# Mathematical Modeling of Capillary Drawing Stability for Hollow Optical Fibers
## 3 Numerical Study of the Capillary Drawing Stability  
### Isothermal Process  
#### 3.1.1 Linear Stability  
Based on the obtained stability model, a software package was developed. Depending on the input parameters of the process, it finds a stationary solution, forms a matrix of coefficients for a given number of discretization nodes, and determines the growth factor. The created complex allowed us to obtain a number of important theoretical and practical results. However, as mentioned earlier, one of the stages of the stability determination algorithm is the use of the finite difference method. Therefore, at first, the influence of the discretization step on the accuracy of calculations was evaluated and the optimal step was chosen. To do this, at fixed system parameters and for a various number of partition points, the damping coefficients of the first modes \(\omega_{i}^{(1)}\) were calculated. The observation was carried out for the relative deviation of the following type:  
\[\Delta\omega_{i,j}^{(1)}=\frac{\omega_{i,j}^{(1)}-\omega_{i,j-100}^{(1)}}{ \omega_{i,j-100}^{(1)}} \tag{10}\]  
where \(j=100\), \(200\),..., \(1000\) is the number of partitions in the numerical experiment. Calculations show that when the number of discretization points is more than \(500\), the calculation result changes by less than \(0.1\%\). All further calculations were done with this accuracy. For the fiber manufacturing process, there is a set of technological parameters that can be adjusted in order to achieve the desired drawing results. Here is a brief overview of the parameters that are key to assessing the stability of the fiber manufacturing process. The most important of them is the drawing ratio \(E\). The influence of this indicator was assessed and the influence of the geometrical parameters of the preform on the stability of the process were simultaneously considered (Figure 5). Figure 5: Dependence of \(\omega_{i}^{(1)}\) on the geometry of the preform at various drawing ratios (\(E\)). It can be said that with an increase in the drawing ratio, the process becomes less stable. In addition, reducing the thickness of the preform while increasing the size of the capillary cavity also leads to buckling. It should be noted that this dependence between the ratio and the stability of drawing was also revealed in the case of a solid fiber [27; 28; 29; 30]. The main interest is the connection between the process stability and the Reynolds number, which characterizes the action of inertia forces and viscous friction. The drawing ratio was fixed at \(E=20\), and stability calculations were made at various values of the Reynolds number and for various geometric parameters of the preforms. In this case, the main parameters of the model were chosen according to the quartz fiber drawing technology [11; 20; 23]. The change in the Reynolds number was carried out by changing the viscosity values within the allowable values due to production. In this case, the Reynolds number took a value of no more than 1.5. In particular, this is consistent with the studies [28; 29; 30] on solid fibers, in which the range of the Reynolds numbers is the same. Based on the results of calculating the coefficients of the first modes (Figure 6), it can be established that with an increase in the Reynolds number, the process stability of drawing a quartz capillary increases. In addition, an increase in the inner radius of the capillary or, what is the same, the radius of its cavity, leads to the stability loss of the drawing process. The analysis performed made it possible to find the stability limits to small perturbations for the drawing process, considering their geometric parameters, ratio, and Reynolds numbers. However, supercritical regions and secondary flows that develop as a result of the main flow stability loss remained unconsidered. The sensitivity level of the process to possible oscillations of the initial and boundary conditions of the system (7) could not be estimated.