# Mathematical Modeling of Capillary Drawing Stability for Hollow Optical Fibers
## 4 Conclusions  
During our research we studied the stability of the process of drawing quartz capillaries for photonic-crystal fibers. A modified PCF drawing model was obtained that considers the forces of inertia, viscous friction, and surface tension, as well as all types of heat transfer. Within the framework of the linear theory of stability, a mathematical model of capillary drawing was developed. The influence of the drawing ratio E and inertia forces (Reynolds number) on the stability of the process under consideration was estimated. At the same time, the influence of the preform geometric parameters did not go unnoticed.  
Figure 14: Dependence of \(\omega_{i}^{(1)}\) on the radius of the furnace and \(h\).  
Figure 15: Dependence of \(\omega_{i}^{(1)}\) on h for various values of \(T_{f1}\).  
For a complete nonlinear model of the isothermal drawing of a capillary, a numerical analysis of the sensitivity to perturbations arising in the boundary conditions was carried out. It was found that in the stability region constructed using the linear theory, perturbations of finite amplitude also fade over a finite period of time and do not significantly affect the quality of the finished product. As for the zone of linear instability, the model has a periodic solution, the amplitude and period of which depend on the drawing ratio.  
The behavior of the secondary flow, which is formed as a result of the process transition to the instability region, is analyzed. It is shown that when the draw ratio is higher than the critical one (unstable mode), the process is not established with time, and the oscillation amplitude depends on the ratio E, and these oscillations are stable. The effect of the relative width of the heating element on stability was studied. It is shown that with an increase in the drawing ratio, the stability of the process is lost. Calculations for various values of the furnace radius showed that stability depends on the distance between the furnace and the preform surface: with the smaller radius, the stability of the process decreases. For the first time, the existence of optimal parameters of the heating element, such as the core width, its temperature, the temperature at the edges of the furnace, and the radius of the furnace, which guaranteed several times the drawing process stability increase, is shown.  
The optimal heating zone, with the most stable fiber formation is revealed.  
Conceptualization, V.P.; methodology, V.P. and D.V.; software, A.D. and D.V.; validation, V.P., D.V. and A.D.; formal analysis, V.P.; investigation, A.D. and D.V.; resources, V.P.; data curation, V.P.; writing--original draft preparation, A.D.; writing--review and editing, D.V. and V.P.; visualization, A.D. and D.V.; supervision, V.P.; project administration, A.D. and V.P.; funding acquisition, V.P.All authors have read and agreed to the published version of the manuscript.  
This research was carried out with the financial support of the Ministry of Science and Higher Education of the Russian Federation in the framework of the program of activities of the Perm Scientific and Educational Center \"Rational Subsoil Use\".  
Data Availability Statement: Data and sources that are not openly available may be provided by the corresponding author on reasonable request.  
The authors declare no conflict of interest.