Viscoelastic Fiber Spinning Model

It is appropriate at this time to introduce viscoelastic flow analysis. Fiber spinning is one of the few processes that can be analyzed using analytical viscoelastic models. Here, we follow the approach developed by Denn and Fisher [4], Neglecting inertia, we can start with the momentum balance by modifying eqn. (6.79) as, 

Using the continuity balance and and setting the take-force to, 

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Data on extensional stresses in fiber melt spinning, and their constitutive modeling for both isothermal and nonisothermal fiber spinning, can be found in the several books referred to earlier. Further modeling elforts are discussed by Denn.(66) However, it is fair to say that no totally acceptable rheological model has so far been found to quantitatively explain viscoelastic fiber-spinning results. The question of rheological constitutive equations is examined briefly in Section 7. 

Figure 6.20 Comparison between experimental and computed velocity profiles during fiber spinning using Denn and Fisher s viscoelastic model [4].

Our analysis of fiber spinning in this chapter will be based on an inelastic rheological model of the stresses. This rheological description appears to be adequate for polyesters and nylons, which comprise the bulk of commercial spinning applications, and our spinning model is essentially the one used in industrial computer codes. This is a process in which melt viscoelasticity can sometimes play an important role, however, and we will revisit the process in Chapter 10. 

The book begins with introductory material and a brief review of fundamentals, after which the first part focuses on analytical treatments of basic polymer processes extrusion, mold filling, fiber spinning, and so forth. The thin gap (lubrication) and thin filament approximations are employed, and all analyses in this part are for inelastic liquids. An introduction to finite element calculation follows, where full numerical solutions are compared to analytical results. Polymer rheology is then introduced, with an emphasis on relatively simple viscoelastic models that have been used with some success to model processing operations. Applications in which melt viscoelasticity is important are then revisited, followed by a chapter on stability and sensitivity that focuses on melt spinning and a chapter on wall slip and extrusion... 

Better agreement with experiments was achieved by Phan-Thien (1978), who solved the fiber-spinning problem using the PTT viscoelastic model (see Eq. 3.45). In this case the constitutive equation was fitted to data for LDPE and PS and the solutions to the fiber-spinning problem were compared to experimental data. 

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