# Numerical Simulation of Polymer Solutions in a Single-Screw Extruder
## 1 Introduction  
The single-screw extruders are the classic equipment in the processing of polymers [1]. Films, sheets, pipes, and various profiles are produced by the extrusion method. Simulation of the extrusion process allows us to study the influence of many parameters (such as geometrical, rheological) on the structure of the polymer melt and predict the final product [2]. It is very difficult to observe the processes occurring in the extruder channel, therefore one of the possible ways to control and determine the rational parameters of the entire extrusion process is mathematical modeling. In [3, 4], it is noted that currently, it is difficult to obtain an exact analytical solution to even a simplified mathematical model of the extrusion process (one-dimensional non-Newtonian flow). Analytical solutions of the problem of the Newtonian fluid isothermal flow in the channel of a single-screw extruder are known, which differ in the way they solve the system of differential equations [5, 6]. Several works are devoted to semi-analytical solutions of a similar problem, where the rheological behavior of the polymer is described by a power law [7, 8, 9]. Experimental studies of polymers (polymer melts and solutions) revealed the nonlinear viscoelastic behavior [10, 11, 12, 13, 14]. Simulation of polymer flow (polymer solutions and melts) including viscoelastic effects is a difficult task. In this regard, in most published works, elastic effects are not taken into account and polymeric media are considered as shear-thinning fluids. The power law model is the most frequently used model for the rheological description of polymer solutions and melts, for example, [2, 15, 16]. Despite the complexity of the problem being solved, the application of numerical methods is an urgent task. Numerical methods make it possible to predict the structure of the polymer melt flow, including the presence of normal stresses, adjust the equipment operating parameters, and reduce the cost of experimental studies. When developing mathematical models, the most commonly used assumptions are those that reduce the original problem to a two-dimensional formulation. For example, the assumption thatthe helical screw channel is unwound and considered as a flat rectangular channel [4; 17; 18; 19; 20; 21; 22]. Thus, plane-parallel fluid flow in a channel of infinite width is considered instead of direct 3D modeling. Also, the principle of reversed motion is often used (tube rotation is considered at the fixed screw) [4; 17; 18; 19; 20; 21; 22]. In all works, polymers are considered as shear-thinning fluid [4; 17; 18; 19; 20; 21; 22]. The main functional zones of the extruder are feeding (solids conveying), melting, and metering. The metering zone is the simplest for modeling in a single-screw extruder, since the flow laws of viscous fluids can be applied to the polymer melt flow [23]. In this zone, the polymer is completely molten [24]. Besides, it is the metering zone that determines the extruder performance [23]. Work [4] describes a simple numerical method for solving the problem of the polymer melt flow in the metering zone of the single-screw extruder. The 2D mathematical model takes into account the presence of circulating motion, non-Newtonian behavior of the processed material, and heating of the polymer due to viscous dissipation. In [3; 18; 19; 21; 22; 25], numerical solutions obtained in a 2D and 3D formulation are approximated by symbolic regression models. The heuristic approach for predicting the main characteristics of the polymer melts flow in the extruder channel is based on genetic programming. Speaking about the evolutionary computation in general, it should be noted that they, like any method using an element of randomness, do not guarantee the detection of a global extremum of the objective function (or optimal solution) for a certain time. Their main advantage is that they allow finding a \"better\" solution to a difficult task in less time than other methods.