Mathematical formulation

This paper proposes a calculation method that requires less calculation time and provides the resin distribution in both the screw axial and circumferential directions. Our method combines the Hele–Shaw flow and 2.5D FEM methods to calculate the pressure distribution from which the velocity, strain rate, temperature, and degree of fill distribution are calculated. The original FAN method is based on a flattened screw and barrel for modeling the flow behavior, and requires a certain approximation when the cylindrical shape of the screw and barrel is transformed to that of a flat plane. As such, it is limited to the degree of fill in the axial direction of the screw. Our approach addresses the screw and barrel as is, in a cylindrical coordinate system. The Hele–Shaw flow model simplifies the equation of continuity and motion. This approximation reduces the number of variables to be determined from four (vr ,v θ, v z, p ) to one (p ) for each finite element. As the number of variables becomes 1/4, the calculation time can be reduced.
The mathematical formulation consists of three parts: (1) The equations of continuity and motion of flow in the annulus are formulated in the cylindrical coordinate system based on the Hele–Shaw flow model. (2) The equations are then converted to equations of the FEM. The difference between the flight and disc screws can be considered by changing the height of the elements. The FEM solves the equations and determines the pressure of each element. (3) The pressure distribution is updated by our newly developed scheme: down-wind pressure update scheme. If the pressure of one element is negative or zero, the element is considered to be empty. If the pressure is positive, the element is filled with resin. The pressure distribution is adjusted to correspond with the theoretical value of the degree of fill as derived from the balance of drag and pressure-driven flow. Finally, it is determined that each finite element is filled or empty, and the resin distribution is obtained.
Meijer et al. described that the fluid moves in a figure ‘8’ motion around the periphery of the screw.26 Therefore, it is reasonable to neglect the flow in the self-wiping section and apply the Hele–Shaw flow approximation to a cylindrical coordinate system.

Hele-Shaw flow in cylindrical coordinates