Viscoelastic Fluid Models

The simplest model for dilute polymer solutions is to idealize the polymer molecule as an elastic dumbbell consisting of two beads connected by a Hookean spring immersed in a viscous fluid (Fig. 2.1). The spring has an elastic constant Hq. Each bead is associated with a frictional factor C and a negligible mass. If the instantaneous locations of the two beads in space are riand r2, respectively, then the end-to-end vector, R = ri — ri, describes the overall orientation and the internal conformation of the polymer molecule. The polymer-contributed stress tensor can be related to the second-order moment of R. There are two expressions namely the Kramers expression and the Giesekus expression, respectively (Bird et al. 1987b)  

For the Hookean spring, the connector force — H t and hence the Kramers expression becomes 

One can simply eliminate -cy from Eqs. 2.52 and 2.54, and obtain the following equation for RiRj)  

2 Finite Extensible Non-linear Elastic Dumbbell Model 

The unrealistic behavior of the UCM results from the fact that the Hookean spring allows extensions to go to infinity. A way of improvement is to use a more realistic force law in the model. Warner (1972) replaced the Hookean spring constant Hq by 

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For viscoelastic fluids, the formalism of a viscous fluid and an elastic solid are mixed [31]. The equations for the effective viscosity, dynamic viscosity, and the creep compliance are given in Table 12.4 for a viscous fluid, an elastic solid, and a visco-elastic solid and fluid. For the viscoelastic fluid model the dynamic viscosity, >j (tu), and the elastic contribution, G (ti)), are plotted as a function of (w) in Figure 12.31. With one relaxation time, X, the breaks in the two curves occur at co. 

B.J. Edwards and A.N. Beris, Remarks concerning compressible viscoelastic fluid models, J. Non-Newtonian Fluid Mech., 36 (1990) 411-417. 

In flow situations where the elastic properties play a role, viscoelastic fluid models are generally needed. Such models may be linear (e.g., Voigt, Maxwell) or nonlinear (e.g., Oldroyd). In general they are quite complex and will not be treated in this chapter. For further details, interested readers are referred to the textbooks by Bird et al. [6] and Barnes et al. [25],... 

An example of creep deflection in a tensile bar for an epoxy at different temperatures is shown in Fig 5.12. It will be noticed that the creep response for a temperature of 155° C still has a positive slope after seven hours. Without knowing the type of material, one might expect the response to be that of a viscoelastic fluid. The creep response for 165° C and 170° C clearly have reached a limit and has the character of a thermoset. Because of the nature of the response, the epoxy could be best characterized by a viscoelastic fluid model such as the four-parameter fluid for both the 155° C and 160° C data. On the other hand, the epoxy could best be characterized by a viscoelastic solid model such as the three-parameter solid for temperatures above 160° C. To characterize the material over all time and temperature ranges would require a generalized model with a large number of elements. Methods to accomplish this will be discussed in subsequent sections. 

Die extmsion, draw down ratio, viscoelastic fluid model, PVDF, post-extrusion shrinkage, tensile yam buckling... 

The present study attempted to numerically predict residual stress and birefringence in injection molded PC specimens with different thickness, 2.0mm and 6.5mm. Numerical simulations have been done based on a viscoelastic fluid model and commercial software MOLDFLOW by three dimensional finite element methods. The former is used to compute flow-induced residual stress, while the latter for combined residual stresses, including thermal-induced and flow-induced stresses. Effects of processing conditions on the residual are considered by the numerical simulations. As for 2.0mm PC injection molded parts, the predicted residual stresses of viscoelastic model show quite precise in accordance with experimental results. But for 6.5mm PC specimen, Moldflow simulated results have less error. 

The present study combines photoelastic experiments and numerical methods to investigate residual stresses in injection molded polycarbonate samples. There are two types of samples are measured to verify the criterion is valid. Numerical simulations are used to separate thermal-induced stresses and flow-induced stresses, while experiment results are used for comparison. Software named PARTMOLDING in which governing equations include viscoelastic fluid model is developed to calculate flow-induced residual stress mainly. Thermal-induced stresses are simulated by MOLDFLOW MPI5.0. Compared with experiments, it is found that the criterion is helpful to choose appropriate simulation theory, improve simulation precision and reliability. 

Depending on the method of analysis, constitutive models of viscoelastic fluids can be formulated as differential or integral equations. 

Johnson, M. W. and Segalman, D., 1977. A model for viscoelastic fluid behaviour which allows non-affine deformation. J. Non-Newtonian Fluid Mech. 2, 255-270. 

In the absence of body force, the dimensionless form of the governing model equations for two-dimensional steady-state incompressible creeping flow of a viscoelastic fluid are written as... 

Therefore the viscoelastic extra stress acting on a fluid particle is found via an integral in terms of velocities and velocity gradients evalua ted upstream along the streamline passing through its current position. This expression is used by Papanastasiou et al. (1987) to develop a finite element scheme for viscoelastic flow modelling. 

Keunings, R., 1989. Simulation of viscoelastic fluid flow. Tn Tucker, C. L. HI (ed.), Computer Modeling for Polymer Proces.sing, Chapter 9, Hanser Publishers, Munich, pp. 403-469. 

In generalized Newtonian fluids, before derivation of the final set of the working equations, the extra stress in the expanded equations should be replaced using the components of the rate of strain tensor (note that the viscosity should also be normalized as fj = rj/p). In contrast, in the modelling of viscoelastic fluids, stress components are found at a separate step through the solution of a constitutive equation. This allows the development of a robust Taylor Galerkin/ U-V-P scheme on the basis of the described procedure in which the stress components are all found at time level n. The final working equation of this scheme can be expressed as... 

Keeping all of the flow regime conditions identical to the previous example, we now consider a finite element model based on treating silicon rubber as a viscoelastic fluid whose constitutive behaviour is defined by the following upper-convected Maxwell equation... 

Solution of the flow equations has been based on the application of the implicit 0 time-stepping/continuous penalty scheme (Chapter 4, Section 5) at a separate step from the constitutive equation. The constitutive model used in this example has been the Phan-Thien/Tanner equation for viscoelastic fluids given as Equation (1.27) in Chapter 1. Details of the finite element solution of this equation are published elsewhere and not repeated here (Hou and Nassehi, 2001). The predicted normal stress profiles along the line AB (see Figure 5.12) at five successive time steps are. shown in Figure 5.13. The predicted pattern is expected to be repeated throughout the entire domain. 

Many investigators beheve that the Bingham model accounts best for observations of electrorheological behavior (116,118), but other models have also been proposed (116,119). There is considerable evidence that ER materials behave as linear viscoelastic fluids while under the influence of electric field (120) thus it appears that these materials maybe thought of as elastic Bingham fluids. 

The Maxwell model is also called Maxwell fluid model. Briefly it is a mechanical model for simple linear viscoelastic behavior that consists of a spring of Young s modulus (E) in series with a dashpot of coefficient of viscosity (ji). It is an isostress model (with stress 5), the strain (f) being the sum of the individual strains in the spring and dashpot. This leads to a differential representation of linear viscoelasticity as d /dt = (l/E)d5/dt + (5/Jl)-This model is useful for the representation of stress relaxation and creep with Newtonian flow analysis. 

The above model assumes that both components are dynamically symmetric, that they have same viscosities and densities, and that the deformations of the phase matrix is much slower than the internal rheological time [164], However, for a large class of systems, such as polymer solutions, colloidal suspension, and so on, these assumptions are not valid. To describe the phase separation in dynamically asymmetric mixtures, the model should treat the motion of each component separately ( two-fluid models [98]). Let Vi (r, t) and v2(r, t) be the velocities of components 1 and 2, respectively. Then, the basic equations for a viscoelastic model are [164—166]... 

One feature of the Maxwell model is that it allows the complete relaxation of any applied strain, i.e. we do not observe any energy stored in the sample, and all the energy stored in the springs is dissipated in flow. Such a material is termed a viscoelastic fluid or viscoelastic liquid. However, it is feasible for a material to show an apparent yield stress at low shear rates or stresses (Section 6.2). We can think of this as an elastic response at low stresses or strains regardless of the application time (over all practical timescales). We can only obtain such a response by removing one of the dashpots from the viscoelastic model in Figure 4.8. When a... 

Lu et al. [7] extended the mass-spring model of the interface to include a dashpot, modeling the interface as viscoelastic, as shown in Fig. 3. The continuous boundary conditions for displacement and shear stress were replaced by the equations of motion of contacting molecules. The interaction forces between the contacting molecules are modeled as a viscoelastic fluid, which results in a complex shear modulus for the interface, G = G + mG", where G is the storage modulus and G" is the loss modulus. G is a continuum molecular interaction between liquid and surface particles, representing the force between particles for a unit shear displacement. The authors also determined a relationship for the slip parameter Eq. (18) in terms of bulk and molecular parameters [7, 43] ... 

When a spring and a dash pot are connected in series the resulting structure is the simplest mechanical representation of a viscoelastic fluid or Maxwell fluid, as shown in Fig. 3.10(d). When this fluid is stressed due to a strain rate it will elongate as long as the stress is applied. Combining both the Maxwell fluid and Voigt solid models in series gives a better approximation for a polymeric fluid. This model is often referred to as the four-parameter viscoelastic model and is shown in Fig. 3.10(e). Atypical strain response as a function of time for an applied stress for the four-parameter model is found in Fig. 3.12. 

Moreover, real polymers are thought to have five regions that relate the stress relaxation modulus of fluid and solid models to temperature as shown in Fig. 3.13. In a stress relaxation test the polymer is strained instantaneously to a strain e, and the resulting stress is measured as it relaxes with time. Below the a solid model should be used. Above the Tg but near the 7/, a rubbery viscoelastic model should be used, and at high temperatures well above the rubbery plateau a fluid model may be used. These regions of stress relaxation modulus relate to the specific volume as a function of temperature and can be related to the Williams-Landel-Ferry (WLF) equation [10]. 

Using a Maxwell model as a constitutive equation for a viscoelastic fluid, one can show that the instantaneous shear stress at the wall is smaller in the viscoelastic fluid than in the corresponding Newtonian fluid. 

As mentioned in the beginning of this review (see Sect. 1), besides the theoretical importance of modelling and experiments in extension of molten polymers, there is an increasing interest in this field of rheology and mechanics of viscoelastic fluids from the technological point of view. This is connected with a wide spectrum of applied problems, the solution of which is based on data on melt extension. Below we shall discuss... 

Helical flow being analyzed as resultant from two independent flows (axial and circular), we may well assume that stable flow parameters (at least the flow rate) are determined primarily by viscous (flow) properties of the system, and the highelasticity effects (at superimposition of two flows) can be neglected in this case with a sufficient degree of accuracy which is reasonable from the point of view of engineering. The above assumtion was checked for correctness in 28,29) in a specific model of a viscoelastic fluid. 

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