# Mathematical Modeling of Capillary Drawing Stability for Hollow Optical Fibers
## 1 Introduction  
One of the modern and promising directions in the development of science-intensive industries is the production of optical fibers [1]. The areas of fiber technologies application are quite wide. These are fiber-optic lines for fast and high-quality communication, navigation, medicine, oil and gas industry, geology [2]. Today, much attention is paid to instrumentation which are using fiber-optic components [3]. Note that the process of manufacturing optical fibers is expensive and rather complicated [4]. At the same time, there is no complete study yet on the production problems. In some cases, this is due to the lack of information on these issues, and in some cases, due to the fact that the practical implementation of modern technologies often outstrips the process of full scientific description and research. Therefore, effective management of fiber production and increasing the stability of technological processes is certainly an urgent task. First of all, the above concerns special quartz optical fibers since they are of the greatest interest for use in various fields. It is well-known that the full production cycle of optical fibers from raw materials to finished fiber can be represented as a number of independent production stages and four main stages can be distinguished among them: extrusion of quartz pipes from quartz raw materials; doping of quartz pipes; jacketing/scaling of quartz pipes; drawing fiber from the preform. The drawing process is the process of converting a quartz preform into fiber. It takes place in a special high-precision mechanical system called a drawing tower. A schematic of the drawing tower is shown in Figure 1. The tower feed mechanism (block B1) places the preform in the center of the precision furnace (B2), inside which the preformmelling process is started. The characteristic temperatures of the process are approximately 2000 \({}^{\circ}\)C (2273.15 K), close to the temperatures of the phase transition state of quartz, at which the viscosity of the substance changes sharply, creating the possibility of changing its shape [5]. When a preform with a diameter of 2-4 cm is transformed into a fiber with a diameter of only a few microns, an important target moment is the complete preservation of the relative geometric characteristics of quartz (in cross section). Following the exit from the furnace, the diameter of the hot quartz fiber is measured (B3). Then, as the fiber cools, it passes through an acrylate or silicone UV coating mechanism (B4) and is wound onto a coil (B5) [6]. It has been shown that the main optical and mechanical characteristics of quartz fibers depend on many factors [7]. First of all, this is the quality of the original preform. Here, an important role is played by the evaluation of its taper, the presence of small internal defects and exceeds [8]. The selected technological modes of the production process are also important factors affecting the stability of the drawing process parameters. The critically important ones include the so-called drawing ratio (the ratio of the speeds of fiber drawing and quartz preform feeding), as well as the parameters of the furnace that heats the preform [9]. In addition, significant differences should be noted in the process of drawing solid and hollow (holey) fibers. In the first case, the control of the stability of the drawing process is reduced only to observe the geometric shape of the outer surface of the fiber. In the second case, it is also important to observe the inner surface in order to avoid collapse or swelling of the capillary [10]. Today, the problems of stability in the production of capillaries (tubes) are of particular interest, as far as capillaries are the main elements of photonic crystal fibers (PCF). A number of works [11; 12; 13] are devoted to mathematical modeling of technological processes of drawing capillaries, with an analysis of mathematical aspects and features of models, as well as numerical results of studies.