# Mathematical Modeling of Capillary Drawing Stability for Hollow Optical Fibers
## 3 Numerical Study of the Capillary Drawing Stability
### Isothermal Process
#### 3.1.2 Numerical Sensitivity Analysis  
To run numerical experiments in the Comsol Multiphysics 5.2 simulation system, a software package was developed. This allows to solve the original nonlinear system describing the drawing process (5). Modeling of the drawing process was made with parameters \(Re=15.2\times 10^{-5}\), \(Fr=3.4\times 10^{-5}\), \(We=0.22\), \(La=0\), \(Ma=144,662\) for a preform with radii \(r_{10}=0.01\), \(r_{20}=0.025\) at drawing ratio \(E=20\), which corresponds to a stable state. The drawing process stability at the chosen values of the parameters means that the shapes of the profiles of the jet radius and the melt velocity approach the shapes corresponding to the stationary states. To analyze the sensitivity of the system, harmonic perturbations were introduced into the boundary conditions (5) for the values of the preform radius or the drawing speed at the inlet:  
\[V(0,\tau)=\frac{1}{E}\cdot(1+A\sin(2\pi\gamma\tau)),V(1,\tau)=1 \cdot(1+A\sin(2\pi\gamma\tau)),\] \[R_{2}(0,\tau)=R_{20}\cdot(1+A\sin(2\pi\gamma\tau)),R_{1}(0,\tau )=R_{10}\cdot(1+A\sin(2\pi\gamma\tau)),\]  
Since the most important indicator of the fibers drawing quality is the exact correspondence of the fiber proportions to the required dimensions (stationary), it is necessary to know what effects possible perturbations will have. Observation of the introduced perturbations was reduced to the control of the geometric characteristics of the finished fiber. The frequency effect of introduced oscillations on the stability of the capillary radius was evaluated. The calculations were carried out at a fixed amplitude \(A=0.05\), the frequency \(\gamma\) varied in the range from 0 to 30. The analysis performed showed that oscillations with frequency \(\gamma=25\) persist for the longest time, while the maximum amplitude at various frequencies is almost the same (Figure 7). In further sensitivity studies, the frequency of the introduced oscillations was taken as 25 in order to identify the worst possible scenario. Next, the influence of the amplitude A of the introduced oscillations on the geometry of the finished fiber was evaluated. As noted earlier, the perturbation frequency \(\gamma=25\) was fixed, and the amplitude A varied from 0.01 to 0.4. The calculation showed that the process  
Figure 7: Oscillations in the values of the fiber radius in time (change in the frequency of harmonic oscillations of the initial conditions from 0 to 25). is not very sensitive to the amplitude of the introduced oscillations (Figure 8). Nevertheless, the highest sensitivity to oscillations with an amplitude of 0.05 was shown. Similarly to the previous one, the value \(A=0.05\) is fixed as the value of the perturbation amplitude, leading to the worst-case scenario in the study of stability. So, the critical values of the amplitude and frequency of harmonic oscillations, leading to the longest process of establishing solutions, have been identified and fixed. Further, under the conditions of acting perturbations, the relative deviations of the outer and inner radii of the fiber from their stationary values \(\Delta R_{2}(1,\tau)\) and \(\Delta R_{1}(1,\tau)\) were calculated. Figure 9 shows the calculated values \(\Delta R_{1}(1,\tau)\) as a function of time for perturbing actions specified at the boundaries of the computational region. Figure 8: Oscillations in the values of the fiber radius in time (change in the amplitude of harmonic oscillations of the initial conditions). Figure 9: _Cont._  
Similar results were also obtained in the calculation of \(\Delta R_{2}(1,\tau)\). The form of the solutions presented in Figure 9 makes it possible to judge the damping of the perturbation amplitude with time. So, we note that when various types of boundary conditions are perturbed by 5%, the relative deviations \(\Delta R_{2}(1,\tau)\), \(\Delta R_{1}(1,\tau)\) change by no more than 1%.