' I zyxwvutsrqponmlkjihgfedcbaZ 198 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 11, NO. 2, FEBRUARY 1993 The Viscous Collapse of Radial Nonsymmetric Composite Tubes Alexander L. Yarin, V. Bernit, Juraj Doupovec, and Pave1 MikloS zyxw Abstruct-We present an analytical method to study the viscous collapse of composite tubes with arbitrary cmss sections. A good agreement between the theoretical description and the experimental observation of the collapse of radial symmetric support tubes with radial nonsymmetric deposited layers inside is found. I. INTRODUCTION ' HE glass tube serves as a main element for the production T of optical fiber performs by the modified chemical vapor deposition method (MCVD). The glass particles are deposited from a gas flow onto an inside surface of such a tube, creating a coating. Afterwards, a tube is heated and begins to collapse, i.e., the creeping flow of highly viscous liquid (glass) directed to the center arises under the action of the surface tension forces, which tend to decrease the free surface filling up the cavity by a material. Thus, the slow viscous flow of the glass during this step is driven by surface tension and differential pressures on the inner and outer tube surfaces [l]. The maintenance of the slight internal overpress in the substrate tube with deposited layers is key to achieving high circularity of the resultant preform [2] on the one hand, and on the other hand, collapsing at the reduced pressure and by careful selection of the softening temperatures of the coreklad and jacket materials, resulting in radial nonsymmetric preform profiles [3]. In other words, by means of collapse conditions, it is possible to obtain preforms for both low and high birefringent fibers. The aim of this work is to theoretically describe the collapse of the substrate tube with the radial-nonsymmetric layers inside, and to compare this result with the experimental observation. zyxwvutsrq 11. THEORETICAL ANALYSIS The objective of the present paper is the theoretical analysis of the collapse of a tube with a coating inside. The simplest model system with a single-layer coating is considered as shown in Fig. 1. The coating domain is marked by 1; the staring tube material domain is marked by 2. The flow is Manuscript received March 26, 1992. A. L. Yarin is with the Faculty of Mechanical Engineering, Technion-Israel V. Bernat and J. Doupovec are with the Institute of Physics, Slovak P. MikloS is with the Faculty of Electrical Engineering, Slovak Technical IEEE zyxwvutsrqpo Log Number 9205553. Institute of Technology, Technion City, Haifa 32 000, Israel. Academy of Sciences, Ddbravski 9, 84228 Bratislava, Czechoslovakia. University, IlkoviEova 3, 81219 Bratislava, Czechoslovakia. Fig. 1. Single-layer coating. Layers 1 and 2 represent deposited domain and substrate tube material domain, respectively. Boundaries zyxw Bo, zyx Bi , and BZ denote inner, medium, and outer interface, respectively. considered as planar and creeping (inertialess), and it is supposed that there is a vacuum both inside and outside a tube. The problem concerns the fact that the inner surface of the deposited coating may differ from the circle, as a result of which the other two interfaces may be rather severely changed. In turn, as a result, the collapsed coating during some period of time may be distinct from the circle. The flow is described by the biharmonic equation for the stream function in each domain, 1 or 2: AA$k = 0, zyxwv IC = 1,2. (1) Here and hereafter, the additional indexes 1 and 2 cor- respond to the domain of the coating and the starting tube material, respectively. The solutions of (1) for the problem at hand have the form $1 = $2 = where C2,i, C3,i, C4,i, A,, B,, D,, E,, MI, and M2 are the constant coefficients, and r and cp are the polar radius and angle of a point. The following velocity components correspond to the solu- tion (2) (index r corresponds to the radial direction, index cp to the azimuthal one): zyxwv 00 UTI= ~(r~+l+C2,1r~-l+c3,1r-n-l+c4,,lr-~+l)n n=2 0733-8724/93$03.00 0 1993 IEEE -~