Giesekus model

A review by Bird and Wiest [6] gives a more complete list of existing viscoelastic models. The upper convective model and the White-Metzner model are very similar with the exception that the White-Metzner model incorporates the strain rate effects of the relaxation time and the viscosity. Both models provide a first order approximation to flows, in which shear rate dependence and memory effects are important. However, both models predict zero second normal stress coefficients. The Giesekus model is molecular-based, non-linear in nature and describes thepower law region for viscosity andboth normal stress coefficients. The Phan-Thien Tanner models are based on network theory and give non-linear stresses. Both the Giesekus and Phan-Thien Tanner models have been successfully used to model complex flows. 

For each r, a constitutive equation must be selected. Dooley and Dietsche [5] evaluated the White-Metzner, the Phan-Thien Tanner-1, and the Giesekus models, given by,... 

Figure 9.41 presents the predicted secondary flow patterns that result from the vicoelastic flow effects. The Giesekus model with one relaxation time was used for the solution presented in the figure. For the simulation, a relaxation time, A, of 0.06 seconds was used along with a viscosity, r], of 8,000 Pa-s and a constant a of 0.80. Similar results were achieved using the Phan-Thien Tanner-1 model. As expected, when the White-Metzner model was used, a flow without secondary patterns was predicted. This is due to the fact that the White-Metzner model has a second normal stress difference, N2 of zero. 

Sepehr et al. [2008] are investigating the viscoelastic Giesekus model [Giesekus, 1982, 1983 Bird et al., 1987] coupled with Eq. (16.41), with Dr described by Doi [1981]. The interactions between polymer and particles were incorporated following suggestions by Fan [1992] and Azaiez [1996]. These authors used Eq. (16.42) with the contribution to stress tensor caused by clay platelets [Eq. (16.43)] and viscoelastic Giesekus matrix expressed as [Fan, 1992]... 

To help understand and quantitatively evaluate the secondary movement shown above, Debbaut et al. [75, 77] augmented this experimental work with a three-dimensional flow simulation that incorporated viscoelastic effects. The finite element method, using a 4-mode Giesekus model as the viscoelastic constitutive equation, was used for the simulation. The polymer used for the experiment and simulation was a low-density polyethylene. Figures 12.20 and 12.21 show the experimental observations and the numerical predictions of the deformations of the interface for the rectangular straight channel [78], and for the teardrop channel [75], respectively. 

The application of the White-Metzner, Phan-Thien Tanner and Giesekus models was done by Dietsche, L.,... 

Phan-Thien-Tanner (PTT) Model and Giesekus Model... 

Doufas et al. (1999, 2000) proposed a two-phase model based on a modified Giesekus model for the amorphous melt phase and a rigid dumbbell model for the semi-crystalline phases. In the modified Giesekus model the relaxation time is a function of relative crystallinity a ... 

There are numerous other constitutive equations of both differential and integral type for polymer melts, and some do a better job of matching data from a variety of experiments than does the PTT equation. The overall structure of the differential equations is usually of the form employed here The total stress is a sum of individual stress modes, each associated with one term in the linear viscoelastic spectrum, and there is an invariant derivative similar in structure to the one in the PTT equation, but with different quadratic nonlinearities in t and Vv. The Giesekus model, for example, which is also widely used, has the following form ... 

Kohler and McHugh s calculations for Young s Run 9 indicated that there was substantial crystallization. The Giesekus model for the melt is still inadequate, despite the inclusion of stress-induced crystallization in the spinline model, and the response does not differ greatly from the PTT simulations without crystallization. 

Christenson and McKinley [ 19] evaluated a generalized linear Maxwell model as well as the upper convected Maxwell model and the Giesekus model. These authors worked with the tensorial forms of these functions which are capable of correctly treating large strain deformations. 

In the Giesekus model, the parameters a, are greater than zero and are called the mobility factors. 

For the generalized linear Maxwell model (GLM) and the upper convected model (UCM), the only material parameters needed are contained in the relaxation time spectrum of the material which can be obtained from simple linear viscoelastic measurements. For the Giesekus model, one needs in addition the mobility factors which Christensen and McKinley obtained by fitting the stress-strain curves of the adhesive. The advantage of the Giesekus model was that it provided them with a better description of the stress-strain curves. This, of course, is to be expected since those curves were used to deduce the parameters of the model. 

Therefore, with the exception of the Giesekus model, the parameters for all of these constitutive equations can be deduced from the relaxation time spectrum of the material which can be obtained from the small strain linear viscoelasticity measurements alone. There are various numerical methods in the literature which allow the determination of this spectrum from measured viscoelastic master curves, such as dynamic modulus, relaxation modulus, and creep compliance. 

For example, Christenson et al. [3,19] performed a detailed study of polyisobutylene-based pressure-sensitive adhesives. Although these authors did not postulate a specific detachment criterion, they did extensive work characterizing the linear viscoelastic properties, the tensile stress-strain properties, and the peel force. In addition, they conducted detailed visualization of the deformation of the adhesive during peel and therefore, could assess the ability to predict the peel force from the mechanical properties of the adhesive and the visually observed detachment strain. In this work, the adhesive consisted of a blend of high and low molecular weight polyisobutylene. They showed that when they used the Giesekus model as the constitutive equation for the adhesive, they could accurately describe the stress-strain curves of the adhesive and the peel force was well predicted by the integral of the stress-strain curve up to the measured detachment strain. Their results are summarized in Table 1. 

Here, G is the shear modulus and S the stress tensor. It is easy to see that the Giesekus model gives a simple relation between all rheological parameters by inducing the anisotropy of a streaming solution. 

This law gives a decreasing viscous resistance with increasing shear rate and it therefore describes shear thinning or pseudo-plastic behavior. It is often convenient to use dimensionless variables in order to compare experimental data with the predictions of the Giesekus model. This can be achieved by dividing the steady-state shear viscosity by the zero shear viscosity and by multipl3dng the shear rate with the relaxation time ... 

It is evident that the Giesekus model can quantitatively describe the shear thinning behavior of the entangled solutions of rod-shaped micelles. The decrease of the viscous resistance is caused by the alignment of the anisometric aggregates in the streaming solutions. Similar conclusions can be drawn from measurements of the first normal stress difference. This parameter is often represented in terms of the first normal stress coefficient ... 

Figure 9.7 Normalized shear viscosity ri (oo,x) as a function of normalized shear rate % compared to the theoretical predictions of the Giesekus model (lines) (T = 20°C, 15%-MDMA0-HN03, pH =...

In Figure 9.8 we compare measured values of the normalized first normal stress coefficient with Equation 9.35. Again, we observe a pretty good agreement between experimental data and the predictions of the Giesekus model. We can thus conclude that this theoiy describes very well the mechanical anisotropy of the streaming viscoelastic surfactant solution. 

Beside these relaxation experiments, the Giesekus model was also applied for different types of rheological experiments.Up to now, no significant deviations between experimental values and theoretical predictions were detected. The Giesekus model thus gives a successful description of the nonlinear rheological properties of viscoelastic surfactant solutions. This holds for an anisotropy factor a = 0.5. ... 

The shear stress of the Giesekus model is represented by Equation 9.34. This parameter can easily be normalized ... 

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