Delayed Hopf bifurcation and control of a ferrofluid interface via a time-dependent magnetic field

TL;DR

This paper develops a reduced-order model to control the emergence of a spinning ferrofluid droplet interface via a time-dependent magnetic field, revealing delayed bifurcation and hysteresis effects.

Contribution

It introduces a novel amplitude equation approach to predict and control interface dynamics in ferrofluids under time-varying magnetic fields.

Findings

Delayed onset of interface instability can be controlled by magnetic field timing.

Hysteresis-like behavior observed during magnetic field reversal.

Reduced-order model accurately predicts dynamic bifurcation outcomes.

Abstract

A ferrofluid droplet confined in a Hele-Shaw cell can be deformed into a stably spinning ``gear,'' using crossed magnetic fields. Previously, fully nonlinear simulation revealed that the spinning gear emerges as a stable traveling wave along the droplet's interface bifurcates from the trivial (equilibrium) shape. In this work, a center manifold reduction is applied to show the geometrical equivalence between a two-harmonic-mode coupled system of ordinary differential equations arising from a weakly nonlinear analysis of the interface shape and a Hopf bifurcation. The rotating complex amplitude of the fundamental mode saturates to a limit circle as the periodic traveling wave solution is obtained. An amplitude equation is derived from a multiple-time-scale expansion as a reduced model of the dynamics. Then, inspired by the well-known delay behavior of time-dependent Hopf bifurcations, we design a slowly time-varying magnetic field such that the timing and emergence of the interfacial traveling wave can be controlled. The proposed theory allows us to determine the time-dependent saturated state resulting from the dynamic bifurcation and delayed onset of instability. The amplitude equation also reveals hysteresis-like behavior upon time reversal of the magnetic field. The state obtained upon time reversal differs from the state obtained during the initial (forward-time) period, yet it can still be predicted by the proposed reduced-order theory.