Slip Effects in Tapered Dies Savvas G. Hatzikiriakos, 1 Evan Mitsoulis 2 1 Department of Chemical and Biological Engineering, University of British Columbia, Vancouver, British Columbia, Canada 2 School of Mining Engineering and Metallurgy, National Technical University of Athens, Athens, Greece Approximate analytical equations are derived for the calculation of pressure drop of power-law fluids for viscous flow through tapered dies for a wide range of wall-slip conditions. The predicted pressure drop val- ues are compared with two-dimensional (2D) finite ele- ment calculations to identify contraction angles for which the analytical equations can be used. It is found that the disagreement increases with increase of the contraction angle and with increase of wall slip. At a given flow rate, the pressure drop from the analytical equations is found to decrease continuously with con- traction angle, which agrees with the 2D calculations only at small contraction angles. At larger contraction angles, the 2D calculations show that pressure drop increases with contraction angle as opposed to the no-slip case where pressure drop saturates. The exis- tence of a minimum pressure at a specific taper angle depends on the rheological parameters of the fluid and the degree of slip (slip-law exponent), and has scien- tific importance for the die designer. POLYM. ENG. SCI., 49:1960–1969, 2009. ª 2009 Society of Plastics Engineers INTRODUCTION Tapered dies are very important in polymer processing, including profile extrusion, molding, and film blowing [1, 2]. The calculation of pressure drop for polymer melt flow through such dies is of considerable practical importance to the design engineer [1]. The importance of converging dies in experimental rheology to assess the extensional rheological properties of polymers was stressed and exam- ined by Cogswell [3, 4]. Vlachopoulos and Scott [1] sum- marize all references studying viscous flow phenomena under no-slip through tapered dies using analytical techni- ques up to 1985. Numerical solutions to the converging flow problem of viscoelastic fluids were also provided to address the effect of extensional viscosity, and in general of viscoelasticity, in converging dies [5–7]. These effects become significant at high entrance angles where the lubrication approximation is not a valid assumption, and extensional rheology plays a crucial role [7]. All the aforementioned studies use the assumption of no- slip at the solid boundary. However, it is known that highly viscous fluids (specifically polymer melts) slip at solid boun- daries when the wall shear stress exceeds a critical value [8– 10]. Therefore, the aforementioned studies are of no use when slip occurs at the wall. This study presents analytical solutions for viscous power-law fluids flowing through tapered capillary (truncated cone) and slit (wedge) dies under slip conditions, ranging from no-slip to severe slip. The results are compared with two-dimensional (2D) finite ele- ment simulations to identify the limits of their applicability. Another motivation for this work was to explain exper- imental results of poly-tetra-fluoro-ethylene (PTFE) paste flow through tapered dies of various contraction angles [11, 12]. The authors have reported that the pressure drop versus contraction angle relationship goes through a defi- nite minimum at an angle of around 30 to 40 degrees. Corresponding data for molten polymers show an initial decrease of the pressure drop, which saturates for angles greater than 30 to 40 degrees in accord with viscous and viscoelastic finite element results [7, 13]. The increase in the pressure drop of paste flow at higher angles was inter- preted as a manifestation of solid-like behavior also experimentally observed by Horrobin and Nedderman [14]. However, solid like pastes exhibit significant slip at solid boundaries. As will be explained here through finite element simulations, viscous fluids in a certain range of severe wall slip conditions may exhibit such an increase of pressure drop at high entrance angles. ANALYTICAL EXPRESSIONS FOR PRESSURE DROP Tapered Capillary Dies We start the analysis for incompressible flow of a power-law fluid through a tapered die under wall slip. No Correspondence to: Savvas G. Hatzikiriakos; e-mail: hatzikir@apsc. ubc.ca Contract grant sponsors: Natural Sciences and Engineering Research Council (NSERC) of Canada, NTUA (KARATHEODORI) of Greece. DOI 10.1002/pen.21430 Published online in Wiley InterScience (www.interscience.wiley.com). V V C 2009 Society of Plastics Engineers POLYMER ENGINEERING AND SCIENCE—-2009