Metal-pad-roll instability theory for small-scale models of reduction cells
Abstract
We present a theoretical model for the metal-pad-roll instability in laboratory-scale experimental models of reduction cells consisting of two stably stratified liquid layers subjected to a vertical electric current and magnetic field. In contrast to most previous studies, we do not rely on the shallow-water approximation and can account for viscous and capillary effects that are prevalent in small-scale experiments. In small cells, dissipation effects are largely attributable to laminar viscous damping, which we calculate analytically using a perturbative formulation of the Stokes boundary layers forming at all container walls and on both sides of the liquid interface. The derived damping rates, which are also useful for modeling liquid-liquid sloshing in rectangular containers, are validated against direct numerical simulations conducted with the coupled level set and volume-of-fluid solver ARCHER. On this basis, we derive analytical solutions for the growth rates and instability onsets of the metal-pad-roll instability, which we compare with different theoretical descriptions and existing experiments. The presented solutions are intended to serve as a foundation for future benchmarking of multiphase solvers.
Related
- Partition-theoretic density formulas: radial Tauberian theory and non-radial instability
- Instability theory of stationary solutions for nonlinear Schrödinger equations
- Algebraic 𝔸1-invariance of K-theory for affinoid Tate adic spaces of good reduction
- Generic small-scale creation in the two-dimensional Euler equation
- Cyclic L3 Models and Holomorphic Reduction in Heterotic G2 Deformation Theory
BibTeX
@article{hegde2026metalpadroll,
title = {{Metal-pad-roll instability theory for small-scale models of reduction cells}},
author = {Pranav Hegde and Wietze Herreman and Jorge César Brändle de Motta and Romain Canu and Marie-Charlotte Renoult and Gerrit Maik Horstmann},
journal = {Physical Review Fluids},
year = {2026},
month = sep,
doi = {10.1103/5ynm-wdlr},
url = {https://hal.science/hal-05740122v1},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Metal-pad-roll instability theory for small-scale models of reduction cells
AU - Pranav Hegde
AU - Wietze Herreman
AU - Jorge César Brändle de Motta
AU - Romain Canu
AU - Marie-Charlotte Renoult
AU - Gerrit Maik Horstmann
PY - 2026
DA - 2026/09/07
JO - Physical Review Fluids
DO - 10.1103/5ynm-wdlr
UR - https://hal.science/hal-05740122v1
AB - We present a theoretical model for the metal-pad-roll instability in laboratory-scale experimental models of reduction cells consisting of two stably stratified liquid layers subjected to a vertical electric current and magnetic field. In contrast to most previous studies, we do not rely on the shallow-water approximation and can account for viscous and capillary effects that are prevalent in small-scale experiments. In small cells, dissipation effects are largely attributable to laminar viscous damping, which we calculate analytically using a perturbative formulation of the Stokes boundary layers forming at all container walls and on both sides of the liquid interface. The derived damping rates, which are also useful for modeling liquid-liquid sloshing in rectangular containers, are validated against direct numerical simulations conducted with the coupled level set and volume-of-fluid solver ARCHER. On this basis, we derive analytical solutions for the growth rates and instability onsets of the metal-pad-roll instability, which we compare with different theoretical descriptions and existing experiments. The presented solutions are intended to serve as a foundation for future benchmarking of multiphase solvers.
SN - 2469-990X
ER -