# Mathematical Modeling of Capillary Drawing Stability for Hollow Optical Fibers
## 2 Mathematical Models
### Mathematical Model of the Non-Isothermal Process Stability of the Quartz Capillary Drawing  
The property of smallness of the perturbation is essential in the definition of stability, and this property can serve as a basis for neglecting the products of perturbations of the desired functions in the process of studying stability. In other words, the analysis of a system linearized in the vicinity of its stationary state can replace the analysis of the original nonlinear system [24; 25; 26]. Accordingly, at the first stage, linearization was performed, in which the parameters determining the state of the system were divided into main \(\overline{F}(z)\) and perturbing ones \(\widetilde{F}(z,\tau)\): \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Symbols** & **Description** & **Symbols** & **Description** \\ \hline \(Re=\frac{\rho v_{L}L}{\mu_{0}}\) & Reynolds number & \(La=\frac{P_{0}L}{\mu_{0}v_{L}}\) & Criterion for the interaction of capillary forces \\ \(Fr=\frac{v_{L}^{2}}{L\underline{S}}\) & [PERSON] number & \(Ma=\frac{\mu_{0}v_{L}}{\widehat{\gamma}}\) & Criterion for the interaction of forces of molecular friction \\ \(We=\frac{\rho Lv_{L}^{2}}{\widehat{\gamma}}\) & [PERSON] number & \(Pe=\frac{\rho C_{p}v_{L}L}{\lambda_{t}}\) & [PERSON] number \\ \(\chi_{1}=\frac{\alpha_{2}e\cdot n_{c}^{2}\sigma_{0}\overline{T}_{a}^{3}}{\rho C _{p}v_{L}}\) & Dimensionless complexes 1, 2 & \(St_{1}=\frac{\pi_{1}}{\rho C_{p}v_{L}}\) & [PERSON]’s criterion \\ \(\chi_{2}=\frac{n_{c}^{2}\sigma_{0}\overline{T}_{a}^{3}}{\rho C_{p}v_{L}}\) & & \(St_{2}=\frac{\pi_{2}}{\rho C_{p}v_{L}}\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Dimensionless quantities. \[F(z,\tau) = \overline{F}(z)\cdot(1+\widetilde{F}(z,\tau)),\] \[F(z,\tau) \in (V(z,\tau)\,R_{1}(z,\tau)\,R_{2}(z,\tau)\,T(z,\tau)\ ),\] \[\overline{F}(z) \in (\overline{V}(z)\,\overline{R}_{1}(z)\,\overline{R}_{2}(z)\, \overline{T}(z)\ ),\] \[\widetilde{F}(z,\tau) \in (\widetilde{V}(z,\tau)\,\widetilde{R}_{1}(z,\tau)\,\widetilde{R}_{2}(z,\tau)\, \widetilde{T}(z,\tau)\ )\]  
The main parameters are stationary solutions of system (3). As a result, a system was obtained that describes the evolution of perturbing influences. To simplify the notation, everywhere below \(\widetilde{F}(z,\tau)=\widetilde{F}\), \(\overline{F}(z)=\overline{F}\). With this remark in mind, the linearized system of equations takes the form  
\[ \begin{split}&\frac{\partial\widetilde{V}}{\partial\tau}=\frac{3}{Re} \cdot\frac{\partial^{2}\widetilde{V}}{\partial z^{2}}+\beta_{1}(z)\frac{ \partial\widetilde{V}}{\partial z}+\beta_{2}(z)\widetilde{V}+\alpha_{1}(z) \frac{\partial\widetilde{R}_{2}}{\partial z}+\alpha_{2}(z)\widetilde{R}_{2}+ \theta_{1}(z)\frac{\partial\widetilde{R}_{1}}{\partial z}+\theta_{2}(z) \widetilde{R}_{1}+\varphi_{1}(z)\frac{\partial\widetilde{T}}{\partial z}+ \varphi_{2}(z)\widetilde{T},\\ &-\frac{\partial\widetilde{R}_{1}}{\partial\tau}=\beta_{3}(z) \frac{\partial\widetilde{V}}{\partial z}+\beta_{4}(z)\widetilde{V}+\theta_{3} (z)\frac{\partial\widetilde{R}_{1}}{\partial z}+\theta_{4}(z)\widetilde{R}_{1 }+\alpha_{3}(z)\widetilde{R}_{2},\\ &-\frac{\partial\widetilde{R}_{2}}{\partial\tau}=\beta_{3}(z) \frac{\partial\widetilde{V}}{\partial z}+\beta_{5}(z)\frac{\partial\widetilde{ R}_{1}}{\partial z}+\theta_{5}(z)\widetilde{R}_{1}+\alpha_{4}(z)\widetilde{R}_{2},\\ &\frac{\partial\widetilde{T}}{\partial\tau}=\frac{\lambda}{Pe} \cdot\frac{\partial^{2}\widetilde{T}}{\partial z^{2}}+\phi_{3}(z)\frac{ \partial\widetilde{T}}{\partial z}+\varphi_{4}(z)\widetilde{T}+\alpha_{5}(z) \frac{\partial\widetilde{R}_{2}}{\partial z}+\alpha_{6}(z)\widetilde{R}_{2}+ \theta_{6}(x)\frac{\partial\widetilde{R}_{1}}{\partial z}+\theta_{7}(z) \widetilde{R}_{1}+\beta_{6}(z)\widetilde{V},\\ &\widetilde{V}(z,0)=0,\ \widetilde{V}(0,\tau)=0,\ \widetilde{V}(1,\tau)=0\, \\ &\widetilde{R}_{1}(z,0)=0,\ \widetilde{R}_{1}(0,\tau)=0,\ \widetilde{R}_{2}(z,0)=0,\ \widetilde{R}_{2}(0,\tau)=0,\\ &\widetilde{T}(z,0)=0,\ \widetilde{T}(0,\tau)=0,\ \frac{\partial \widetilde{T}}{\partial z}\big{|}_{z=1}=0.\end{split}