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

Published online by Cambridge University Press:  26 October 2020

Benjamin M. Sloman ,
Colin P. Please  and
Robert A. Van Gorder Open the ORCID record for Robert A. Van Gorder [Opens in a new window]
Benjamin M. Sloman
Affiliation:
Elkem ASA, Technology, Fiskaaveien 100, 4621Kristiansand, Norway
Colin P. Please
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, OxfordOX2 6GG, UK
Robert A. Van Gorder*
Affiliation:
Department of Mathematics and Statistics, University of Otago, P.O. Box 56, Dunedin9054, New Zealand
*
†Email address for correspondence: rvangorder@maths.otago.ac.nz
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Abstract

We consider the flow of a thermoviscous fluid within a vertical tube which is heated from below, modelling a scenario where a fluid melts, flows and eventually drips due to a temperature-dependent viscosity. To do so, we develop a two-dimensional axisymmetric model comprising three regions, a solid granular upper region (modelled as a region of hydrostatic pressure), a middle highly viscous ‘crust’ region which flows and a lower cavity region within which the material can drip. New material is continuously added to the top, yet the highly viscous middle region can slow mass transfer from the top region to the cavity if it becomes too thick or does not drip fast enough. In the limit of a tall, thin geometry, akin to what is often seen in industrial applications, the resulting model comprises a moving boundary problem governed by an energy equation with a Stefan condition, both subject to a non-local radially averaged convective term. We carry out numerical simulations and an asymptotic analysis of the model in this tall, thin limit, for a variety of physically relevant parameter regimes. Our results reveal a variety of qualitatively different behaviours, and enable us to explore how various parameter regimes influence the salient features of the flow, including the ‘crust’ thickness and the flux of material through the lower moving boundary.

JFM classification

Phase change: Solidification/melting
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 905 , 25 December 2020 , A16
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

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