# Numerical Simulation of Polymer Solutions in a Single-Screw Extruder
## 2 Simulation Methodology  
### Geometric Representation, Governing Equations, and Boundary Conditions  
This work considers a stationary laminar three-dimensional flow of shear-thinning fluid in the metering zone of a single-screw extruder. The part of the extruder under investigation is presented in the form of a channel with a screw twisted counterclockwise (Figure 1). Mathematical modeling was performed under the assumption that gravity force is negligible and the fluid is incompressible [21]. In this regard, the distributions of hydrodynamic characteristics are identical in areas #1 and #2. Therefore, further, area #1 is considered as the area under investigation (Figure 2). In this paper, the screw rotation is simulated by the rotation of the outer pipe, this method is widely used for similar problems in the two-dimensional setting [19].  
The computational area includes a straight section of length \(L_{0}\) and a channel section with a screw of length \(L_{1}\). It should be noted that the outer wall rotation is organized in section \(L_{1}\). Here \(L\) is the pitch of the screw channel, i.e., the length of the channel part that corresponds to the rotation of the screw flights on 360 degrees, \(h\) is the flight thickness, \(r_{1}\), \(r_{2}\) are inner and outer radii of the channels.  
The governing equations in tensor form are as follows:  
\[\rho(\
abla\mathbf{v}\cdot\mathbf{v})=\
abla\cdot\left(-\rho\mathbf{I}+\eta \big{(}\
abla\mathbf{v}+(\
abla\mathbf{v})^{T}\big{)}\right) \tag{1}\]  
\[\
abla\cdot\mathbf{v}=0 \tag{2}\]  
where \(p\)--pressure, Pa; **I**--unit tensor; \(\eta\)--effective viscosity, Pa-s; \(\rho\)--fluid density, kg/m\({}^{3}\); \(\mathbf{v}=\big{\{}v_{r},v_{\varphi},v_{z}\big{\}}\)--velocity vector.  
Boundary conditions:  
(1) At the inlet, a developed velocity profile in the coaxial channel is used [30]  
\[v_{z}=2v_{a}\frac{\left(1-(r/r_{2})^{2}\right)\log(r_{1}/r_{2})-\left(1-(r_{1} /r_{2})^{2}\right)\cdot\log(r/r_{2})}{\left(1+(r_{1}/r_{2})^{2}\log(r_{1}/r_{2 })+\left(1-(r_{1}/r_{2})^{2}\right)\right)},\ v_{r}=0,\ v_{\varphi}=0 \tag{3}\]  
Figure 1: Transversal (**a**) and longitudinal (**b**) channel section.  
Figure 2: Computational area.  
where \(v_{a}\)--average velocity, defined to be the volumetric flow rate divided by the cross-sectional area of flow;  
(2) The no-slip conditions of the fluid are assumed on the channel walls of straight section (\(L_{0}\)) and on the surfaces of the screw flight and screw shaft (\(r=r_{1}\)) [16; 19; 21; 29; 31];  
(3) On the tube wall (\(r=r_{2}\)):  
\[v_{\varphi}=(-1)\cdot r_{2}\cdot 2\cdot\pi\cdot N \tag{4}\]  
where \(N\)--a number of full rotations per sec, therefore \(N^{*}=N\cdot 60\)--a number of full rotations per min. Let us note that because of the adopted orientation of the coordinate system (Figure 2), the \(v_{\varphi}\) component is taken with a negative sign to implement the correct direction of rotation of the outer pipe.  
The numerical solution of the problem is implemented in the commercial package COMSOL Multiphysics(r). It is known that, in this package, the basic Equations (1) and (2) include only the pressure gradient while the pressure function itself is absent. Thus, the pressure is determined accurately up to a constant value. Based on the above, the condition that pressure is equal to zero is specified as the boundary condition at the channel outlet [22].