data. 

The relaxation processes described above apply to linear viscoelastic behavior. If the deformation is not small or slow, the orientation of the chain segments may be sufficiently large to cause a nonlinear response. We will see that this effect alone can be accounted for in rheological models by simply replacing the infinitesimal strain tensor by one able to describe large deformations no new relaxation mechanism needs to be invoked. Nonlinear effects related to orientation, such as normal stress differences, can be described in this manner. 

The second normal stress difference has been found to be negative with a magnitude less than that of Wj. It is quite sensitive to assumptions used in deriving a tube model of rheological behavior. A useful material function is the normal stress ratio it,f) defined as follows ... 

It is clear that viscoelastic fluids require a constitutive equation that is capable of describing time-dependent rheological properties, normal stresses, elastic recovery, and an extensional viscosity which is independent of the shear viscosity. It is not clear at this point exactly as to how a constitutive equation for a viscoelastic fluid, when coupled with the equations of motion, leads to the prediction of behavior (i.e., velocity and stress fields) which is any different from that calculated for a Newtonian fluid. As the constitutive relations for polymeric fluids lead to nonlinear differential equations that cannot easily be solved, it is difficult to show how their use affects calculations. Furthermore, it is not clear how using a constitutive equation, which predicts normal stress differences, leads to predictions of velocity and stress fields which are significantly different from those predicted by using a Newtonian fluid model. Finally, there are numerous possibilities of constitutive relations from which to choose. The question is then When and how does one use a viscoelastic constitutive relation in design calculations especially when sophisticated numerical methods such as finite element methods are not available to the student at this point For the... 

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