Solitary waves on a ferrofluid jet

Blyth, Mark; Părău, Emilian

doi:10.1017/jfm.2014.275

Cited by 15 publications

(33 citation statements)

References 21 publications

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“…This system admits a homoclinic solution which corresponds to a solitary wave of elevation for c 1 > 0 and depression for c 1 < 0; the wave is symmetric with an oscillatory decaying tail. For m 1 (1) close to the critical value 8 3 we write m 1 (1) = 1 3 (8 + 1 144 √ 6κ µ 2 ) with 0 <κ 1 and find that small-amplitude solitary waves are given by η(z) = 1 2 µ 2 P 1 (µz) + O(µ 3 ), where (Q, P ) is a homoclinic solution of the reversible Hamiltonian systeṁ Q 1 = −P 1 + 2 3 (1 + δ)P 2 + 4 9 (1 + δ) 2 P 1 +κP 2 1 + 4d 1 P 3 1 + O(µ), Q 2 = P 2 + 2 3 (1 + δ)P 1 + O(µ), P 1 = Q 2 + O(µ), (1) . For d 1 > 0 this system admits a a pair of homoclinic solutions which correspond to symmetric solitary waves with oscillatory decaying tails; one is a wave of depression, the other a wave of elevation.…”

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confidence: 93%

Spatial Dynamics Methods for Solitary Waves on a Ferrofluid Jet

Groves

Nilsson

2018

J. Math. Fluid Mech.

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This paper presents existence theories for several families of axisymmetric solitary waves on the surface of an otherwise cylindrical ferrofluid jet surrounding a stationary metal rod. The ferrofluid, which is governed by a general (nonlinear) magnetisation law, is subject to an azimuthal magnetic field generated by an electric current flowing along the rod.The ferrohydrodynamic problem for axisymmetric travelling waves is formulated as an infinite-dimensional Hamiltonian system in which the axial direction is the time-like variable. A centre-manifold reduction technique is employed to reduce the system to a locally equivalent Hamiltonian system with a finite number of degrees of freedom, and homoclinic solutions to the reduced system, which correspond to solitary waves, are detected by dynamical-systems methods. * Fachrichtung 6.We consider an incompressible, inviscid ferrofluid of unit density in the regionbounded by the free interface {r = R + η(θ, z, t)} and a current-carrying wire at {r = 0}, where (r, θ, z) are cylindrical polar coordinates. The fluid is subject to a static magnetic field and the surrounding regionis a vacuum (see Figure 1). Travelling waves move in the axial direction with constant speed c and without change of shape, so that η(θ, z, t) = η(θ, z − ct). We are interested in particular in axisymmetric solitary waves for which η does not depend upon θ and η(z − ct) → 0 as z − ct → ±∞. Waves of this kind for ferrofluids with a linear magnetisation law have been investigated using a weakly nonlinear approximation by Rannacher & Engel [18], experimentally by Bourdin, Bacri & Falcon [4] and numerically by Blyth & Parau [3]. In this paper we present a rigorous existence theory for small-amplitude solitary waves and consider fluids with a general (nonlinear) magnetisation law. Our starting point is a formulation of the hydrodynamic problem as a reversible Hamiltonian systemin which the axial coordinate z plays the role of time,φ is a variable related to the fluid velocity potential φ and ω,ζ are the momenta associated with the coordinates η,φ. The spatial Hamiltonian system (1.1) is derived from a variational principle for the governing equations in Section 3; it depends upon two dimensionless physical parameters α and β (see equation (2.6) for precise definitions) and the (dimensionless) magnitude m 1 (|H 1 |) of the magnetic intensity corresponding to the magnetic field H 1 in the ferrofluid. Homoclinic solutions of (1.1) (solutions with (η, ω,φ,ζ) → 0 as z → ±∞) are of particular interest since they correspond to solitary waves. We detect such solutions using a technique known as the Kirchgässner reduction (Section 4), in which a centre-manifold reduction principle is used to show that all small, globally bounded solutions of a spatial (Hamiltonian) evolutionary system solve a (Hamiltonian) system of ordinary differential equations, whose solution set can in principle be determined. In this fashion we reduce (1.1) to a Hamiltonian system with finitely many degrees of freedom which can be treated...

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confidence: 93%

Spatial Dynamics Methods for Solitary Waves on a Ferrofluid Jet

Groves

Nilsson

2018

J. Math. Fluid Mech.

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“…Blyth & Pȃrȃu [7] (referred to as BP thoughout) performed a numerical investigation of solitary wave solutions to the one-layer model in the fully nonlinear regime for arbitrary values of d. They found that, for 1 < B < B 1 (d), solitary waves bifurcating from zero amplitude are elevation waves, while for B 1 (d) < B < B 2 (d) these solutions are depression waves. This is in good agreement with Rannacher & Engel's KdV equation, who found B 1 = 3/2 and B 2 = 9 when d = 0.…”

Section: Introductionmentioning

confidence: 99%

Travelling wave solutions on an axisymmetric ferrofluid jet

Doak

Vanden‐Broeck

2019

J. Fluid Mech.

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We consider a potential flow model of axisymmetric waves travelling on a ferrofluid jet. The ferrofluid coats a copper wire, through which an electric current is run. The induced azimuthal magnetic field magnetises the ferrofluid, which in turn stabilises the well known Plateau–Rayleigh instability seen in axisymmetric capillary jets. This model is of interest because the stabilising mechanism allows for axisymmetric magnetohydrodynamical solitary waves. A numerical scheme capable of computing steady periodic, solitary and generalised solitary wave solutions is presented. It is found that the solution space for the model is very similar to that of the classical problem of two-dimensional gravity–capillary waves.

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“…Recent experiments by Bourdin et al [7] have confirmed the existence of axisymmetric depression and elevation solitons that follow KdV dynamics. In a recent theoretical study by Blyth and Parau [6] solitary waves of arbitrary amplitude were computed numerically and the elevation and depression weakly nonlinear solitons found in [17] were calculated, along with new branches of solitary waves. Large amplitude waves can develop to form toroidal trapped bubbles as seen by Grandison et al [11] in a different physical and mathematical setup.…”

Section: Introductionmentioning

confidence: 99%

Korteweg–de Vries solitons on electrified liquid jets

2015

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The propagation of axisymmetric waves on the surface of a liquid jet under the action of a radial electric field is considered. The jet is assumed to be inviscid and perfectly conducting, and a field is set up by placing the jet concentrically inside a perfectly cylindrical tube whose wall is maintained at a constant potential. A nontrivial interaction arises between the hydrodynamics and the electric field in the annulus, resulting in the formation of electrocapillary waves. The main objective of the present study is to describe nonlinear aspects of such axisymmetric waves in the weakly nonlinear regime, which is valid for long waves relative to the undisturbed jet radius. This is found to be possible if two conditions hold: the outer electrode radius is not too small, and the applied electric field is sufficiently strong. Under these conditions long waves are shown to be dispersive and a weakly nonlinear theory can be developed to describe the evolution of the disturbances. The canonical system that arises is the Kortweg-de Vries equation with coefficients that vary as the electric field and the electrode radius are varied. Interestingly, the coefficient of the highest-order third derivative term does not change sign and remains strictly positive, whereas the coefficient α of the nonlinear term can change sign for certain values of the parameters. This finding implies that solitary electrocapillary waves are possible; there are waves of elevation for α > 0 and of depression for α < 0. Regions in parameter space are identified where such waves are found.

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Solitary waves on a ferrofluid jet

Cited by 15 publications

References 21 publications

Spatial Dynamics Methods for Solitary Waves on a Ferrofluid Jet

Spatial Dynamics Methods for Solitary Waves on a Ferrofluid Jet

Travelling wave solutions on an axisymmetric ferrofluid jet

Korteweg–de Vries solitons on electrified liquid jets

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