The mechanism of vortex instability in electromagnetically driven flow in an annular thin layer of electrolyte

Abstract

A circumferential flow of a conducting fluid in an annular channel can be created by the action of a Lorentz force arising as a result of the interaction between an applied vertical magnetic field and a radial electric current flowing through the electrolyte. Quite unexpectedly, experiments revealed that a robust vortex system appears near the outer cylindrical wall in such flows. McCloughan and Suslov (J. Fluid Mech. 887:A23, 2020) (McCS) reported comprehensive linear stability results of such a flow for variable Lorentz forcing. Here we complement that study by investigating the flow structure as a function of the channel aspect ratio. Remarkably, despite the completely different physical nature of parametric dependences, dynamic in McCS and purely geometric here, we show that in both scenarios vortices appear on a background of a steady axisymmetric flow at the boundary between two counter-rotating toroidal structures and have a similar energy distributions. The two studies demonstrate the robustness of the mechanism responsible for the vortex formation: Rayleigh's inviscid centrifugal instability aided by radial shear in the boundary layer near the outer cylindrical wall. References P. A. Davidson. An introduction to magnetohydrodynamics. Cambridge University Press, 2nd edition, 2017. doi:10.1017/CBO9780511626333. J. McCloughan and S. A. Suslov. Linear stability and saddle–node bifurcation of electromagnetically driven electrolyte flow in an annular layer. J. Fluid Mech., 887:A23.1–30, 2020. doi:10.1017/jfm.2020.29. J. Perez-Barrera, J. E. Perez-Espinoza, A. Ortiz, E. Ramos, and S. Cuevas. Instability of electrolyte flow driven by an azimuthal Lorentz force. Magnetohydrodynamics, 51(2):203–213, 2015. http://mhd.sal.lv/contents/2015/2/MG.51.2.4.R.html. S. A. Suslov, J. Perez-Barrera, and S. Cuevas. Electromagnetically driven flow of electrolyte in a thin annular layer: Axisymmetric solutions. J. Fluid Mech., 828: 573–600, 2017. doi:10.1017/jfm.2017.551.

Swirling electrolyte. Part 1. Hierarchy of catastrophic transitions in steady annular flows

“…This naturally leads to the hypothesis that free-surface vortices result from an instability of an azimuthally uniform steady basic flow, and that they are effectively perturbations superposed on such a basic flow. This led us to perform a linear stability analysis of Type 1 and 2 solutions reported in McCloughan & Suslov (2020 a , b ) for cases when either the Reynolds number or the electrolyte layer depth were varied. It revealed that only Type 2 flows can be unstable with respect to various azimuthally periodic perturbations whereas Type 1 flows are always linearly stable.…”

Analysis of Electromagnetically Driven Flow in an Annular Layer of Conducting Fluid

“…As the electric current is increased a second counter-rotating toroidal structure suddenly appears, and we refer to such a flow as Type 2. Subsequently, 2 J. McCloughan [2] vortices develop on the background of such a two-torus basic flow near the corner of the flow domain formed by the outer cylinder and the free surface. As the electric current is increased further these two steady basic flow types become topologically indistinguishable and eventually disappear, suggesting the existence of a saddle-node bifurcation.…”

“…This results in more energetic collisions of the basic flow and perturbations near the free surface and, subsequently, past the fold point the vortices still arise on the two-torus background, even though it originates from a different branch of the fold. Some of this research has been published in [1,2].…”