In order to reduce the cost of large three-dimensional calculations of steady state free surfaces, we have combined a time-dependent approach, a decoupling algorithm and a conjugate gradient solver along the lines introduced earlier by Gresho and Chan. The free surface is calculated separately by applying the kinematic condition to a number of faces defined on the undeformed surface. For the pseudo-time-marching technique we show that it is economical to adopt different time steps for the free surface calculation and the other fields. The accuracy of the method is tested on the well-known circular die problem; the method is then used to reveal the effects of inertia and shear thinning on square and rectangular dies.
KEY WORDS Free surfaces Extrusion Conjugate gradients Finite elements Three-dimensional free surfaces
1. Introduction
The goal of the present paper is the development of a cost-effective method for calculating threedimensional extrusion flows of generalized Newtonian fluids, together with the prediction of the associated free surfaces.
The problem of three-dimensional free surface calculation has been considered by several authors over the last few years. Tran-Cong and Phan-Thien' have applied the boundary element method to extrusion processes and later extended their technique to viscoelastic flow problems.' In particular, Tran-Cong and Phan-Thien3 were able to calculate the shape of a die required by an imposed cross-section of the free jet.
The use of finite elements for such problems was introduced by Shiojima and S h i m a ~a k i . ~. ~ Their papers, however, contain little information about the numerical method and the meshupdating technique.
Finite element techniques for three-dimensional extrusion problems have been introduced by Karagiannis et aL6 They have extended to three dimensions the concepts introduced by Scriven and his collaborators (see e.g. Reference 7) for calculating free surfaces in two-dimensional flows, i.e. the use of spines for locating the free surface and for updating the mesh, together with the Newton-Raphson method for simultaneously solving the flow and the free surface. Nonisothermal flows have been considered in Reference 8, while coextrusion flows have been calculated in Reference 9. The method of spines produces some difficulties when the dies have sharp edges; it has been slightly modified in Reference 8 for calculating such flows.