Rheologica Acta Rheol Acta 26:414-417 (1987) Extrudate swell of Newtonian fluids from converging and diverging annular dies E. Mitsoulis and F.L. Heng Department of Chemical Engineering, University of Ottawa, Ottawa, Ontario (Canada) Abstract: Finite element results are presented for the extrudate swell of Newtonian fluids from converging and diverging annular dies. Numerical calculations for a va- riety of diameter ratios and taper angles show the dependance of diameter and thickness swell on the angle. For diverging dies a thickness contraction occurs for angles greater than 30 degrees, while the diameter swell increases rapidly. For con- verging dies the design is limited to angles that do not allow contact of the inner free surfaces. The present results show that the diameter swell is highest for the diverg- ing, followed by the straight and then the converging dies. Key words: _Extrudate_swell,_tapered annular die I. Introduction In the continuing effort to gather more information about the swelling phenomenon in polymer flows through extrusion dies, both experiments and numer- ical simulation have been used. Results for Newtonian fluids extruded from capillaries show a very good agreement between theory and experiment [1]. For polymer melts the simulations have been pursued with a variety of constitutive equations with unclear results so far [2]. Meanwhile experimental research has shifted from the simplest die design of a straight capillary tube to more sophisticated annular dies for pipe formations [3]. Converging or diverging annular dies are common- ly used in the plastics industry for extrusion. In order to decouple the effect of geometry and non-Newtonian viscoelastic nature of the fluids, it is essential to consider first the Newtonian fluid behaviour in such geometries. The case of Newtonian extrudate swell from a straight annulus has been examined recently [4]. Newtonian swell from converging and diverging capillaries has also been studied [5]. The experimental work of Orbey and Dealy [3] has been performed on a variety of tapered annular dies to study the effect of die design on the swelling of different HDPE melts. It is the purpose of this work to provide the corresponding results from numerical calculations on Newtonian fluids of various diameter ratios and convergence angles in flow through tapered annular dies. 219 2. Mathematical formulation and method of solution A schematic diagram of extrusion through a tapered annular die is given in figure 1, along with the notation used. Two independent swell ratios can be defined in the case of an annular extrudate: the diameter swell B1, and the thickness swell, B2, defined by B1 = Dp , B2 =--hP (l), (2) Do h0 " A third swell ratio, the inner diameter swell, B3, follows from the above definitions B3 Dp - 2 hp (3) Do - 2h0 " The flow is governed by the general equations for conservation of mass and momentum. For an incom- pressible fluid under isothermal, creeping flow condi- tions (Re = 0) we have V. v = 0, (4) 0=-Vp+V.~. (5) where r is the velocity vector, • is the extra stress tensor and p is the scalar pressure. The constitutive equation for a Newtonian fluid is given by =/~ ~ (6) where p is a constant viscosity and f is the strain-rate tensor. The flow domain is axisymmetric and cylin- drical coordinates r, z, 0 are used.