*4.2. Slow Drawing Ratios*

In this section, we first show representative results obtained by solving the system of Equations (36), (37), (43) and (45) for the case DR54-15. Figure 3a shows that the axial velocity profile assumes values close to zero at the beginning of drawing until *z* ≈ 0.4. Afterwards, it varies abruptly and suddenly increases, reaching the final value of one at *z* ≈ 0.8. To satisfy the mass conservation equation, the inner and outer surfaces greatly reduce their size before reaching the final dimensions; see Figure 3b. The radii significantly change their sizes and the axial velocity steeply increases only in a small portion of the furnace length measured by Voyce et al. [22]. They termed this part of the oven "hot zone", where the viscosity is low enough to enable the glass to deform. In Figure 3c, we contrast the evolution of the temperature profiles of the inner and outer surfaces of the capillary with that of the furnace along the drawing direction. At the beginning of the drawing, where the speed of the internal and the external surfaces of the capillary are very close to the feed speed, the temperature profiles of the glass surfaces are very close to the temperature distribution of the furnace, indicating thermal equilibrium between the furnace and the capillary. Starting from approximately *z* ≈ 0.5, the temperature profiles of the capillary surfaces deviate from that of the furnace, resulting in lower values of the glass temperature. Two possible causes lie in the reduced radiative heat exchange between the furnace and the fibre due to the smaller surface and in the significant increase of the convective heat exchange due to the higher values the capillary speed achieves toward the end of the drawing. Furthermore, we observe that the differences between the temperature profiles of the inner and outer radii of the capillary are negligible (see Figure 3c), indicating strong heat diffusion inside the glass due to the radiative contribution to the thermal conductivity. This is clear from the temperature contour displayed in Figure 4, which

shows a substantial axial temperature variation across the fibre length and an imperceptible temperature change in the whole radial direction. In Figure 5, we can compare the final external diameters of the capillaries computed by numerically solving Equations (36), (37), (43) and (45) with the ones obtained experimentally by Luzi et al. [20] for the case of three slow drawing ratios without internal pressurization, that is, DR 36-1, DR 54-15, and DR 72-2, as reported in Table 1. In Figure 6, we compare their final air-filling fractions, that is, the ratio of their final internal radii *h*<sup>1</sup> *<sup>f</sup>* over their final external radii *h*<sup>2</sup> *<sup>f</sup>* . By doing this, we vary the peak temperature from 1850 ◦C up to 2050 ◦C in steps of 25 ◦C. The values of both the outer and inner final radii at the end of the drawing process gradually increase with decreasing furnace peak temperature for all three drawing ratios, as the higher viscosity at lower temperatures values hinders the capillary from further decreasing its size. With regard to the final external diameter, the numerical and experimental results are in excellent agreement, with a difference ranging from a minimum of approximately 0.06% up to a maximum of approximately 2%. With regard to the final air-filling fractions of the capillaries, the agreement between the numerical simulations and experimental results is excellent only in the temperature range between 1850 ◦C and 1975 ◦C, with a maximum difference of approximately only 2.8%. In the temperature range between 2000 ◦C and 2050 ◦C, the discrepancy between the numerical and experimental results varies between approximately 5% and 48%. This maximum extreme difference may be due to too low a value of the surface tension used in the simulations, that is, *γ* = 0.25 N/m. Another reason for the discrepancy may rely on the possible partial collapse of the inner surface of the capillary. The temperature profile may not be accurately described by the functional form provided by Equation (26) in the case of high peak temperatures, as it has been determined by fitting experimental data obtained at lower peak temperatures. This has a direct impact on the collapse of the capillary, as a simple order of magnitude analysis demonstrates that the collapse time of the inner capillary surface depends on both the viscosity and the surface tension of the molten glass [48]. Because the viscosity greatly varies with the temperature, a different temperature correlation may be required to approximate the furnace temperature profile at high peak temperatures. In addition, the insufficient information concerning the dependency of the surface tension with the temperature makes the estimation of the time required for collapse a very challenging task. In Figures 7 and 8 we examine cases with internal pressurization. First, we keep the pressure constant and vary the peak temperature, then we maintain the temperature and vary the internal pressure. In Figure 7a, we compare the final external diameter of the capillary obtained numerically with the experimental data of Luzi et al. [20], while in Figure 7b we compare the final air-filling fractions obtained numerically and experimentally. In both cases, we keep the internal pressure constant at *po* = 9 mbar and vary the peak temperature of the furnace from 1850 ◦C up to 2050 ◦C in steps of 25 ◦C. The numerical data match very well with the experimental data, with the maximum discrepancy between the final external diameters computed numerically and experimentally being approximately 11% in the case of *Tpeak* = 2050 ◦C. We obtain similar agreement for the final external diameters and the final air-filling fractions in the case where we fix the peak temperature at 1950 ◦C and vary the internal pressure between 0 and 25 mbar; see Figure 8a,b. In this case, the maximum difference between the experimental values and the numerical computations is approximately 8%. The internal pressure counteracts the effects of the surface tension by promoting an enlargement of the inner hole, thereby increasing the size of the internal and external radii. The maximum deviations between the experimental and numerical results are again obtained for high values of the peak temperatures, where major uncertainties concerning the furnace temperature profile and the value of the surface tension and viscosity prevail.

**Figure 3.** (**a**) Evolution of the axial component of the velocity *w* against the axial distance *z*. The inner and outer surfaces of the capillary, *h*<sup>1</sup> and *h*2, respectively, are plotted against the axial distance *z* in (**b**). The temperature profiles of the inner and outer capillary surfaces are plotted against the axial distance *z* together with the temperature profile of the furnace in (**c**). The inset in (**c**) magnifies the difference among the plotted temperature profiles of capillary surfaces and the temperature profile of the furnace.

**Figure 4.** Fibre temperature contour in the case of DR54-15, *Tpeak* = 2050 ◦C and no internal pressurization.

**Figure 5.** *Cont*.

**Figure 5.** Comparison between the final external diameter of the capillary obtained numerically and experimentally by Luzi et al. [20] for the three different drawing ratios (**a**) DR36-1, (**b**) 54-15, and (**c**) 72-2 at different furnace peak temperatures.

**Figure 6.** *Cont*.

**Figure 6.** Comparison between the final air-filling fraction of the capillary obtained numerically and experimentally by Luzi et al. [20] for the three different drawing ratios (**a**) DR36-1, (**b**) 54-15, and (**c**) 72-2 at different furnace peak temperatures.

**Figure 7.** Comparison between the final outer diameter (**a**) and air-filling fraction (**b**) of the capillary obtained numerically and experimentally by Luzi et al. [20] for different furnace peak temperatures with a fixed value of internal pressurization.

**Figure 8.** Comparison between the final outer diameter (**a**) and air-filling fraction (**b**) of the capillary obtained numerically and experimentally by Luzi et al. [20] for different values of internal pressurization and a fixed value of the furnace peak temperature.

#### *4.3. High Drawing Ratios*

In this section, we present exemplary results for the case of the high drawing ratio DR 1-102. This is relevant for fibre drawing, as the final dimensions of the capillary are of the same order of magnitude as commercially available optical fibres. Because the drawing speed is approximately ten times higher compared to the cases of the slow drawing ratios analyzed before, a smaller step size in the axial direction is required. Furthermore, because we realized that the temperature variations in the radial direction were not significant for the cases of the slow drawing ratios, we utilized a larger step size in the radial direction in the case of this high drawing ratio. This allowed us to choose a step size in the axial direction smaller than the one used with the slow drawing ratios, while it is not excessively small due to the constraint imposed by the Courant number for the stability of an explicit method. Moreover, we utilize Equation (33) to compute the Nusselt number *β* in the case of the high drawing ratio, as Equation (32) delivers values that are extremely high, promoting an unreasonable convective heat exchange that cools down the capillary too much and makes the numerical computations unstable. The high values of *β* are due to the high values of the axial velocity and the low values of the capillary size attained at the end of the drawing stage. Figure 9a shows that the axial velocity profile abruptly increases from a value approximately close to zero at *z* ≈ 0.4 up to about one at *z* ≈ 0.6. The increase of the axial velocity profile is steeper and the final value is attained at a shorter distance compared to the slow drawing ratios. In addition, the inner and outer surfaces of the capillary shrink considerably during the drawing process, and the final size of the capillary is very thin; see Figure 9b and inset. With regard to the temperature distribution within the capillary, we notice the absence of a significant variation of the temperature with the radial direction; see the inset of Figure 9c. Furthermore, in this case the temperature distribution in the axial direction is very close to that of the furnace until *z* ≈ 0.45. Afterwards, the temperature profile of the capillary significantly departs from that of the furnace, assuming lower values until the exit of the furnace is reached; see Figure 9c. The dimensionless values of the temperature of the capillary at different radial positions at the exit of the oven are approximately equal to 0.7, while the dimensionless value of the furnace temperature at the exit of the furnace is approximately equal to 0.82. In Figure 10, we compare the values of the final external diameters and those of the air volume fractions obtained numerically and experimentally for different values of the peak temperature without internal pressurization. The deviations between experiments and numerical simulations reach a maximum of approximately 12 % (see Figure 10a) and is again achieved at a high peak temperatures, that is, *Tpeak* = 2050 ◦C.

**Figure 9.** *Cont*.

**Figure 9.** (**a**) Axial component of the velocity *w* against the axial distance *z*, with *h*<sup>1</sup> and *h*<sup>2</sup> plotted against the axial distance *z* in (**b**) and the temperature profiles of the inner and outer capillary surfaces and the temperature profile of the furnace plotted against the axial distance *z* in (**c**). The inset in (**c**) shows the difference among the temperature profiles.

**Figure 10.** Comparison between the final outer diameter (**a**) and air-filling fraction (**b**) of the capillary obtained numerically and experimentally by Luzi et al. [20] for different furnace peak temperatures.
