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When liquid metal batteries are charged or discharged, strong electrical currents are passing through the three liquid layers that we find in their interior. This may result in the metal pad roll instability that drives gravity waves on the interfaces between the layers. In this paper, we investigate theoretically metal pad roll instability in idealised cylindrical liquid metal batteries that were simulated previously by Weber et al. ( Phys. Fluids , vol. 29, no. 5, 2017 b , 054101) and Horstmann et al. ( J. Fluid Mech. , vol. 845, 2018, pp. 1–35). Near the instability threshold, we expect weakly destabilised gravity waves, and in this parameter regime, we can use perturbation methods to find explicit formulas for the growth rate of all possible waves. This perturbative approach also allows us to include dissipative effects, hence we can locate the instability threshold with good precision. We show that our theoretical growth rates are in quantitative agreement with previous and new direct numerical simulations. We explain how our theory can be used to estimate a lower bound on cell size beneath which metal pad roll instability is unlikely.
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Steady translating hollow vortex pair in weakly compressible flow
The weakly compressible translating hollow vortex pair is examined using the Imai–Lamla formula and a direct conformal mapping approach applied to a Rayleigh–Jansen expansion in the Mach number, 𝑀, taken to be small. The incompressible limit has been studied by Crowdy et al. (Eur. J. Mech. B Fluids 37 (2013), 180–186) who found explicit formulas for the solutions using conformal mapping. We improve upon their results by finding an explicit formula for the conformal map which had been left as an integral in Crowdy et al. (2013). The weakly compressible problem requires the solution of two boundary value problems in an annulus. This results in a linear parameter problem for the perturbed propagation velocity and speed along the vortex boundary. We find that two additional constraints are required to solve for the perturbed parameters. We require that the perturbation in vortex area and the perturbation in centroid separation to vanish. Three possible centroid definitions are given and the results worked out for each case. It is found that the correction to the propagation velocity is always negative, so that the vortex always slows down at first order due to compressible effects. Numerical results are consistent in the limit of small vortex size with the previous result by Leppington (J. Fluid Mech. 559 (2006), 45–55) that the propagation parameter is unchanged to 𝑂(𝑀2).
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- Award ID(s):
- 1706934
- PAR ID:
- 10476562
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Physica D: Nonlinear Phenomena
- Volume:
- 457
- Issue:
- C
- ISSN:
- 0167-2789
- Page Range / eLocation ID:
- 133943
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1016/j.physd.2023.133943
