## 2 Materials and Methods
### Governing Equations  
The conservation equations for a compressible, generalized Newtonian fluid (GNF) in 3-D transient nonisothermal motion are [1]  
\[\frac{\partial\rho}{\partial t}+\
abla\cdot\rho\mathbf{v}=0 \tag{1}\]  
\[\frac{\partial}{\partial t}(\rho\mathbf{v})+\
abla\cdot(\rho\mathbf{vv}+ \boldsymbol{\tau})=-\
abla p+\rho\mathbf{g} \tag{2}\]  
\[\rho\hat{C}_{p}\bigg{(}\frac{\partial T}{\partial t}+\mathbf{v}\cdot\
abla T \bigg{)}=\
abla\cdot(k\
abla T)+\eta\dot{\gamma}^{2} \tag{3}\]  
where \(\mathbf{v}\) is the velocity vector, \(\boldsymbol{\tau}\) is the total stress tensor, \(p\) is the pressure, \(\mathbf{g}\) is the acceleration vector of gravity, \(\eta\) is the shear viscosity, and \(\dot{\gamma}\) is the shear rate (the magnitude of the rate-of-strain tensor \(\dot{\boldsymbol{\gamma}}\)). For a polymer melt modeled as a GNF, the stress tensor can be expressed as  
\[\boldsymbol{\tau}=-\eta\Big{(}\
abla\mathbf{v}+\
abla\mathbf{v}^{\mathsf{T}} \Big{)} \tag{4}\]  
The Cross viscosity model is employed to describe the polymer viscosity [38]  
\[\eta=\frac{\eta_{0}}{1+\left(\frac{\eta_{0}\dot{\gamma}}{\tau^{*}}\right)^{1-n}} \tag{5}\]  
where \(\eta_{0}\) is the zero-shear-rate viscosity, \(n\) is the power-law index, and \(\tau^{*}\) is the critical stress upon which shear thinning commences. The temperature dependence of \(\eta_{0}\) is expressed as the William-Landel-Ferry (WLF) equation, which has been found to hold for a wide variety of polymers [39; 40]  
\[\eta_{0}=\eta_{0,r}\exp\left[\frac{-A_{1}(T-T_{\mathrm{r}})}{A_{2}+(T-T_{ \mathrm{r}})}\right] \tag{6}\]  
where \(\eta_{0,r}\) is the zero-shear-rate viscosity at the reference temperature \(T_{\mathrm{r}}\), and \(A_{1}\) and \(A_{2}\) are the WLF coefficients. For simplicity, \(T_{\mathrm{r}}\) is taken to be the glass-transition temperature and \(A_{2}=51.57\) is used [1]. The conservation equations of mass, momentum, and energy (Equations (1)-(3)) can be simultaneously solved, along with appropriate boundary conditions, to obtain the temperature, pressure, and velocity profiles for a compressible fluid under nonisothermal conditions by Moldex3D flow solver. We assumed fully developed flow at the inlet plane of the barrel and 1 atmosphere pressure at the outlet plane of the capillary exit (Figure 1). Two types of thermal boundary conditions were studied: (i) The isothermal wall boundary condition and (ii) the heat transfer boundary condition. The former assumes sufficiently effective cooling of isothermal die walls, while the latter accounts for the contact resistance between the die and the melt by applying [PERSON]'s law of cooling with a heat transfer coefficient \(h\) (HTC).  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Sample** & \(\boldsymbol{T_{0}}\) (\({}^{\circ}\)C) & \(\rho\hat{C}_{p}\) **(J/m\({}^{3}\cdot\)K)** & \(k\) **(W/m\(\cdot\)K)** & **MFI (g/10 min) \({}^{\dagger}\)** & \(\lambda\) **(s)** & \(\boldsymbol{\alpha_{q}}\) **(1/K)** \\ \hline HIPS & 220 & \(2.17\times 10^{6}\) & 0.256 & 5.5 & 0.178 & \(1.51\times 10^{-2}\) \\ GPPS & 210 & \(2.08\times 10^{6}\) & 0.259 & 8.0 & 0.250 & \(1.53\times 10^{-2}\) \\ ABS & 210 & \(2.27\times 10^{6}\) & 0.235 & 1.6 & 0.222 & \(1.68\times 10^{-2}\) \\ PP & 210 & \(2.05\times 10^{6}\) & 0.279 & 14.5 & 0.109 & \(1.14\times 10^{-2}\) \\ Filled PA66 & 290 & \(2.68\times 10^{6}\) & 0.282 & 20.8 & 0.015 & \(3.44\times 10^{-2}\) \\ \hline \hline \end{tabular} \({}^{\dagger}\) HIPS, GPPS, and ABS: 200 \({}^{\circ}\)C/5.0 kg; PP: 230 \({}^{\circ}\)C/2.16 kg; filled PA66: 290 \({}^{\circ}\)C/5.0 kg.  
\end{table}
Table 1: Fluid properties of polymer melts.