## 2 Materials and Methods
### Temperature Profiles and Capillary Pressure Drops  
To validate the computed temperature profiles for capillary flow with viscous heating, we compared our simulation result with the analytical solution of a power-law fluid [1; 3], assuming (i) the die wall is maintained at a fixed temperature \(T_{0}\); (ii) the fluid is described adequately by a power-law viscosity, \(\eta=m\dot{\gamma}^{n-1}\), with \(m\) and \(n\) independent of temperature; (iii) the velocity profile is fully developed at the die entrance; (iv) \(\rho\), \(\dot{C}_{p}\), and \(k\) do not vary  
Figure 1: 3-D and partial view of the two meshes used in the capillary flow simulations for (**a**) \(L/D=0.2\) (\(D=1\) mm and \(2\phi=90^{\circ}\)) and (**b**) \(L/D=30\) (\(D=1\) mm and \(2\phi=180^{\circ}\)). with temperature or pressure. An example of such calculation is provided for the filled PA66 in Figure 2a. The melt had the following physical properties at \(T_{0}=290\)\({}^{\circ}\)C (Table 1): \(\rho\hat{\mathcal{C}}_{p}=2.68\times 10^{6}\,\mathrm{J/m^{3}\cdot K}\); \(k=0.282\,\mathrm{W/m\cdot K}\); \(m=4.05\times 10^{3}\,\mathrm{J/Pa\cdot s^{n}}\); \(n=0.50\). The flow curve in Figure 2c manifests a power-law flow regime \(\eta\sim\dot{\gamma}^{-(0.50\pm 0.05)}\). At a fixed axial distance \(z=15\,\mathrm{mm}\) (i.e., midpoint of the 30-mm-long capillary die depicted in Figure 1b), the temperature profiles at \(\dot{\gamma}_{\mathrm{a}}=50-5000\,\mathrm{s^{-1}}\) are plotted in Figure 2a. As our simulation (Simulation A) is subject to the same assumptions (i\(-\)iv) as in the analytical power-law model [1; 3], the agreement between Simulation A (filled symbols) and the analytical solution (dash-dot lines) is remarkably good, except that the simulation predicts an extra temperature rise by \(\sim\)0.5\({}^{\circ}\)C near the tube center. The small discrepancy is caused by a small viscous heating contribution when the fluid undergoes a sudden contraction upon entering the tube (i.e., entrance effects). At high shear rates, there is a peak in the temperature profile near the wall where the velocity gradient and also the viscous heating are large. In this isothermal-wall case, where the heat transfer at the wall is extremely effective, the temperature rise can still be as large as \(\sim\)12\({}^{\circ}\)C. It is thus evident that viscous heating can produce nonignorable temperature rises in actual capillary rheometry. Figure 2: Calculated temperature profiles (\(T-T_{0}\)) under the (**a**) isothermal wall and (**b**) finite heat transfer boundary conditions for the filled PA66 melt at \(T_{0}=290\)\({}^{\circ}\)C. The power-law fit to the uncorrected shear viscosity \(\eta\) (\(\dot{\gamma}\)) and the calculated capillary pressure drop \(\Delta P_{\mathrm{cap}}\) are, respectively, shown in (**c**,**d**). Simulation A: \(\eta\
eq\eta(T)\) and isothermal wall; Simulation B: \(\eta=\eta(T)\) and isothermal wall; Simulation C: \(\eta=\eta(T)\) and finite heat transfer at the wall. In contrast to the analytical power-law model where the temperature-independent viscosity is assumed, we next took into account the temperature dependence of \(m\) in the same figure (Figure 2a, Simulation B). Since the temperature rise can now result in a lower viscosity (and hence less viscous heating), we find in Figure 2a that the temperature rises from Simulation B (open symbols) begin to fall below those from the analytical power-law model above a critical shear rate 1000 s\({}^{-1}\). The departure from the analytical power-law model indicates that the viscosity change, however, cannot be ignored if the temperature rise is more than a few degrees (\(\sim\)2 \({}^{\circ}\)C) [1]. Later, we demonstrate that the onset of viscous heating can be reliably predicted by the Nahme number [25; 40; 41], whose magnitude gives an indication of the relative importance of the rate of heat generation by viscous dissipation and the rate of heat removal by conduction at the capillary wall.