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We investigate the performance of the finite volume method in solving viscoplastic flows. The creeping square lid-driven cavity flow of a Bingham plastic is chosen as the test case and the constitutive equation is regularised as proposed by Papanastasiou [J. Rheology 31 (1987) 385-404]. It is shown that the convergence rate of the standard SIMPLE pressure-correction algorithm, which is used to solve the algebraic equation system that is produced by the finite volume discretisation, severely deteriorates as the Bingham number increases, with a corresponding increase in the non-linearity of the equations. It is shown that using the SIMPLE algorithm in a multigrid context dramatically improves convergence, although the multigrid convergence rates are much worse than for Newtonian flows. The numerical results obtained for Bingham numbers as high as 1000 compare favorably with reported results of other methods.

2Department of Chemical EngineeringThe University of British Columbia 221 6 Main Mall Vancouuer, BC 6VT 124, Canadu Froduct quality and, in general, processing windows and rates of production in the extrusion of molten polymm are limited by various flow instabilities (mainly melt fi-acture) that are observed at high production rates. While there still are unresolved issues in comprehending these phenomena, industry demands for process optimization dictate the employment of processing aids (PAS) for product quality improvement and energy requirement reduction. In this review paper, most commercially used processing aids (PAS) are discussed, namely, fluoroelastomers. stearates, small PAS on the rheologid properties and the instability (melt fracture] elimination is also discussed. The factors that determine the additive performance, such as concentration, dispersion quality. and interactions between additives, additive and polymer, and additive and die surface, are also examined. The various mechanisms by which Merent types of processing aids help eliminate the instabilities are reviewed as well. Section 2 discusses the various melt fi-acture phenomena in 'Dafieated tothe lnemoryof Dr. Charles w. Stewart, an imnstlgator* made *Author rrsponsiMe for conrspdmce. emaIk h a t z i k J & h~. u b c . c a 8orrTBl CXCCPUOIld eonttibutblU3 the -Of alds. & ADDITIVE TECHIVOLOGY, MARCH 2002, Yo/. 8, No. 1 7 JOURNAL OF VINYLwhere L is the length of the capillary, A P is the total pressure drop driving the flow, and AP,, is the end correction for the pressure drop. Figure 1 also represents typical behavior of a linear polymer in general. Polymers such as LDPE and PP do not exhibit the stick-slip flow regime as explained below. Four distinct flow regimes can be identified in Rg. 1. Initially, the flow is stable, and the shear stress increases "linearly" (log-log plot) with apparent shear rate (power-law behavior), while the extrudate has a smooth and glossy appearance. When the shear stress reaches a first critical value (crcl), small amplitude, periodic distortions m. 1. Qpicclrjbw curue for UnearpolyethyIene acbuded in apiston driven cqpillnry tllustmting the various regions of instability.

Numerical simulations of the flow in an extrusion damper are performed using a finite volume method. The damper is assumed to consist of a shaft, with or without a spherical bulge, oscillating axially in a containing cylinder filled with a viscoplastic material of Bingham type. The response of the damper to a forced sinusoidal displacement is studied. In the bulgeless case the configuration is the annular analogue of the well-known lid-driven cavity problem, but with a sinusoidal rather than constant lid velocity. Navier slip is applied to the shaft surface in order to bound the reaction force to finite values. Starting from a base case, several problem parameters are varied in turn in order to study the effects of viscoplasticity, slip, damper geometry and oscillation frequency to the damper response. The results show that, compared to Newtonian flow, viscoplasticity causes the damper force to be less sensitive to the shaft velocity; this is often a desirable damper property. The bulge increases the required force on the damper mainly by generating a pressure difference across itself; the latter is larger the smaller the gap between the bulge and the casing is. At high yield stresses or slip coefficients the amount of energy dissipation that occurs due to sliding friction at the shaft-fluid interface is seen to increase significantly. At low frequencies the flow is in quasi steady state, dominated by viscoplastic forces, while at higher frequencies the fluid kinetic energy storage and release also come into the energy balance, introducing hysteresis effects.