Mathematical modelling of a viscida network

Abstract

We develop a general model to describe a network of interconnected thin viscous sheets, or viscidas, which evolve under the action of surface tension. A junction between two viscidas is analysed by considering a single viscida containing a smoothed corner, where the centreline angle changes rapidly, and then considering the limit as the smoothing tends to zero. The analysis is generalized to derive a simple model for the behaviour at a junction between an arbitrary number of viscidas, which is then coupled to the governing equation for each viscida. We thus obtain a general theory, consisting of $N$ partial differential equations and $3J$ algebraic conservation laws, for a system of $N$ viscidas connected at $J$ junctions. This approach provides a framework to understand the fabrication of microstructured optical fibres containing closely spaced holes separated by interconnected thin viscous struts. We show sample solutions for simple networks with $J=2$ and $N=2$ or 3. We also demonstrate that there is no uniquely defined junction model to describe interconnections between viscidas of different thicknesses.

Melting and dripping of a heated material with temperature-dependent viscosity in a thin vertical tube

“…Natural applications include convection of planetary mantles (Fowler 1985), planetary core formation (Ockendon et al 1985), the dynamics of ice sheet flows (Fowler 1992) and the melting of thin films (Zippelius, Halperin & Nelson 1980). Industrial applications include the manufacture of glass and optical fibre, with related models applied to the flow of molten glass (Stokes 1998;Griffiths & Howell 2008), the pulling of a hollow glass tube that is being radiatively heated (Huang et al 2007) and the fabrication of microstructured optical fibres (Tronnolone et al 2016;Mavroyiakoumou, Griffiths & Howell 2019). The winterisation of biodiesel at temperatures below 273 K involves heating of a solid back to a liquid before the biodiesel can be used, and the temperature-dependent viscosity of biodiesel was described by Kerschbaum & Rinke (2004).…”

“…2007) and the fabrication of microstructured optical fibres (Tronnolone et al. 2016; Mavroyiakoumou, Griffiths & Howell 2019). The winterisation of biodiesel at temperatures below 273 K involves heating of a solid back to a liquid before the biodiesel can be used, and the temperature-dependent viscosity of biodiesel was described by Kerschbaum & Rinke (2004).…”