Levitation of non-magnetizable droplet inside ferrofluid

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 coupled set of Maxwell's magnetostatic equations and the flow dynamic equations is integrated computationally, utilizing a conservative finite-volume based second order pressure projection algorithm combined with the front-tracking algorithm for the advection of the interface of the droplet. The dynamics of the droplet is studied under both the constant ferrofluid magnetic permeability assumption as well as for more realistic field dependent permeability described by the Langevin's non-linear magnetization model. Due to the non-homogeneous nature of the magnetic field, unique shapes of the droplet during its levitation, and at its steady state, are realized. The complete spatio-temporal response of the droplet is a function of the Laplace number La, the magnetic Laplace number La m , and the Galilei number Ga; through detailed simulations we separate out individual roles played by these non-dimensional parameters. 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 compare 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.

Direct numerical simulation of magneto-Archimedes separation of spherical particles

“…Singh et al. (2018) followed a similar approach to investigate the motion of a water droplet in a saturated ferrofluid.…”

“…A dynamical analysis of the linearization of system (5.2) reveals that depending on the problem parameters, the nature of motion of particle in the vicinity of its equilibrium position is monotonic or oscillatory (for details of the dynamical analysis, the reader is referred to appendix A). Singh et al (2018) followed a similar approach to investigate the motion of a water droplet in a saturated ferrofluid.…”

“…When a particle is allowed to move freely under the combined effect of the magnetic buoyancy force and gravity, new features can occur in the particle motion. Singh, Das & Das (2018) used a one-fluid numerical model to explore the centre of mass and interface dynamics of an almost neutrally buoyant droplet (mass density ratio of ) levitating in a ferrofluid. By approximating the vertical motion of the droplet by a simple dynamical model, the authors showed that depending on the parameters of the considered magnetofluidic set-up, the behaviour of vertical droplet motion can be monotonic or oscillating.…”

Magnetofluidic mixing of a ferrofluid droplet under the influence of a time-dependent external field

“…The passive approach entirely depends on the molecular diffusion between the phases to be mixed and can be tuned by altering the surface morphology (for example, a patterned wettability controlled surface) towards achieving the desired controllability of the mixing (Xing, Harake & Pan 2011;de Groot et al 2016). While the active approach, which is more prevalent due to its reconfigurable flexibility, primarily relies on external force fields such as an electric field, magnetic field, acoustic waves and light energy for manoeuvring the flow velocities inside the droplet domain (Shang, Cheng & Zhao 2017;Meng & Colonius 2018;Singh, Das & Das 2018;Behera, Mandal & Chakraborty 2019;Grassia 2019).…”

“…While the active approach, which is more prevalent due to its reconfigurable flexibility, primarily relies on external force fields such as electric field, magnetic field, acoustic waves, and light energy for maneuvering the droplet flow field (Behera et al 2019;Grassia 2019;Meng & Colonius 2018;Shang et al 2017;Singh et al 2018).…”