## 4 Modeling of the Filling Phase  
For the purpose of this investigation, the same geometric configuration is employed in both the simulation and the experiments, cf. Figure 2. The mold filling simulation is configured and executed within the Ansys CFX framework. In this simulation, an inhomogeneous multiphase model [20] was employed to represent the filling process. The following boundary conditions are employed to reproduce the real process properties of the mold filling during injection molding, based on the research by [PERSON] and [PERSON] [9] and [PERSON] et al. [10]. Two different boundary conditions are implemented to account for the complex interactions at the cavity wall boundaries. At the melt boundary, called the \"melt wall\" \(A_{\text{melt wall}}\), a no-slip condition is imposed, characterized by a velocity vector \(\mathbf{U}=0\). A free slip condition is imposed at the air boundary, called the \"air wall\" \(A_{\text{air wall}}\). This is represented mathematically by \(\mathrm{T}\cdot\mathbf{n}=0\), where \(\mathrm{T}\) is the stress tensor and \(n\) is the normal vector to the surface. The inlet surface, \(A_{\text{inlet}}\), is subjected to a constant mass flow rate of the melt, \(\dot{m}_{\text{melt}}\). In order to enhance computational efficiency, a symmetrical halved geometry is utilized. Furthermore, all variables are initialized with each prescribed initial boundary condition or appropriate estimated values [9]. The inlet conditions are specified with a constant mass flow rate of \(\dot{m}_{\text{melt}}=0.015996\,\mathrm{kg/s}\) for the halved geometry. The outlet is configured as an opening and defined with a static pressure of \(p_{\text{stat}}=0\,\mathrm{bar}\). The inlet temperature is maintained at \(T_{\text{inlet}}=280\,^{\circ}\mathrm{C}\), while the wall temperature is fixed at \(T_{\text{wall}}=80\,^{\circ}\mathrm{C}\). The CFD simulation in Ansys CFX employed a volumetric mesh comprising pyramids, tetrahedra, and wedges. The total number of elements in the model is \(29,911\). The mesh was prepared with five inflation layers at the walls and a maximum element size of \(0.5\,\mathrm{mm}\). The simulation employs an adaptive time step control with a maximum time step of \(10^{-3}\,\mathrm{s}\). For all equations, a residual convergence target of \(10^{-5}\) is employed, with a limit of 50 iterations. The material properties of the air and the polymer PA66, designated as Schulamid 66 SK 1000 from Lyondell-Basell, are presented in Table 1[10; 14]. The viscosity behavior of the rheological models was fitted based on data points extracted from the CAMPUS database. For this purpose, the nonlinear least-squares Levenberg-Marquardt (nlsLM) fitting algorithm was employed in the software R. In order to ascertain these data points, the viscosity values were considered in relation to the shear rate at three distinct temperatures. The data points are given in Table 2[58]. Figure 2: Description of the geometric configuration. The melt is conceptualized as a viscous fluid with a non-Newtonian, isothermal, and incompressible flow. The rheological properties of non-Newtonian fluids are described by several different mathematical models, including the Power-Law, Second-Order, Herschel-Bulkley, Carreau, Carreau-WLF, Carreau Winter, Carreau-WLF Winter, Bird-Carreau, Cross and Cross-WLF. Table 3 presents the parameters of the rheological models, as well as the coefficient of determination (\(R^{2}\)) and the Root Mean Square Error (RMSE) between rheological model and the data points of the CAMPUS database the determined parameters. The evaluation of various rheological models based on their RMSE and \(R^{2}\) values indicates significant differences in their predictive accuracy and quality of fit. The Second-Order model demonstrates the lowest RMSE (7.7517) and the highest \(R^{2}\) value (0.9992), indicating superior predictive accuracy and an excellent fit to the data.