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

Abstract

This study extends our previous work (McCloughan & Suslov, J. Fluid Mech., vol. 887, 2020, A23), where the existence of a saddle-node bifurcation of steady axisymmetric electrolyte flows driven by the Lorentz force in a shallow annular domain was first reported. Here we perform further weakly nonlinear analysis over a wider range of the governing parameters to demonstrate that the previously reported saddle-node bifurcation is a local feature of a global fold catastrophe, which, in turn, is a part of cusp catastrophe occurring as the thickness of the fluid layer increases. The amplitude equation characterising multiple flow solutions in the finite vicinity of catastrophe points is derived. The sensitivity of its coefficients and solutions to the distance from the catastrophe points is assessed demonstrating the robustness of the used analytical procedure. The asymptotic flow solution past the catastrophe point is subsequently obtained and its topology is explored confirming the existence of the secondary circulation in the bulk of flow (two-tori background flow structure). The latter is argued to lead to the appearance of experimentally observable vortices on the fluid surface. The rigorous justification of this conjecture is to be given in Part 2 of the study.