# Mathematical Modeling of Capillary Drawing Stability for Hollow Optical Fibers
## 2 Mathematical Models  
### Mathematical Model of Quartz Capillaries Drawing  
Before proceeding directly to the stability analysis, it is necessary to consider the mathematical model of heat and mass transfer during drawing of quartz capillaries. The mathematical model of such process is described by a system of partial differential equations, namely, the equations of motion, continuity, and energy. It should be noted that the first works in this area were related to the drawing of solid fibers, and now there is a wide range of such models that allow us to consider problems in various formulations, as well as various types of heat transfer [21]. As for the mathematical models of capillary drawing (Figure 2), they were proposed later in the early 2000s. In 2000s, a study [20] presenting a quasi-one-dimensional model of capillary drawing was published. In this work, a modified pipe-drawing model was proposed. The model was obtained under the same assumptions as the model of the authors of the article [20]. These assumptions are as follows: (1) Flow and heat transfer are axisymmetric; (2) quartz melt is a viscous incompressible liquid; (3) the capillary wall thickness is small compared to its radius; (4) due to its smallness, the radial velocity is not taken into account; (5) changes in the axial velocity and temperature along the cross section of the capillary are small  
Figure 2: Configuration of the calculated area. compared to their changes along the \(z\) axis. The modified model, as well as the [20] model, takes into account inertial forces, viscous friction forces, and surface tension, as well as the combined action of all types of heat transfer: heat conduction, convection, and radiation. The modification of the model is caused by taking into account a radiant heat transfer more strictly. In the proposed work, a modified model of tube drawing is proposed, which differs from the above model (radiant heat transfer is more strictly considered). Let us look at this aspect in more detail. When deriving the energy equation, considering radiant heat transfer, the laws of [PERSON], [PERSON], and [PERSON] are used [22]. It is known that the radiation spectrum of a blackbody with temperature \(t\) is determined by [PERSON]'s law:  
\[\overline{E}_{\
u}=2\pi\frac{\overline{h}\
u^{3}}{c^{2}}\cdot\frac{1}{\exp( \overline{h}\
u/k_{B}t)-1}\, \tag{1}\]  
where \(c\) is the speed of light, \(\lambda\) is the wavelength, \(\
u\) is the frequency, at that \(c=\lambda\cdot\
u\), \(\overline{h}\) is [PERSON]'s constant, \(k_{B}\) is [PERSON] constant. For the integral flux density, according to the [PERSON] law, we have (\(\sigma_{0}\) is the Stefan-Boltzmann constant):  
\[\overline{E}=\int\limits_{0}^{\infty}\overline{E}_{\
u}d\
u=\sigma_{0}t^{4} \tag{2}\]  
Radiant heat transfer between bodies is determined by the [PERSON] law: the amount of energy emitted by a surface element \(dA_{2}\) in the direction of element \(dA_{1}\) (Figure 3) is proportional to the energy emitted along the normal \(E_{n}dA_{2}\) multiplied by the value of the elementary solid angle \(d\Omega\) and \(cos\alpha_{2}\), i.e.,  
\[dQ=E_{n}dA_{2}d\Omega\ \cos\alpha_{2}^{\prime}\,\ \ \ E_{n}=\overline{E}/\pi. \tag{3}\]  
Since real materials do not completely absorb radiation, to consider this fact, the concept of emissivity \(\varepsilon\), (\(\varepsilon<1\)) is used and the laws of [PERSON] and [PERSON] are applied with a given correction factor \(\varepsilon\). Let us consider \(n_{1}\) as the normal to the point \(A_{1}\) of the surface \(\omega_{1}\) of the quartz fiber, and \(A_{1}A_{2}\) as the segment connecting the points \(A_{1}\) and  
Figure 3: Radiant heat transfer during capillary drawing. \(A_{2}\). The resulting radiation flux at a surface point is the sum of the intrinsic radiation flux and the absorbed flux.