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Perturbation theory for metal pad roll instability in cylindrical reduction cells

Published online by Cambridge University Press:  18 September 2019

W. Herreman Open the ORCID record for W. Herreman [Opens in a new window] ,
C. Nore Open the ORCID record for C. Nore [Opens in a new window] ,
J.-L. Guermond ,
L. Cappanera ,
N. Weber  and
G. M. Horstmann Open the ORCID record for G. M. Horstmann [Opens in a new window]
W. Herreman*
Affiliation:
LIMSI, CNRS, Université Paris-Sud, Université Paris-Saclay, Orsay, F-91405, France
C. Nore
Affiliation:
LIMSI, CNRS, Université Paris-Sud, Université Paris-Saclay, Orsay, F-91405, France
J.-L. Guermond
Affiliation:
TAMU, Texas A&M, College Station, TX 77843, USA
L. Cappanera
Affiliation:
CAAM, Rice University, 6100 Main St., Houston, TX 77005-1827, USA
N. Weber
Affiliation:
Helmholtz-Zentrum Dresden - Rossendorf, Bautzner Landstr. 400, 01328 Dresden, Germany
G. M. Horstmann
Affiliation:
Helmholtz-Zentrum Dresden - Rossendorf, Bautzner Landstr. 400, 01328 Dresden, Germany
*
†Email address for correspondence: wietze@limsi.fr
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Abstract

We propose a new theoretical model for metal pad roll instability in idealized cylindrical reduction cells. In addition to the usual destabilizing effects, we model viscous and Joule dissipation and some capillary effects. The resulting explicit formulas are used as theoretical benchmarks for two multiphase magnetohydrodynamic solvers, OpenFOAM and SFEMaNS. Our explicit formula for the viscous damping rate of gravity waves in cylinders with two fluid layers compares excellently to experimental measurements. We use our model to locate the viscously controlled instability threshold in cylindrical shallow reduction cells but also in Mg–Sb liquid metal batteries with decoupled interfaces.

JFM classification

Multiphase and Particle-laden Flows: Multiphase flow

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Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 878 , 10 November 2019 , pp. 598 - 646
Copyright
© 2019 Cambridge University Press 

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