# Mathematical Modeling of Capillary Drawing Stability for Hollow Optical Fibers
## 2 Mathematical Models
### Mathematical Model of Quartz Capillaries Drawing
As a result, the system of equations describing the motion and heat transfer during capillary drawing, taking into account the initial and boundary conditions, has taken the following form:  
\[\begin{array}{l}\frac{\partial r_{1}^{2}(\overline{z},\overline{\tau})}{ \partial\overline{\tau}}+\frac{\partial(v(\overline{z},\overline{\tau})r_{1} ^{2}(\overline{z},\overline{\tau}))}{\partial\overline{z}}=\frac{p_{1}r_{2} ^{2}(\overline{z},\overline{\tau})r_{2}^{2}(\overline{z},\overline{\tau})- \widetilde{\tau}r_{1}(\overline{z},\overline{\tau})r_{2}(\overline{z}, \overline{\tau})(r_{1}(\overline{z},\overline{\tau})+r_{2}(\overline{z}, \overline{\tau}))}{\overline{\eta}(t(\overline{z},\overline{\tau}))\cdot(r_{ 2}^{2}(\overline{z},\overline{\tau})-r_{1}^{2}(\overline{z},\overline{\tau})) },\\ \frac{\partial r_{2}^{2}(\overline{z},\overline{\tau})}{\partial\overline{\tau }}+\frac{\partial(v(\overline{z},\overline{\tau})r_{2}^{2}(\overline{z}, \overline{\tau}))}{\partial\overline{z}}=\frac{p_{1}r_{2}^{2}(\overline{z}, \overline{\tau})r_{2}^{2}(\overline{z},\overline{\tau})-\widetilde{\tau}r_{1 }(\overline{z},\overline{\tau})r_{2}(\overline{z},\overline{\tau})(r_{1}( \overline{z},\overline{\tau})+r_{2}(\overline{z},\overline{\tau}))}{\overline {\eta}(t(\overline{z},\overline{\tau}))(r_{2}^{2}(\overline{z},\overline{ \tau})-r_{1}^{2}(\overline{z},\overline{\tau}))},\\ \rho(r_{2}^{2}(\overline{z},\overline{\tau})-r_{1}^{2}(\overline{z},\overline{ \tau}))\left(\frac{\partial v(\overline{z},\overline{\tau})}{\partial \overline{\tau}}+v(\overline{z},\overline{\tau})\frac{\partial v(\overline{z}, \overline{\tau})}{\partial\overline{z}}-g\right)=\frac{\partial}{\partial \overline{z}}\left(3\eta(t(\overline{z},\overline{\tau}))\cdot(r_{2}^{2}( \overline{z},\overline{\tau})-r_{1}^{2}(\overline{z},\overline{\tau}))\frac{ \partial v(\overline{z},\overline{\tau})}{\partial\overline{z}}+\widetilde{ \tau}(r_{2}(\overline{z},\overline{\tau})+r_{1}(\overline{z},\overline{\tau}) )\right),\\ (r_{2}^{2}(\overline{z},\overline{\tau})-r_{1}^{2}(\overline{z},\overline{\tau }))\left(\frac{\partial t(\overline{z},\overline{\tau})}{\partial\overline{ \tau}}+v(\overline{z},\overline{\tau})\frac{d(\overline{z},\overline{\tau})}{ \partial\overline{z}}\right)\rho C_{p}=\frac{\partial}{\partial\overline{z}} \left(\lambda_{f}(r_{2}^{2}(\overline{z},\overline{\tau})-r_{1}^{2}(\overline{z },\overline{\tau}))\frac{\partial t(\overline{z},\overline{\tau})}{\partial \overline{z}}\right)\\ -2r_{1}(\overline{z},\overline{\tau})\sqrt{1+r_{1}^{2}(\overline{z},\overline{ \tau})}\cdot\overline{\tau}_{1}\cdot(t(\overline{z},\overline{\tau})-t_{1n})-2 r_{2}(\overline{z},\overline{\tau})\sqrt{1+r_{2}^{2}(\overline{z},\overline{\tau})} \cdot[\omega_{2R}\cdot r_{2}^{2}\omega_{0}\cdot(t^{4}(\overline{z},\overline{ \tau})-t_{out}\,^{4})+\overline{\pi}_{2}(t(\overline{z},\overline{\tau})-T_{s} )]\\ +4n_{2}^{2}c_{0}\cdot r_{2}(\overline{z},\overline{\tau})\cdot r_{f}\cdot(f_{f}-r _{2}(\overline{z},\overline{\tau}))\cdot\int\limits_{\omega}^{L}\frac{ \left(\overline{k}_{f}r_{f}^{4}(\overline{z},\overline{\tau})-c^{4}(\overline{z },\overline{\tau})\right)\left((r_{f}-r_{2}(\overline{z},\overline{\tau}))+ \left[\zeta_{2}^{2}(\overline{z},\overline{\tau})\right]\left(\overline{z}- \overline{\tau}\right)\right)}{\left((q-\overline{z})^{2}+(f_{f}-r_{2}(\overline {z},\overline{\tau}))^{2}\right)^{2}}d\eta,\\ v(\overline{z},0)=v_{z}(\overline{z}),\ v(0,\overline{\tau})=v_{0},\ v(L, \overline{\tau})=v_{L},\\ r_{1}(\overline{z},0)=r_{1k}(\overline{z}),\ r_{1}(0,\overline{\tau})=r_{10},\ r_{2}(\overline{z},0)=\ r_{2k}(\overline{z}),\ r_{2}(0, \overline{\tau})=r_{20},\\ t(\overline{z},0)=t_{s}(\overline{z}),\ t(0,\overline{\tau})=t_{0},\ \ \frac{\partial t}{\partial\overline{z}}\Big{|}_{z-L}=0.\]  
Here \(v_{s}(\overline{z})\) is the initial speed, \([\text{m}/\text{s}]\); \(v_{0}\) is the fiber feed rate, \([\text{m}/\text{s}]\); \(v_{L}\) is the fiber drawing speed, \([\text{m}/\text{s}]\); \(r_{1s}(\overline{z})\), \(r_{2s}(\overline{z})\) are initial values of the inner and outer radius of the capillary, \([\text{m}]\); \(r_{10}\), \(r_{20}\) are the inner and outer radius of the preform, \([\text{m}]\); \(t_{s}(\overline{z})\) is the initial temperature, \([\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\texttexttexttexttexttexttexttexttexttexttexttext{ \text{\text{\text{\text{\text{\texttext{\texttext{\textTo check the adequacy of the modified mathematical model (5), the numerical calculations were compared with the experimental data given in [23].