Simulation of Giesekus fluid flow in extruder using helical coordinate system

In the present work, it is proposed to use a helical coordinate system for mathematical modeling of non-isothermal Giesekus fluid flow in the extruder in the metric zone. The convenience of using a helical coordinate system is to reduce the three-dimensional problem to two-dimensional without the use of additional assumptions. The helical coordinate system is applicable only for steady flow conditions. Particular results of the numerical implementation of the use of a helical coordinate system are presented on the example of a Giesekus fluid flow in an extruder with a single flight, the results are compared with the flow of Newtonian fluid.


Introduction
The screw is a key element of the extruder and its most important part, therefore its design is crucial for the successful operation of any extrusion system [1]. The optimal design of the screw will provide a homogeneous melt with desired characteristics [2]. A detailed study of the hydrodynamics of the polymer flow in the extruder channel is the basis for modeling and understanding the extrusion process as a whole [3]. Experimental studies of liquid polymers showed a nonlinear viscoelastic nature of their behavior; therefore, almost at the same time, analytical solutions to the same problem for non-Newtonian fluids appeared. Analytical solutions of the Couette -Poiseuille flow of a viscoelastic fluid for the helical flow are known, as well as for Poiseuille flow (simpler solution for helical flow) [4]. In a number of works, for the rheological description of polymer flows, models were also used that take into account their viscoelastic properties, for example, FENE-P [5], Pom-Pom [6], Giesekus [7], PTT [8], the Leonov model [9], etc. In [10], it was proved that the PTT and Giesekus models can capture the complex nature of the polymer relaxation time at high Weissenberg numbers. Despite the complexity of the problem being solved, the use of numerical methods is important for analyzing the flow of polymers in the channels of extrusion equipment. For example, based on numerical results [11] the influence of the screw characteristics (length, diameter, pitch, normal pitch, clearance, twist angle and number of turns of the screw) and physical properties of the material (density, viscosity) on the main process were analyzed. In the general case, the velocity field of a moving fluid in a channel with a screw insert depends on three independent variables. Typically, mathematical models are used to calculate hydrodynamic fields, in which the basic system of equations is written in a Cartesian coordinate system, sometimes in a cylindrical one, whose axis coincides with the axis of symmetry of the channel. This approach is not based on helical symmetry, which is possessed by both a system of equations together with uniqueness conditions and a geometric regiona helical channel, and requires a large amount of computer memory and time for calculations in a three-dimensional setting. Using a helical coordinate system 2 allows you to get the distribution of hydrodynamic fields, depending not on three, but on two variables. This significantly reduces both the memory costs and the time needed to obtain results. In this work, we used a helical coordinate system to study the processes of the Giesekus fluid flow in a channel with a non-rotating extruder.

Helical coordinate system
In this paper, we consider the flow of an incompressible viscoelastic fluid in a channel of a singlescrew extruder (non-rotating) in the metric zone ( fig. 1). It is assumed that the fluid motion is stationary, laminar, with a formed velocity profile at the inlet to the pipe. The temperature of the liquid does not change during its movement. It is proposed to use the following helical coordinate system 1 2 3 ,, where ,, x y z -Cartesian coordinate system, channel. In eq. (1), the "+" sign is selected in the case when the fluid flow swirling clockwise, and the "-" sign is selected in the case of the flow swirling counterclockwise. Note that the coordinate system (1) has no singular points, is non-orthogonal, and compared with the coordinate system presented in previous work [12] has a larger number of nonzero components of the metric tensor. It is advisable to use it in the case where the use of the system presented in the work [12] complicated by the presence of a singular point near the flow region, which leads to difficulties in numerical calculations.

Giesekus model
To describe the rheological behaviour of viscoelastic fluid it is used the Giesekus model [13] , 2 , 2, where

Governing equation
Based on assumption, the system of equations for the transfer of momentum and continuity in the coordinate system (1) can be written as follows ( )

Results and discussion
The numerical implementation of the problem was carry out in the «Comsol Multiphysics». The package allows you to rewrite the governing equations in the helical coordinate system (1