Observation of Axisymmetric Solitary Waves on the Surface of a Ferrofluid

Abstract

We report the first observation of axisymmetric solitary waves on the surface of a cylindrical magnetic fluid layer surrounding a current-carrying metallic tube. According to the ratio between the magnetic and capillary forces, both elevation and depression solitary waves are observed with profiles in good agreement with theoretical predictions based on the magnetic analogue of the Korteweg-deVries equation. We also report the first measurements of the velocity and the dispersion relation of axisymmetric linear waves propagating on the cylindrical ferrofluid layer that are found in good agreement with theoretical predictions.PACS numbers: 47.65.Cb,47.35.Fg Solitary waves or solitons are localized nonlinear waves that propagate almost without deformation due to the balance between the nonlinearity and the dispersion. Since the first observation of a solitary wave on the freesurface of water by Russel [1], and its interpretation using the Korteweg-de Vries equation (KdV) [2], it has been shown that the KdV equation describes a large class of solitons observed in various situations: acoustic waves on a crystal lattice, plasma waves, hydrodynamics internal or surface waves, elastic surface waves, and waves in optical fibers [3]. Most of them involve a localized elevation disturbance propagating within a quasi-one-dimensional plane system. Observations of axisymmetric solitary waves governed by the KdV equation are scarce [4], and mainly concern waves in rotating fluids confined in a tube or on vortex lines. More recently, Bashtovoi et al. derived a KdV equation with an axisymmetric solitary waves solution propagating on the surface of a cylindrical magnetic fluid layer submitted to a magnetic field [5,6]. Without gravity, the stability of the cylindrical magnetic fluid layer is governed by the ratio between the magnetic force and the capillary one. According to its ratio, axisymmetric elevation (hump-like) or depression (hole-like) solitary waves are predicted with a subsonic or supersonic velocity [5,6]. To our knowledge, direct observation of axisymmetric magnetic solitary waves have never been reported.In this Letter, we report the first observation of axisymmetric solitary waves on the surface of a cylindrical ferrofluid layer submitted to an azimuthal magnetic field. Depending on the strength of the field, elevation or depression solitary waves are observed on the ferrofluid surface. A ferrofluid is a stable suspension of nanometric magnetic particles diluted in a carrier liquid (water or oil) that responds to an external applied magnetic field [7,8]. Although the solitary waves are damped by viscous dissipation, we have shown that they keep the self-similar profile given by the solution of the KdV equation on a propagation length larger than their typical scale. Moreover, we also report the first measurement of the velocity and dispersion relation of axisymmetric magnetic linear waves in this system in good agreement with the theoretical predictions [9].The experimental setup is shown in Fig. 1. It consists...

Azimuthal field instability in a confined ferrofluid

“…We want to examine the family of planar curves whose curvature satisfies the differential equation (18). These curves are the stationary solutions that balance the competing mag netic and capillary forces at the ferrofluid interface.…”

“…ufpe.br tjm e@ df.ufpe.br the azimuthal magnetic field configuration produced by a straight current-carrying wire has also been used to examine the dynamics of solitons propagating on cylindrical ferrofluid surfaces surrounding the wire. There is a recent example in the literature in which theoretical predictions [15][16][17] about solitary wave propagation in ferrofluids has been realized experimentally [18].…”

Solitary waves on a ferrofluid jet

“…Alternatively the limiting profile may correspond to a smooth solution of the static problem. Bourdin et al (2010), henceforth referred to as BBF, reported experimental observations of axisymmetric solitary waves on a ferrofluid. The experimental set-up is essentially identical to the theoretical scenario studied by RE and also studied in the present work.…”

“…When B ≥ 1, the wave speed c is real and neutral waves exist for arbitrary wave number k. It follows that a magnetic field of sufficient intensity can stabilise the Rayleigh capillary mode responsible for jet break-up under normal conditions. This result underpins the studies of RE and Bourdin et al (2010) as it provides a firm basis for both the theoretical and experimental search for solitary wave solutions, which appear as bifurcations from the neutral wave branch. A second important observation from (3.3) is that a minimum, c M , appears in the dispersion curve when B > B 2 , where B 2 depends on the radius of the axial rod b.…”

“…We consider a more general set-up in which the magnetic field is generated by a current-carrying rod of arbitrary radius. One advantage of this is that it allows us to compare results with the experimental observations which have recently been reported by Bourdin et al 2010. The structure of the remainder of the paper is as follows.…”