Polymer flow model

Numerous examples of polymer flow models based on generalized Newtonian behaviour are found in non-Newtonian fluid mechanics literature. Using experimental evidence the time-independent generalized Newtonian fluids are divided into three groups. These are Bingham plastics, pseudoplastic fluids and dilatant fluids. 

Weighted residual finite element methods described in Chapter 2 provide effective solution schemes for incompressible flow problems. The main characteristics of these schemes and their application to polymer flow models are described in the present chapter. 

As already discussed, in general, polymer flow models consist of the equations of continuity, motion, constitutive and energy. The constitutive equation in generalized Newtonian models is incorporated into the equation of motion and only in the modelling of viscoelastic flows is a separate scheme for its solution reqixired. 

The convection term in the equation of motion is kept for completeness of the derivations. In the majority of low Reynolds number polymer flow models this term can be neglected. 

Non-Newtonian characteristics are introduced by expressing the wall shear in the capillary tube as an equivalent shear derived from a rheological model such as the power-law model (Equation 1) or the Carreau Model A (Equation 2). Derivations of polymer flow models based upon power-law and Carreau Model A are found in references 6 and 7. Equation 7... 

Fig. 7. The variation of relative drag reduction with time for mixtures of pure xanthan gum (10 ppm) and polyacrylamide (10 ppm) x, Theoretical (Brostow Polymer Flow Model) , experimental (59).

Computer modelling provides powerful and convenient tools for the quantitative analysis of fluid dynamics and heat transfer in non-Newtonian polymer flow systems. Therefore these techniques arc routmely used in the modern polymer industry to design and develop better and more efficient process equipment and operations. The main steps in the development of a computer model for a physical process, such as the flow and deformation of polymeric materials, can be summarized as ... 

The practical and computational complications encountered in obtaining solutions for the described differential or integral viscoelastic equations sometimes justifies using a heuristic approach based on an equation proposed by Criminale, Ericksen and Filbey (1958) to model polymer flows. Similar to the generalized Newtonian approach, under steady-state viscometric flow conditions components of the extra stress in the (CEF) model are given a.s explicit relationships in terms of the components of the rate of deformation tensor. However, in the (CEF) model stress components are corrected to take into account the influence of normal stresses in non-Newtonian flow behaviour. For example, in a two-dimensional planar coordinate system the components of extra stress in the (CEF) model are written as... 

The majority of polymer flow processes are characterized as low Reynolds number Stokes (i.e. creeping) flow regimes. Therefore in the formulation of finite element models for polymeric flow systems the inertia terms in the equation of motion are usually neglected. In addition, highly viscous polymer flow systems are, in general, dominated by stress and pressure variations and in comparison the body forces acting upon them are small and can be safely ignored. 

The majority of polymer flow processes involve significant heat dissipation and should be regarded as nou-isothermal regimes. Therefore in the finite element modelling of polymeric flow, in conjunction with the equations of continuity... 

Iterative solution methods are more effective for problems arising in solid mechanics and are not a common feature of the finite element modelling of polymer processes. However, under certain conditions they may provide better computer economy than direct methods. In particular, these methods have an inherent compatibility with algorithms used for parallel processing and hence are potentially more suitable for three-dimensional flow modelling. In this chapter we focus on the direct methods commonly used in flow simulation models. 

The mathematical model of a single die hole consists of equations which describe the polymer flow and heat transfer. In deriving this model the following assumptions are made ... 

Finite Element Modeling of Polymer Flow and Heat Transfer in Processing Equipment... 

Formation damage caused by clay migration may be observed when the injected brine replaces the connate water during operations such as water-flooding, chemical flooding including alkaline, and surfactant and polymer processes. These effects can be predicted by a physicochemical flow model based on cationic exchange reactions when the salinity decreases [1665]. Other models have also been presented [345,1245]. 

Case materials, in electronic materials packaging, 77 837-840 Case nitrided steels, properties of, 76 207 Case polymers, uses for, 25 481 82 Cash flows, 9 540-542 CASLINK software, 73 250 Caspian Sea, 5 784 CAS Registry, 73 242, 246 Cassegrain condenser, reflective, 74 233 Cassia, 23 165-166 Cassie-Baxter equation, 22 112, 113 Cassiterite, 24 783, 791 Casson-Asbeck plots, 27 709 Casson fluid flow model, 27 705 CASS test, 9 790... 

Keywords. Viscoelasticity, Molecular rheology. Branched polymers. Tube model, Non-Newtonian flow... 

The nature of the liquid in contact with a surface is also very important, with respect to boundary conditions. Although slip has long been observed for highly non-Newtonian, viscoelastic liquids such as polymer flows and extrusions, many recent studies have reported slippage of Newtonian liquids under a variety of experimental conditions. This clearly indicates that care must be taken when modeling any type of micro- or nanofluidic system, no matter which liquid is employed. 

E. A. J. F. Peters, Polymers in Flow Modelling and Simulation, Ph.D. thesis, Technische Universiteit Delft (2000). 

Evidently a fluid polymer cannot be considered in the model the deformation approaches to a limit. For a solid polymer the model seems more appropriate, though is represents neither a spontaneous elastic deformation nor permanent flow. Therefore a combination of a Kelvin-Voigt element with a spring and with a dashpot in series is, in principle, more appropriate. 

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