Stability and nonlinear evolution of circular flows in free electrolyte films under steady and time-periodic electromagnetic forcing

Circular flow can be induced electromagnetically in a horizontal unsupported film of an electrolyte fluid attached to concentric cylindrical electrodes and placed in an external orthogonal magnetic field. In the presence of a surfactant, the film remains approximately undeformed, and the flow is effectively two-dimensional. We study the linear stability of the base azimuthal flow and nonlinear flow dynamics induced by direct and alternating sinusoidal currents. For a direct current, precessing vortices are formed for Reynolds numbers above the critical. We follow such nonlinear solutions by numerical continuation in parameter space and determine the metastability boundary for the base flow. The classical plane Poiseuille flow (PPF) is recovered in the limit of infinite annulus radius. In channels with a small width-to-radius ratio, we find a signature of temporal intermittency in a fully developed nonlinear flow. Contrary to PPF, subcritical transition to turbulence in a circular channel is not found. In the case of a sinusoidal current, the time-periodic base flow is calculated analytically, and its stability is studied using the Floquet theory. The first azimuthal mode that gives rise to a single precessing vortex is known to be linearly stable for a dc forcing. However, we show that this mode can be destabilized under the ac forcing. The linear stability of azimuthal modes with larger wavenumbers follows a qualitatively similar scenario as known for periodically driven PPF: high frequency forcing stabilizes the base flow, while moderate and low frequency forcing is destabilizing.