Viscoelasticity in Processing Flows

Another useful dimensionless measure follows from a traditional dimensional analysis approach. Consider the Maxwell equation. 

From the Song of Deborah, Judges 5 5, The mountains quaked (sometimes translated as flowed ) at the presence of the Lord. The concept of different types of deformation on different time scales and the name of the dimensionless group were introduced by Marcus Reiner in 1964, although Reiner s definition of De was the inverse of the definition now in use. 

We suppose that there is only one characteristic length in the system, L, and one characteristic velocity, F. We further assume that the only characteristic time is L/F. We then define dimensionless quantities, denoted by an asterisk ( ), in the usual way  

There are two possible choices of the characteristic stress, G and A.GF/L. The former is the modulus, while the latter is proportional to the shear stress. (Recall that the viscosity = A.G for a Maxwell fluid.) We choose to normalize with the shear 

Substitution of these new variables into Equation 10.1 then leads immediately to the 

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The lubrication approximation as previously derived is valid for purely viscous Newtonian fluids. But polymer melts are viscoelastic and also exhibit normal stresses in shearing flows, as is discussed in Chapter 3 nevertheless, for many engineering calculations in processing machines, the approximation does provide useful models. 

Fig. 12.16 Entrance flow patterns in molten polymers, (a) Schematic representation of the wine glass and entrance vortex regions with the entrance angle. [Reprinted by permission from J. L. White, Critique on Flow Patterns in Polymer Fluids at the Entrance of a Die and Instabilities Leading to Extrudate Distortion, App/. Polym. Symp., No. 20, 155 (1973).] (b) Birefringence entrance flow pattern for a PS melt. [Reprinted by permission from J. F. Agassant, et al., The Matching of Experimental Polymer Processing Flows to Viscoelastic Numerical Simulation, Int. Polym. Process., 17, 3 (2002).]...

N2 values are always lower than Nj values, see e.g. [40]. Therefore for many processes taking into consideration only Nj will suffice. The normal stress differences are independent of the direction of flow and, in laminar flow (low y), are proportional to y2. In following p = x/y for a Newtonian fluid, normal stress coefficients ipi = Nj/y2 and ip2 = N2/y2 are occasionally used. Their dependence on the shear rate i j(y) describes the non-linear viscoelastic behavior of the fluid. 

Polymeric fluids are the most studied of all complex fluids. Their rich rheological behavior is deservedly the topic of numerous books and is much too vast a subject to be covered in detail here. We must therefore limit ourselves to an overview. The interested reader can obtain more thorough presentations in the following references a book by Ferry (1980), which concentrates on the linear viscoelasticity of polymeric fluids, a pair of books by Bird et al. (1987a,b), which cover polymer constitutive equations, molecular models, and elementary fluid mechanics, books by Tanner (1985), by Dealy and Wissbrun (1990), and by Baird and Dimitris (1995), which emphasize kinematics and polymer processing flows, a book by Macosko (1994) focusing on measurement methods and a book by Larson (1988) on polymer constitutive equations. Parts of this present chapter are condensed versions of material from Larson (1988). The static properties of flexible polymer molecules are discussed in Section 2.2.3 their chemistry is described in Flory (1953). 

In the last twenty years, major advances in the characterisation of polymer melt viscoelasticity has taken place and in addition applied mathematicians have produced numerical codes that enable viscoelastic fluids to be modelled for processing conditions. Within the last few years it has become possible to reasonably accurately predict the way in which a viscoelastic polymer will flow into, within and out of an extrusion die. The accurate prediction of die swell is nearly possible and advances are being made to predict the onset of extrusion instabilities. 

In many materials, the mechanical response can show both elastic and viscous types of behavior the combination is known as viscoelasticity. In elastic solids, the strain and stress are considered to occur simultaneously, whereas viscosity leads to time-dependent strain effects. Viscoelastic effects are exhibited in many different forms and for a variety of structural reasons. For example, the thermoelastic effect was shown earlier to give rise to a delayed strain, though recovery of the strain was complete on unloading. This delayed elasticity is termed anelastic-ity and can result from various time-dependent mechanisms (internal friction). Figure 5.9 shows an example of the behavior that occurs for a material that has a combination of elastic and anelastic behavior. The material is subjected to a constant stress for a time, t. The elastic strain occurs instantaneously but, then, an additional time-dependent strain appears. On unloading, the elastic strain is recovered immediately but the anelastic strain takes some time before it disappears. Viscoelasticity is also important in creep but, in this case, the time-dependent strain becomes permanent (Fig. 5.10). In other cases, a strain can be applied to a material and a viscous flow process allows stress relaxation (Fig. 5.11). 

Upon a large shear rate, the polymer flow exhibits nonlinear viscoelasticity. In this case, the Boltzmann superposition principle becomes invalid, and the fluid appears as a non-Newtonian fluid. A typical treatment is to consider the nonlinear resptmse as separate processes at two different time scales the first one is the rapid elastic recovery in association with the shear rate, which can relax part of the stress instantaneously the second one is the slow relaxation of the rest stress in associa-ti(Mi with time. Thus, the nonlinear relaxation modulus can be expressed as... 

It is well known that viscoelastic stress fields are developed in the flow of molten polymer. When such a flowing polymer is cooled a molecular orientation process may lead to anisotropic mechanical properties. Studies relating... 

Work on designing profile extrusion dies is complicated by the effect known as die swell. In capillary flow, elastic effects cause the diameter of the extrudate to be greater than the capillary diameter. This effect depends on the length of the capillary as well as the processing conditions and must be taken into account when designing extrusion dies. To model such an effect requires a viscoelastic constitutive equation. There is a lack of appropriate models for which data is readily available and this has hindered the use of computer simulation in this field. Nevertheless, a great deal of literature exists on simulation. ... 

Recently, significant progress has been made in the development of numerieal algorithms for the stable and accurate solution of viseoelastio flow problems, whieh exits in processes like electrospinning process. A limitation is made to mixed finite element methods to solve viscoelastic flows nsing constitutive equations of the differential type [31]. 

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