Collapsing of glass tubes: analytic approaches in a hydrodynamic problem with free boundaries

Abstract

We present an analytic study to support the understanding of collapsing of glass tubes released by local heating and driven by the surface tension only. Our results provide a reliable analytic base for data evaluation if viscosity and surface tension of molten glasses are measured through collapsing. We complete existing 1D approaches to arrive at a consistent 2D description for axial symmetrical arrangements. We focus on advancing, steady-state collapsing profiles for heat sources moving with a constant axial velocity. The glass is considered an incompressible Stokes fluid with temperature-dependent viscosity of Arrhenius type that is ad hoc modeled by an axial, steady-state viscosity course comoving with the heat source. The analysis is carried out in comoving coordinates. Stringent scaling properties of collapsing are derived. We assume two distinct governing length scales L andh in axial and radial directions, respectively, as the base of a joint asymptotic multiscale analysis (AMSA) of the Stokes equation, boundary conditions, and kinematics for smallh/L. Numerical studies of collapsingrelevant parameters in comparison with finite element reference calculations are outlined for axial courses of the reciprocal viscosity idealized as Gaussian. For arbitrary axial viscosity courses, AMSA results in a consistent analytical description of collapsing for smallh/L. Forh/L < 1, precise formulae for collapsing-relevant geometry parameters are outlined.

A two-dimensional asymptotic model for capillary collapse

“…and, on substituting Klupsch & Pan (2017), and the 1-D model, for φ I = 0.7, 0.9. Table 2 compares the collapse as given by the AM, 2DT+ and 1-D models, for φ I = 0.7 and 0.9 and the given parameters u, α for the AM model and the equivalent parameter V for the 2DT+ model.…”

“…To determine and compare the change in the capillary shape over −1/2 x 1/2 for the models of § § 2 and 3, we need to specify a viscosity profile and, following Klupsch & Pan (2017), we assume the scaled viscosity profile to be Gaussian, given by…”

“…The first 2-D axisymmetric model developed explicitly for capillary collapse, accounting for motion in both radial and axial directions, seems to be that of Klupsch & Pan (2017). For a capillary held fixed at both ends, and assuming that the length scale in the radial direction is small relative to the length scale in the axial direction, they performed an 'asymptotic multiscale analysis' to obtain a 2-D description of capillary collapse.…”

“…Extensive attention has been given over three decades to mathematical modelling of the drawing of slender microstructured optical fibres, incorporating both axial and cross-sectional flow (Yarin et al 1989;Yarin, Gospodinov & Rusinov 1994;Fitt et al 2002;Griffiths & Howell 2008;Stokes et al 2014;Chen et al 2015;Xue et al 2015a,b;Chen et al 2016b;Stokes, Wylie & Chen 2019), and the drawing of slender axisymmetric tubes is a particular case. Thus, 2-D models exist which are applicable to capillary collapse and which predate that of Klupsch & Pan (2017). For a radial length scale that is much smaller than the axial length scale, as assumed by Klupsch & Pan, the asymptotic fibre-drawing model of Stokes et al (2014Stokes et al ( , 2019) may be used for capillary collapse and yields analytic formulae relating the extent of collapse to the harmonic mean of the viscosity and the surface tension of the glass.…”

“…In the experimental set-ups of Kirchhof (1980Kirchhof ( , 1985a, where heating is by a moving oxy-hydrogen torch and the capillary is rotated in a glass working lathe to achieve uniform heating around the perimeter of a cross-section, it is clear that the ends of the capillary are held fixed. Klupsch & Pan (2017) assume that the cold ends of the capillary are fixed in the laboratory reference frame. However, it is possible that, with use of a travelling axisymmetric heater around the tube, rotation would not be necessary and the capillary ends could be supported but allowed to slip axially.…”