Tuning a magnetic field to generate spinning ferrofluid droplets with controllable speed via nonlinear periodic interfacial waves

TL;DR

This paper demonstrates how magnetic fields can be used to control the shape and rotation speed of ferrofluid droplets in a Hele-Shaw cell through nonlinear interface waves, combining linear instability analysis with nonlinear theory.

Contribution

It introduces a method to manipulate ferrofluid droplet shapes and motion using combined magnetic fields, advancing control of nonlinear interfacial phenomena.

Findings

Magnetic fields can induce and control nonlinear traveling waves on ferrofluid droplets.

Nonlinear theory predicts droplet deformation and rotation speed based on magnetic field parameters.

The shape and speed of droplets depend on the most unstable linear mode and magnetic field strength.

Abstract

Two dimensional free surface flows in Hele-Shaw configurations are a fertile ground for exploring nonlinear physics. Since Saffman and Taylor's work on linear instability of fluid--fluid interfaces, significant effort has been expended to determining the physics and forcing that set the linear growth rate. However, linear stability does not always imply nonlinear stability. We demonstrate how the combination of a radial and an azimuthal external magnetic field can manipulate the interfacial shape of a linearly unstable ferrofluid droplet in a Hele-Shaw configuration. We show that weakly nonlinear theory can be used to tune the initial unstable growth. Then, nonlinearity arrests the instability, and leads to a permanent deformed droplet shape. Specifically, we show that the deformed droplet can be set into motion with a predictable rotation speed, demonstrating nonlinear traveling waves on the fluid-fluid interface. The most linearly unstable wavenumber and the combined strength of the applied external magnetic fields determine the traveling wave shape, which can be asymmetric.