## 4 Conclusions  
We have systematically investigated the viscous heating correction (or the temperature thinning effect) and extracted its contribution to the capillary pressure drop in thermally developing capillary flows for a series of injection molding grade thermoplastic melts (amorphous, crystalline, and filled polymers). To highlight the importance of applying the temperature correction to the shear viscosity, we begin by showing that the power-law model based on the uncorrected viscosity substantially underestimates the capillary pressure drop \(\Delta P_{\rm cap}\) (Figure 2d), and that the trend in the observed deviation is universal for all the melts investigated (Figure 7a-e). A viscous heating correction method for determining the equivalent viscosity is then developed to take into account the considerable temperature rises. As suggested by the theoretical derivation and the 3-D nonisothermal simulations, the average shear viscosity \(\overline{\eta}\) has to be evaluated by using the average fluid temperature at the wall \(\overline{T}_{\rm w}\), rather than the bulk temperature, as previously assumed. Only by performing the simulations is it possible to obtain the detailed temperature profiles over the capillary die, which would be otherwise difficult to access experimentally.  
The subtraction of the entrance pressure drop \(\Delta P_{\rm ent}\) from the total pressure drop \(\Delta P_{\rm total}\) allows us to investigate the viscous heating contribution to the capillary pressure drop \(\Delta P_{\rm cap}\) in the shear-dominated capillary flow. To assess the generality of the temperature correction, we applied the approach to several polymer melts, and the predictions of the viscous Cross-WLF model are found to be in good to excellent agreement with experimental \(\Delta P_{\rm cap}\). In every case, the onset of viscous heating is observed to coincide with \(\rm Na\approx 1\) and corresponds to a temperature rise of a few degrees (\(\sim 2\)\({}^{\circ}\)C). We also report a strong correlation between the dimensionless fluid temperature rise and the Nahme number that can be summarized by a near material-independent scaling relationship: \(\Theta\sim\) Na\({}^{0.72}\), from which the fluid temperature rise due to viscous heating as well as the resultant viscosity change can be predicted. The inferred correlation may not only be directly applied to correct the melt viscosities from a capillary die having similar dimensions, but also lend additional support to the idea that \(\overline{\eta}\) has to be evaluated by using \(\overline{T}_{\rm w}\).  
Judging from its success in retrieving the reliable equivalent shear viscosity from the 3-D nonisothermal capillary flow simulation, the methodology described in this study may be applicable to dies of different lengths or radii, or of an annular or slit cross-section, where the viscous heating becomes an issue. As the calculated fluid temperature rise at the wall highly depends on the thermal boundary conditions specified, it would be important to combine experimental and numerical approaches to determine the heat transfer coefficient at the polymer-mold interface, as well as to study its influence on the capillary flow data.  
Conceptualization, Y.-H.W., C.-C.W., C.-H.H. and R.-Y.C.; software, Y.-H.W. and G.-S.C.; formal analysis, Y.-H.W. and C.-C.W.; investigation, Y.-H.W.; writing--original draft preparation, Y.-H.W.; writing--review and editing, Y.-H.W., C.-C.W., G.-S.C., R.-H.K., C.-H.H. and R.-Y.C. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
The data presented in this study are available on request from the corresponding author.  
The authors declare no conflict of interest.