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

• Published in: Journal of engineering mathematics 2017-10, Vol.106 (1), p.143-168
• Main Authors: Klupsch, Thomas, Pan, Zhiwen
• Format: Article
• Language: English
• Subjects: Applications of Mathematics; Boundary conditions; Computational fluid dynamics; Computational Mathematics and Numerical Analysis; Finite element method; Fluid flow; Free boundaries; Heat sources; Incompressible flow; Kinematics; Mathematical and Computational Engineering; Mathematical Modeling and Industrial Mathematics; Mathematical models; Mathematics; Mathematics and Statistics; Multiscale analysis; Steady state; Stokes law (fluid mechanics); Surface tension; Theoretical and Applied Mechanics; Tubes; Viscosity
• ISSN: 0022-0833; 1573-2703
• DOI: 10.1007/s10665-017-9897-7

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 ~ and h ~ 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 small h ~ / L ~ . Numerical studies of collapsing-relevant 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 small h ~ / L ~ . For h ~ / L ~ < 1 , precise formulae for collapsing-relevant geometry parameters are outlined.