Hydrodynamics of Collapsing Glass Tubes and Measuring of Glass Viscosities: Analytic Results beyond Asymptotic Approaches for Rapidly Varying Viscosities

Abstract

We present novel analytic solutions of the axial-symmetric boundary value problem of the Stokes equation for incompressible liquids with rapidly varying viscosity, which cover the hydrodynamics of collapsing glass tubes with moving torch. We meet requirements to optimize the contactless measuring of dynamical viscosities and surface tensions of molten glasses through collapsing for tools working with sharply peaked axial temperature courses. We study model solutions for axial courses of the reciprocal viscosity specified as Gaussians extended on small distances compared to the outer tube radius, and we neglect the boundary inclination, corresponding to measuring conditions for large torch velocities. The surface tension is assumed to be constant across the collapsing zone. The boundary value problem becomes disentangled, changing to a gradually independent hierarchy of streaming function, vorticity, and pressure. Axial Fourier transforms are introduced to focus on solutions for infinitely extended tubes. Beyond the predictions of the asymptotic collapsing theory, a successively increasing steepness of the reciprocal viscosity induces an increasing radial pressure gradient that acts against the surface tension and diminishes the collapsing efficiency. The arising systematic error in evaluating the viscosity from experimental data in virtue of the asymptotic collapsing theory is corrected. Error estimations regarding deviations from the specified viscosity course, the neglected boundary inclination, and heat conduction within the tube wall are outlined, and preconditions to simplify the measuring of surface tensions through collapsing are discussed.