## 3 Numerical Modeling of the Filling Phase
### Rheological Models
#### 3.2.4 Carreau  
The Carreau model is commonly employed to characterize the flow behavior of polymers. This model originated from the seminal works of [PERSON] and [PERSON] [33; 34], and further contributions by [PERSON] et al. [35]. It can be described mathematically as follows:  
\[\eta=\frac{P_{1}}{\left(1+\left(\dot{\gamma}P_{2}\right)^{2}\right)^{P_{3}}} \tag{14}\]  
The coefficients \(P_{1}\), \(P_{2}\), and \(P_{3}\) each serve specific functions in characterizing the flow behavior of the material. Specifically, \(P_{1}\) corresponds to zero viscosity, \(P_{2}\) represents the transition region between Newtonian flow and structurally viscous behavior, and \(P_{3}\) indicates the slope of the function within the structurally viscous region. It is important to note that all three coefficients are positive, with \(P_{3}\) being less than 1 [36; 37].  
In their study, [PERSON] et al. [38] adapted this model by incorporating the Williams-Landel-Ferry (WLF) temperature shift factor \(\alpha_{T}\). This modification resulted in the Carreau-WLF model, whose equation is expressed as follows [39]:  
\[\eta=\frac{P_{1}\alpha_{T}}{\left(1+\alpha_{T}\dot{\gamma}P_{2}\right)^{2} \right)^{P_{3}}} \tag{15}\]  
The Williams-Landel-Ferry (WLF) equation, which was initially proposed by [PERSON] [40] and subsequently refined by [PERSON] [41], provides the most accurate description of the temperature dependence of viscosity in amorphous thermoplastics at lower temperatures. This equation expresses the viscosity of the polymer at a given temperature relative to a reference viscosity, which is defined at a specified reference temperature.  
The viscosity of the polymer \(\eta\) is described in terms of its relative viscosity at a specific temperature \(T\), in comparison to a reference viscosity at a reference temperature \(T_{s}\)[30; 42].  
\[\log\frac{\eta_{0}(T)}{\eta_{0}(T_{s})}=-\frac{C_{1}(T-T_{s})}{C_{2}+T-T_{s}} \tag{16}\]  
This equation applies to polymers only within the zero-shear viscosity range. The glass transition temperature \(T_{g}\) is commonly used as the reference temperature \(T_{s}\), with constants \(C_{1}=17.44\) and \(C_{2}=51.6\) K. As indicated in [40; 43]\(T_{g}\) is significantly lower than typical processing temperatures. In contrast to other proposed alternatives, [PERSON] [44] presents an alternative formulation for \(T_{s}\) that is more suitable for the given context. The alternative formulation is expressed as \(T_{s}=T_{g}+43\) K, resulting in the values of \(C_{1}=8.86\) and \(C_{2}=101.6\) K [45]. However, it is not typical to directly measure the viscosity of a polymer at the reference temperature \(T_{s}\). Instead, the measurement is conducted at a temperature within the processing temperature range \(T^{*}\). This necessitates the application of a secondary shift between the measurement or processing temperature \(T^{*}\) and the reference  
The approach proposed by [PERSON] [47] is often employed in the denominator of Equation (14). This approach results in the term \((1+\dot{\gamma}P_{2})^{P_{3}}\) being commonly used [48; 49; 50; 51]. This results in the approaches of the Carreau Winter model, as presented in Equation (18), and the Carreau-WLF Winter model, as described in Equation (19).  
\[\eta=\frac{P_{1}}{(1+\dot{\gamma}P_{2})^{P_{3}}} \tag{18}\]  
\[\eta=\frac{P_{1}\alpha_{T}}{(1+\alpha_{T}\dot{\gamma}P_{2})^{P_{3}}} \tag{19}\]