## 3 Results and Discussion
### Rheological Properties at Wall in Nonisothermal Flow  
Figure 5 shows the results of nonisothermal capillary flow simulations for HIPS, based on shear viscosities with and without temperature correction. Because, in both cases (\(n\) vs. corrected \(n\)), the trends in the rheological properties with axial coordinate \(z\) are very similar, we therefore focused on the simulated results based on the temperature-corrected shear viscosity (solid line) in this illustrative example. The simulation results for the other test polymers were qualitatively the same as those observed for HIPS and are hence not shown here. We can see in Figure 5a that, in the contraction region, the iso-contours of pressure become virtually independent of \(r\) after a small entrance length \(\sim\)0.2 mm. This clear evidence is in support of the assumption made in the derivation of Equation (13) that only the axial variation in pressure is present (i.e., \(P=P(z)\)), whereas the radial variation is small enough to ignore (i.e., \(dP/dr\approx 0\)). As there is a sharp corner in the contraction regime, the steep stress gradient at \(z\approx 0\) (see Figure 5b) is then associated with the singularity (i.e., high stresses generated at the entrance to the smaller channel). In Figure 5b, the shear stress \(\tau_{\rm w}\) and fluid temperature \(T_{\rm w}\) at the wall are plotted versus \(z\) for \(\dot{\gamma}_{\rm a}=5000\) s\({}^{-1}\) and \(T_{0}=220\)\({}^{\circ}\)C (\(z=0\) corresponds to the die entrance), where it can be observed that \(T_{\rm w}\) increases in the flow direction due to viscous heating, while \(\tau_{\rm w}\) shows the opposite due to a progressively smaller pressure gradient (Figure 5c). The fluid temperature profiles \(T(r,\,z)\) at different \(z\) are plotted in Figure 5d, where it can be seen that the fluid reaching one-third of the die length (i.e., \(z=10\) mm) has attained a fluid temperature close to its average (\(\overline{T}_{\rm w}=234\)\({}^{\circ}\)C), therefore \(T_{\rm w}\) does not vary in a linear fashion with \(z\). Furthermore, the viscous heating causes a fluid temperature rise as large as \(\sim\)20 \({}^{\circ}\)C at the tube exit (\(z=30\) mm), clearly indicating the necessity of the temperature correction to the shear viscosity. Figure 4: Steady-state shear viscosity \(\eta\) (\(\dot{\gamma}\)) with and without temperature correction for **(a)** HIPS, **(b)** GPPS, **(c)** ABS, **(d)** PP, and **(e)** filled PA66. The open and filled symbols represent, respectively, the uncorrected and the equivalent \(\eta\) (\(\dot{\gamma}\)). The Cross-WLF model fits based on the corrected \(n\) are shown by lines. When \(\tau_{\rm w}\) is plotted versus \(T_{\rm w}\) in Figure 5e, we see a nearly linear relation between the two quantities, especially when the entrance length has been exceeded (i.e., \(z>0.2\) mm). Figure 5: Graphs elucidating the simulated rheological properties along the capillary wall for HIPS at \(\dot{\gamma}_{\rm a}=5000\) s\({}^{-1}\) and \(T_{0}=220\)\({}^{\circ}\)C, based on \(\eta\) (\(\dot{\gamma}\)) with and without temperature correction: (**a**) Pressure iso-contours in the contraction region and axial variations of (**b**) shear stress \(\tau_{\rm w}\) and fluid temperature \(T_{\rm w}\), (**c**) capillary pressure drop \(\Delta P_{\rm cap}\) and polymer viscosity \(\eta\), (**d**) fluid temperature profile \(T(r,\ z)\), and (**f**) shear rate \(\dot{\gamma}\). The data of (**b**) are replotted against \(T_{\rm w}\) in (**e**) to reveal an approximately linear relationship between \(\tau_{\rm w}\) and \(T_{\rm w}\). Thus, it explains why, in going from Equation (11) to (12), we have used the mean shear stress \(\overline{\tau}_{\rm w}\) and fluid temperature \(\overline{T}_{\rm w}\) in determining the mean shear viscosity \(\overline{\eta}\). In Figure 5c, the change in viscosity after the correction is \(\sim 15\%\), therefore one can see a similar increment in the overall \(\Delta P_{\rm cap}\) in the same figure, in accordance with Equation (13).