Typical information about viscoelastic fluid properties includes the following:

This information is not enough to estimate the importance of viscoelasticity in a given process. You will also need to characterize the flow itself and compare a characteristic (relaxation) time of the material to a characteristic time of the flow. In many situations, the flow can be characterized by a critical shear rate. The critical shear rate should be understood as a wall shear rate in a region of high gradient. In an extrusion process, for example, a critical shear rate will occur at the wall in the vicinity of the die exit. In a contraction or expansion flow, the critical shear rate will occur in the smallest section.

It should also be noted that, due to technological limitations of some rheometry equipment, it is not always possible to obtain viscoelastic data in the range of shear rates (or angular frequencies) where the process operates. In this case, the only option is to extrapolate experimental data for higher shear rates or angular frequencies, and select a model on a qualitative basis. See the Ansys Polymat User’s Guide for information about using the Ansys Polymat module for this purpose.

The knowledge of both the fluid properties and the flow field can help you identify the viscoelastic character of the flow. A dimensionless number used to determine the viscoelasticity of a flow is the Weissenberg number , which is the product of the relaxation time by a typical shear rate :

(11–38)

For an axisymmetric flow problem, a first estimate of the shear rate is given by

(11–39)

where is the volumetric flow rate and is the radius.

When is low, generalized Newtonian models are sufficient to describe the flow. Only at higher values are viscoelastic models required to characterize memory effects. A high value of means that a problem is difficult to solve numerically, because of the nonlinear nature of the system.

When considering the way in which viscoelastic simulations are affected by the value of the Weissenberg number, the following facts are relevant: viscoelastic constitutive relationships often produce high stress values, in particular in the vicinity of geometric singularities; the equations involve an important advective component. The combination of both of these ingredients is often considered as the source of numerical challenges, which in the present context are referred to as the High Weissenberg Number Problem (HWNP).

It is difficult to decide which assumptions should be made about the origin of HWNP, because it can be caused by a variety of factors. Note that the relevance of a high Weissenberg number is certainly questionable. Indeed, this number is the product of a typical shear rate in the flow and a typical time constant of the fluid. Yet, it is not necessarily straightforward that the fluid behaves with the same time constant when experiencing a fast shear rate rather than a slow shear rate.

While it can be an exciting challenge to reach the highest Weissenberg number, it is probably more interesting to obtain reasonable or relevant solutions. Actually, interesting phenomena do already develop at Weissenberg numbers of 2 or 3, as opposed to higher numbers such as 10 or 100. There exist several scientific papers showing comparisons between predictions and data; and here it is not necessary to look for the highest Weissenberg number, but to show one or another interesting phenomenon.

The Weissenberg number is sometimes understood as the ratio between a normal-stress difference and a shear stress. Besides a few biological fluids, the majority of macromolecular fluids would not sustain such a high stress.

The following authors published articles in the 1980s that explore the HWNP. Note that this list is by no means exhaustive.

In addition to viscoelasticity, inertia terms can also affect the flow behavior. This can be the case for low-viscosity polymer calculations. To characterize the influence of inertia, the Reynolds number is used:

(11–40)

The combination of the Weissenberg number and the Reynolds number leads to the viscoelastic Mach number:

(11–41)

For viscoelastic flows characterized by a Mach number greater than 1 (that is, with inertia forces present), the unknown fields (stresses and velocity) are simultaneously controlled by viscoelastic forces and inertia. From a mathematical point of view, both stress and velocity fields are convected. This usually increases the level of nonlinearity. From a physical viewpoint, the combination of viscoelasticity and inertia can lead to phenomena that are not observed under other circumstances. The delayed die swell is a typical example. In this case, the swelling of the fluid does not occur immediately at the die exit, but some distance beyond, as if the fluid is not "aware" that it has entered a free region.

A particular strategy is required for high-Mach-number flows. In a physical experiment, the geometry is fixed and the flow rate will increase from a low value to the prescribed value. This leads to a simultaneous increase of viscoelastic and inertia effects (of both and ). Therefore, you should take these effects into account from the beginning of the calculation. In the frame of an evolution strategy, this means that and should simultaneously increase. You can accomplish this by increasing the flow rate or by increasing the relaxation time and fluid density through the same evolution function.

When data for the storage and loss moduli and are available, their intersection (occurring at a shear rate ) is often a reasonable choice for selecting a typical relaxation time. Flows characterized by a typical shear rate lower than are essentially dominated by viscous forces, while viscoelastic effects may play an important role in flows characterized by a shear rate greater than .

The viscoelastic model you select should reproduce the experimental behavior for shear rates in the vicinity of the characteristic shear rate of the flow to be simulated. This may lead to inappropriate model properties for low or very high shear rates, but such extremes are usually not encountered in real flow conditions. The FENE-P model is sometimes better than the others at covering the shear-thinning behavior of real polymers, at least for a single relaxation time. However, other models, such as Giesekus or DCPP, can also provide results with a similar level of quality, with one or multiple relaxation modes. Finally, for filled materials, the Leonov model can be considered. A single-mode representation is able to reproduce realistic properties, such as shear thinning.

In many circumstances, linear properties are easier to obtain than nonlinear ones. The use of empirical rules can be considered for evaluating other properties, when possible. Alternatively, it may be that only the shear viscosity is available. Usually, when the choice is permitted, shear viscosity should be considered before the linear properties.

The default viscoelastic model is the Maxwell model. As noted in Theory and Equations, the Maxwell model is the simplest viscoelastic constitutive equation from the point of view of the mathematics, but also the most difficult numerically. It exhibits a constant viscosity and a quadratic first normal-stress difference, as well as a high elongation viscosity. Due to its rheological simplicity, it is recommended only when little information about the fluid is available, or when a qualitative prediction is sufficient. It can be a reasonable candidate for the simulation of strain-hardening material in an elongation-dominated flow. While it is suitable in the aforementioned cases, it is generally recommended that you switch to at least the Oldroyd-B model for any real polymer.

The Oldroyd-B model is, like the Maxwell model, one of the simplest viscoelastic constitutive equations. As compared to the Maxwell model, it allows for the inclusion of a purely viscous component into the extra stress, which can lead to better behavior of the numerical scheme. It exhibits a constant viscosity and a quadratic first normal-stress difference, as well as a high elongation viscosity. The Oldroyd-B model can be a reasonable candidate for the simulation of strain-hardening material in an elongation-dominated flow.

The White-Metzner model is able to reproduce viscometric features such as the shear thinning and non-quadratic first normal-stress difference that characterize most fluids. The White-Metzner model also provides additional flexibility, by allowing the use of different functions for the shear-rate dependence of the viscosity and relaxation time. When experimental data about the shear viscosity and first normal-stress difference are available, you can easily obtain the material parameters for the White-Metzner model by curve fitting. First, fit the shear viscosity data. Then select the function for the relaxation time on the basis of the first normal-stress difference in simple shear flow.

The Bird-Carreau law is recommended for the relaxation time, because it yields a constant (and bounded) relaxation time at low shear rates, and enables an easier and successful start of an evolution strategy. The power-law dependence for the relaxation time should be avoided, because it leads to high relaxation times for low shear rates.

Despite its interesting features from a viscometric viewpoint, the White-Metzner model sometimes causes spurious oscillations in the solution at high shear rates. Although it may sound a bit academic, it is worth mentioning that under specific boundary conditions, the White-Metzner constitutive equation combined with the momentum equation may behave in a way that violates evolution. As a result, this model is not generally recommended; the PTT, Giesekus, and FENE-P models are preferred instead.

The PTT model is one of the most realistic differential viscoelastic models. In addition to shear-thinning viscosity, it exhibits a non-quadratic first normal-stress difference at high shear rates, as well as a nonzero second normal-stress difference. A nonzero value for will lead to a bounded, steady extensional viscosity.

However, the PTT model requires a purely viscous component to be added to the extra-stress tensor for stability reasons. See Setting the Viscosity Ratio for details. The PTT model provides poor control of the shear viscosity at high shear rates when used with a single relaxation time. The use of multiple relaxation times can improve the control.

The PTT model is a good candidate for the flow simulation of polymer melts in industrial processes such as extrusion. Care must be taken with high Weissenberg numbers (see The Weissenberg Number), and it is recommended that you focus on the model properties that are relevant within the range of the shear rate being investigated. It is also worth mentioning that viscoelasticity sometimes produces only a local or second-order effect.

Like the PTT model, the Giesekus model is one of the most realistic differential viscoelastic models. In addition to shear-thinning viscosity, it exhibits a non-quadratic first normal-stress difference at high shear rates, as well as a nonzero second normal-stress difference. A nonzero value for will lead to a bounded steady extensional viscosity.

The Giesekus model requires a purely viscous component to be added to the extra-stress tensor for stability reasons, when the parameter is larger than 0.5. See Setting the Viscosity Ratio for details.

The Giesekus model is a good candidate for the flow simulation of polymer melts in industrial processes such as extrusion. Care must be taken with high Weissenberg numbers (see The Weissenberg Number), and it is recommended that you focus on the model properties that are relevant within the range of the shear rate being investigated. It is also worth mentioning that viscoelasticity sometimes produces only a local or second-order effect.

The FENE-P model is derived from molecular theories and is based on the assumption that the macromolecules can be described as dumbbells, each consisting of two spheres linked together by a spring. Unlike in the Maxwell model, the springs are allowed only a finite extension, so that the energy of deformation of the dumbbell becomes infinite for a finite value of the spring elongation. This model predicts a realistic shear thinning of the fluid and a first normal-stress difference that is quadratic for low shear rates and has a lower slope for high shear rates.

In practice, it has been observed that the FENE-P model accurately models viscometric properties for a number of fluids. In view of the kinetic background, the FENE-P model is a good candidate for the flow simulation of low viscosity polymers and dilute solutions.

The pom-pom macromolecule consists of a backbone to which arms are connected at both extremities. In a flow, the backbone may orient in a Doi-Edwards reptation tube consisting of the neighboring macromolecules, while the arms may retract into that tube.

The concept of the pom-pom macromolecule makes the model suitable for describing the behavior of branched polymers. The approximate differential form of the model is based on the equations of macromolecular orientation and macromolecular stretching in connection with changes in orientation. In this construction, the pom-pom macromolecule is allowed only a finite extension, which is controlled by the number of dangling arms. In particular, the strain-hardening properties are dictated by the number of arms. Beyond that, the model predicts realistic shear thinning behavior, as well as a first and a possible second normal-stress difference.

The pom-pom model is a good candidate for the flow simulation of polymer melts in industrial processes such as extrusion. Care must be taken with high Weissenberg numbers (see The Weissenberg Number), and it is recommended that you focus on the model properties that are relevant within the range of the shear rate being investigated. It is also worth mentioning that viscoelasticity sometimes produces only a local or second-order effect.

Rubber compounds consist of an elastomer matrix filled with carbon black and/or silicate. From the point of view of morphology, elastomer macromolecules at rest are trapped by particles of carbon black, via electrostatic forces of the van der Waals type. Under deformation, these electrostatic bonds can break and macromolecules become free. A reverse mechanism develops when the deformation ceases. In general, elastomer systems consist of trapped and free macromolecules, with a reversible transition from one state to the other.

The Leonov model for filled elastomers is able to predict the macroscopic behavior of free and trapped chains under deformation, as well as the transition from one state to the other. It involves two tensor quantities and a scalar one. One of the tensor quantities focuses on the behavior of the free macromolecular chains of the elastomer, while the other focuses on the trapped macromolecular chains. The model is intrinsically nonlinear, since the nonlinear response develops and is observable at early deformations.

The Leonov model is a good candidate for the simulation of filled materials in rheometric flows, as well as in industrial processes such as extrusion. Note that it is computationally expensive. Care must be taken with high Weissenberg numbers (see The Weissenberg Number), and it is recommended that you focus on the model properties that are relevant within the range of the shear rate being investigated. It is also worth mentioning that viscoelasticity sometimes produces only a local or second-order effect.