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

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 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 small ˜h/˜L . For ˜h /˜L < 1, precise formulae for collapsing-relevant geometry parameters are outlined.