## 3 Results and Discussion
### Material-Independent Scaling Relation  
In capillary rheometry, it is of practical importance to know the onset of nonisothermal flow. A dimensionless group was used for characterizing the severity of viscous heating in the individual melts. In fact, the equation of motion (Equation (2)) and the equation of energy (Equation (3)) are coupled through the temperature-dependent viscosity. The extent of the coupling increases with the value of the Nahme number [25; 40; 41; 63]  
\[\rm Na=\overline{\
u}_{z}^{2}\alpha_{y}\eta^{0}/k \tag{14}\]  
Figure 8: Temperature rise at the wall (\(\overline{T}_{\rm w}-T_{0}\)) vs. apparent shear rate \(\dot{\gamma}_{\rm a}\) for (**a**) HIPS, (**b**) GPPS, (**c**) ABS, (**d**) PP, and (**e**) filled PA66. Lines are predictions of Equation (18). where \(\overline{v}_{z}\) is the average axial velocity (\(=4Q/\pi R^{2}\)), \(\alpha_{\eta}\) is the temperature coefficient of viscosity (\(=-\partial\eta/\eta\partial T\)), and \(\eta^{0}\) is the characteristic viscosity evaluated at a characteristic shear rate (\(\overline{v}_{z}/R\)) and characteristic temperature (\(T^{0}\)). Equation (14) can be rewritten in terms of the familiar \(\dot{\gamma}_{\mathrm{a}}\) as \(\mathrm{Na}=R^{2}\dot{\gamma}_{\mathrm{a}}^{2}\alpha_{\eta}\eta^{0}/k\). It can be seen from the relation that, at a given \(\dot{\gamma}_{\mathrm{a}}\), \(\mathrm{Na}\sim R^{2}\), whereas, at a given \(Q\), \(\mathrm{Na}\sim R^{-4}\). The former case has been demonstrated for flow of a polymer melt through two capillaries of different diameters but the same \(L/R\)[41], while the later indicates that the viscous heating occurs to a significant degree when \(R\) is very small. For a self-consistent analysis of the calculated temperature rises in Figures 8 and 9, the Na number has been determined using \(\alpha_{\eta}\) and \(\eta^{0}\) that are numerically evaluated from the temperature-corrected viscosity, not the uncorrected one. The Na number compares the viscous dissipation term and the conduction term in Equation (3). For values of Na greater than 0.5-1 (depending on the particular die geometry and thermal boundary conditions), the viscous dissipation can cause nonignorable viscosity changes. Indeed, when the temperature rises of Figure 8 are replotted in terms of the Na number, as in Figure 9a, they begin to increase rapidly upon the onset of viscous heating at \(\mathrm{Na}\approx 1\). Thus, the transition is correctly captured by our simulation. Among the materials studied, ABS and filled PA66 can attain a comparatively large Na number \(\sim\)70 at the highest shear rate, therefore the viscosity augmentation (hence increased pressure loss), after the viscous heating correction, is the most significant (Figure 4c,e or Figure 7c,e). By contrast, due to the small Na number \(\sim\) 7 it attains, PP requires a small correction to the viscosity. Since the Nahme number is a ratio of the temperature rise characteristic of the viscous heating in the capillary flow problem to the temperature change necessary to alter the viscosity [1,41], we next introduced a quantity, \(\Theta=\alpha_{\eta}\big{(}\overline{T}_{\mathrm{w}}-T_{0}\big{)}\), made dimensionless with \(\alpha_{\eta}\), and the data of Figure 9a are replotted in Figure 9b. Interestingly, we find that the effect of viscous heating on temperature rise can be summarized in Figure 9b by a near material-independent scaling relation  
\[\Theta=c_{1}\mathrm{Na}^{c_{2}} \tag{15}\]  
where \(c_{1}=0.039\) and \(c_{2}=0.72\). This finding may imply a strong correlation between the dimensionless temperature rise and the Na number for all the thermoplastic melts investigated in this study.