**5. Discussion**

In this work, we built a full asymptotic extensional-flow model to describe the evolution of a capillary during the drawing process. To this end, we analyzed and revisited the work of Taroni et al. [24] and integrated a new asymptotic energy equation into the

momentum and mass conservation equations of Fitt et al. [19]. To check the validity of the model, we compared the numerical outcomes with the experimental data of Luzi et al. [20] for the case of low and high drawing ratios with and without internal pressurization. In the derivation of the energy equation, we considered conductive, convective, and radiative heat exchange, assuming that the capillary is optically thick. In addition, we included the effects of viscous dissipation in our model, as it may significantly affect the reduction of the size of the capillary [49]. Yin and Yaluria [39] pointed out that viscous dissipation must not be neglected, as it becomes important in the stage of the drawing where the size of the capillary shrinks due to the viscosity assuming large values and the velocity gradients being significant. To consider a general situation, we allow the temperature to depend on the axial and radial position of the capillary, thus choosing an approach where the convection is balanced by the transport across the capillary and by viscous dissipation effects. In the case of slow drawing ratios, the asymptotic model forecasts a slight reduction of the fibre temperature profile once the capillary size starts to decrease. In the case of high-speed ratios, however, the model predicts a sharp decrease of the capillary temperature profile toward the end of the drawing process compared to the initial phase, where thermal equilibrium prevails and the temperature of the capillary closely matches that of the furnace. The trend of this temperature profile of the capillary resembles the asymmetric fibre temperature profile utilized by Luzi et al. [28] to model the drawing process of a six-hole optical fibre. This fibre temperature profile sharply decreases when the size of the preform contracts and the axial velocity increases. In the contribution of Luzi et al. [28], it is shown that the use of an asymmetric temperature profile produces numerical results that are in good accordance with the experimental outcomes. On the other hand, the choice of a symmetric Gaussian temperature profile with a large width may produce completely incorrect results, even leading to the explosion of a capillary [20]. The asymmetric capillary temperature profile is produced by the selected functional forms of the Nusselt number *β*, that is, Equations (32) and (33). They result from the choice of the functional forms of the convective heat exchange *kh*, i.e., Equations (29) and (30). Yin and Jaluria [39] remark that the heat transfer due to the radiative heat exchange is small at the end of the drawing, as it is directly proportional to the surface area of the fibre. This is reduced by approximately 1000 times from the beginning to the end of the drawing process. Thereby, the convective heat transfer may become the prevailing heat transfer mechanism because of the increase of the velocity at the end of the drawing. To compute *kh*, we utilized the correlation by Patel et al. [43] in the case of high drawing ratios, and the one suggested by Geyling and Homsy [42] in the case of slow drawing ratios. The correlation suggested by Geyling and Homsy [42] produces extremely high values of the maximum Nusselt number *βmax* in the case of the high drawing ratio. In fact, *βmax* ≈ 28.04 if the correlation of Geyling and Homsy is utilized, while *βmax* ≈ 5.02 when the correlation by Patel et al. [43] is applied. This implies that *khmax* <sup>≈</sup> <sup>257</sup> *<sup>W</sup> m*2*K* using the correlation of Geyling and Homsy, and *khmax* <sup>≈</sup> <sup>46</sup> *<sup>W</sup> m*2*K* employing the correlation by Patel et al. [43]. This result is close to the values obtained by Choudhury and Yaluria [50], who numerically computed the heat transfer coefficient while investigating the drawing process of solid glass fibres. The results of our simulations reveal that the temperature profile of the capillary does not vary significantly in the radial direction due to the small size of the preform. A similar result was obtained by Taroni et al. [24] within approximately 1 cm of a solid fibre preform employing the surface radiation parameter *γ<sup>R</sup>* = 5. In our case, *γ<sup>R</sup>* = 5.3 indicates

that the bulk radiation is predominant and the temperature distribution is approximately constant across the capillary. The results of our numerical simulations match very well overall with the experiments by Luzi et al. [20], except for a few isolated cases where the discrepancies are very high. This happens for high values of the peak temperature, where the temperature profile may not be well approximated by the relation (26). As a result, the partial collapse of the inner radius of the capillary resulting from the competition between viscosity and surface tension may not be captured. The agreement between experiments and simulations could be improved by utilizing a different correlation to model the furnace temperature profiles at high peak temperatures and an adequate correlation for the surface tension dependency on the temperature. Measurements of the furnace temperature profile are very difficult and prohibitive at high values of the furnace peak temperature [22], as thermocouples lose their accuracy and are inclined to melt. In addition, experimental and numerical investigations could be carried out to find different correlations for the convective heat transfer coefficient, as the models utilized to date have been developed for a solid thread [43] and for flow past a cylinder [42]. In addition, the present mathematical framework serves as a basis for modeling the drawing process of polymer optical fibres (POFs). As a first approximation, if the assumption of a Newtonian fluid is retained [25], the present mathematical formulation can be used in a straightforward way to model the drawing process of polymer axis-symmetric capillaries, as only a few process parameters need to be adjusted. However, because an adequate nonlinear viscoelastic model is necessary to accurately model the viscosity variations during the drawing of POFs, the governing equations need to be modified in a later stage of the research. Although the present asymptotic energy equation is only valid for annular capillaries and not for the general case of PCFs with arbitrary cross-sections, it provides a basis to improve simple models developed for more complicated geometries. For instance, the effects of viscous diffusion and the optically thick approximation can be included in the one-dimensional thermal model by Stokes et al. [34]. Moreover, this model can be extended to three dimensions, and work in this direction is currently ongoing.

## **6. Conclusions**

In this work, we developed a novel asymptotic energy equation for the drawing process of an annular capillary. By coupling it with the mass, momentum, and evolution equations for the inner and outer surfaces by Fitt et al. [19], we built a complete asymptotic model for the capillary drawing process. The asymptotic system of equations is considerably simpler to solve numerically than the original full three-dimensional system of equations while being able to consider the main effects of heat transport across the capillary by convection, diffusion, and viscous dissipation. Heat is exchanged between the furnace and the capillary by convective and radiative mechanisms. To consider radiation within the capillary, we utilized the optically thick approximation and added an extra term to the conductive heat transfer coefficient. Although the validity of an optically thick capillary is likely to break down when the size of the capillary becomes very small, it nonetheless allows for quick and reliable computations of the geometry of the capillary compared to more involved approaches. Comparisons with experimental results for slow and high drawing ratios comprehensively show very good agreement for both pressurized and unpressurized capillaries. Discrepancies between experiments and simulations may be alleviated by a better choice of the value of the surface tension and a different correlation for the heat transfer coefficient.

**Author Contributions:** Supervision, B.G. and A.D.; Writing—original draft, S.L.; Writing—review & editing, G.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by a National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (NRF-2021R1F1A1050103).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.
