A finite element method is proposed for solving two dimensional flow problems in complex geometrical configurations commonly encountered in polymer processing. The method is applicable to flow in relatively narrow gaps of variable thickness and any desired shape. It was developed for analyzing flow in injection molding dies and certain extrusion dies. The fluid can be any non-Newtonian fluid which is incompressible, inelastic, and time independent. The flow field is divided into an Eulerian mesh of cells. Around each node, located at the center of the cell, a local flow analysis is made. The analysis around all nodes results in a set of linear algebraic equations with the pressures at the nodes as unknowns. The simultaneous solution of these equations results in the required pressure distribution, from which the flow rate distribution is obtained. Solution for the isothermal Newtonian flow problem is obtained by a one-time solution of the equations, whereas solution of a non-Newtonian problem requires iterative solution of the equations.

The flow analysis network (FAN) method previously developed for die design is adapted to the problem of the cavity filling process in injection molding. The method is applicable to relatively narrow gap cavities of any shape. It permits the computation of the advancing front of melt at any time, as well as prediction of weld-line location. The method was extended to nonisothermal flow in which solidification and “skin” formation during filling time was approximately accounted for. The nonisothermal analysis allows prediction of the possibility of a “short shot” situation. The analysis is applicable to any prescribed pressure or flow rate at the gate. Both can be arbitrary functions of time.

An improved theoretical model was derived for the solids conveying zone of a plasticating extruder. The model makes possible calculations in variable channel depth section. It also allows for a bulk density which is a function of pressure and for the non‐isotropic pressure distribution in the solid plug. An expression for maximum flow rate was also derived. Results simulated by the model on a computer indicate the effect of variables on extruder performance. The power consumption terms in the solids conveying zone of a plasticating extruder were also derived. Total power consumption is the sum of power consumptions on the barrel surface, screw surfaces and those due to pressure rise. Their relative importance was analyzed by computations. The effect of operating conditions and coefficients of friction on the various power terms was also analyzed.

Isothermal solids conveying theories have been developed in the past. However, due to friction, the surface temperature of the solid plug does increase. This change in temperature will strongly affect the temperature sensitive coefficients of friction and consequently also the pressure that develops. The surface temperature of the solid plug is also an important variable on its own because, when it reaches the melting point, the solids conveying zone is terminated. A mathematical model has been developed to calculate the temperature profile in the solid plug together with the strongly interacting pressure profile. Calculations indicate that high pressure in the solids conveying zone can practically be obtained only by very efficient cooling of the barrel in this zone.

This paper treats two cases of polymer melt solidification in rectangular geometry. The cases treated are the one of static solidification and that of solidification during flow in a narrow gap channel. Both cases are solved using the method of Dussinberre, which reduces the two-phase moving boundary case to a single phase problem, simplifying the mathematics considerably. The numerical solutions are based on a combination of the concept of flow analysis network (FAN), a finite element method developed for solving polymer flow problems, with a Crank-Nicolson implicit finite difference scheme. The methods may be used in computing the cooling down period and preventing "short shot" conditions in injection molding dies. Examples of solidification of high density polyethylene illustrate the applicability of the method.