Levitation of non-magnetizable droplet inside ferrofluid

TL;DR

This study demonstrates stable levitation of a non-magnetizable droplet inside ferrofluid using complex magnetic fields, revealing unique interfacial features and stability conditions through experiments, simulations, and a dynamical model.

Contribution

It introduces the first detailed analysis of non-magnetizable droplet levitation inside ferrofluid, including stability criteria, surface singularities, and a predictive dynamical model.

Findings

Stable levitation is achievable with proper magnetic field modulation.

Surface cusps and singularities can develop at certain viscosity ratios.

The dynamical model accurately predicts the droplet's vertical trajectory and stability transitions.

Abstract

The central theme of this work is that a stable levitation of a denser non-magnetizable liquid droplet, against gravity, inside a relatively lighter ferrofluid -- a system barely considered in ferrohydrodynamics -- is possible, and exhibits unique interfacial features; the stability of the levitation trajectory, however, is subject to an appropriate magnetic field modulation. We explore the shapes and the temporal dynamics of a plane non-magnetizable droplet levitating inside ferrofluid against gravity due to a spatially complex, but systematically generated, magnetic field in two dimensions. The effect of the viscosity ratio, the stability of the levitation path and the possibility of existence of multiple-stable equilibrium states is investigated. We find, for certain conditions on the viscosity ratio, that there can be developments of cusps and singularities at the droplet surface; this phenomenon we also observe experimentally and compared with the simulations. Our simulations closely replicate the singular projection on the surface of the levitating droplet. Finally, we present an dynamical model for the vertical trajectory of the droplet. This model reveals a condition for the onset of levitation and the relation for the equilibrium levitation height. The linearization of the model around the steady state captures that the nature of the equilibrium point goes under a transition from being a spiral to a node depending upon the control parameters, which essentially means that the temporal route to the equilibrium can be either monotonic or undulating. The analytical model for the droplet trajectory is in close agreement with the detailed simulations. (See draft for full abstract).