The present work is concerned with the mathematical modelling and numerical simulation of three-dimensional (3-D) bicomponent extrusion. The objective is to provide an understanding of the flow phenomena involved and to investigate their impact on the free surface shape and interface configuration of the extruded article. A finite element algorithm for the 3-D numerical simulation of bicomponent stratified free surface flows is described. The presence of multiple free surfaces (layer interface and external free surfaces) requires special free surface update schemes. The pressure and viscous stress discontinuity due to viscosity mismatch at the interface between the two stratified components is handled with both a double node ( u -v -w -P 1 -P 2 -h l -h 2) formulation and a penalty function ( u -v -w -P -h 1-h2) formulation.The experimentally observed tendency of the less viscous layer to encapsulate the more viscous layer in stratified bicomponent flows of side-by-side configuration is established with the aid of a fully 3-D analysis in agreement with experimental evidence. The direction and degree of encapsulation depend directly on the viscosity ratio of the two melts. For shear thinning melts exhibiting a viscosity crossover point, it is demonstrated that interface curvature reversal may occur if the shearing level is such that the crossover point is exceeded. Extrudate bending and distortion of the bicomponent system because of the viscosity mismatch is shown. For flows in a sheath-core configuration it is shown that the viscosity ratio may have a severe effect on the swelling ratio of the bicomponent system.Modelling of the die section showed that the boundary condition imposed at the fluid/fluid/wall contact point is critical to the accuracy of the overall solution.

A three-dimensional (3-D) non-isothermal study of viscous free-surface flows with exponential dependence of viscosity on temperature is presented. The effects of non-isothermal conditions and/or geometry on the extrudate shape are investigated with a fully three-dimensional finite element/Galerkin formulation. Apart from the well known thermally induced extrudate swelling phenomenon, bending and distortion of the extrudate may occur because of temperature differences and/or geometric asymmetries. A temperature difference across the die can be imposed by heating or cooling the die walls, but can also arise because of asymmetric viscous heat generation due to the die geometry. Temperature differences affect velocity profiles because of the temperature dependence of viscosity and lead to extrudate bending, an effect known as "kneeing" in the fiber spinning industry. It is also shown numerically and confirmed experimentally that the die geometry induces extrudate bending even in the case of isothermal Newtonian flows.

The unidirectional flow of two immiscible fluids with different viscosities in a long die of arbitrary shape is considered. Mathematically, the problem has a continuum of solutions corresponding to arbitrarily prescribed interface shapes, but experimental evidence indicates the existence of a unique interface shape with the less viscous fluid encapsulating the more viscous fluid. With the introduction of the minimum viscous dissipation principle, which postulates that the amount of viscous dissipation is minimized for a given flow rate, the problem becomes a nonlinearly constrained optimization problem. A generalized reduced gradient/finite element method combination is used to predict the interface shape when two inelastic fluids are considered. The effect of the viscosity ratio and flow‐rate ratio on the interface shape is examined for different die geometries. Inner layer breakup phenomena are predicted and explained for complex die geometries.

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SUMMARYThe solution of the non-linear set of equations arising from the application of the finite element method to non-Newtonian fluid flow problems often requires large amounts of computer time. Four iteration schemes (Picard, Newton-Raphson, Broyden and Dominant Eigenvalue method) are compared in three different flow geometries using a shear-thinning fluid model. Points of comparison involve the computer time necessary to converge the equations, ease of implementation, radius of convergence and rate of convergence.KEY WORDS Finite elements Shear-thinning flow Convergence study Newton-Raphson Picard Dominant eigenvalue Broyden