Solution of the adjoint problem for instabilities with a deformable surface: Rosensweig and Marangoni instability

Bohlius, Stefan; Pleiner, Harald; Brand, Helmut R.

doi:10.1063/1.2757709

Cited by 13 publications

(13 citation statements)

References 30 publications

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“…38,39 Another inherent problem is the simultaneous presence of many length scales ranging from the atomic or molecular scales of the polymer matrix over the mesoscopic scale of the magnetic particles to the macroscopic scales of a block of material. While there has been quite a number of previous theoretical works on the macroscopic scale by using concepts from hydrodynamics [40][41][42] and elasticity theory, 43,44 there are only very few micro-and mesoscopic studies on ferrogels that explicitly incorporate the magnetic particles.…”

Section: Introductionmentioning

confidence: 99%

Hardening transition in a one-dimensional model for ferrogels

Annunziata

Menzel²,

Löwen³

2013

The Journal of Chemical Physics

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We introduce and investigate a coarse-grained model for quasi one-dimensional ferrogels. In our description the magnetic particles are represented by hard spheres with a magnetic dipole moment in their centers. Harmonic springs connecting these spheres mimic the presence of a cross-linked polymer matrix. A special emphasis is put on the coupling of the dipolar orientations to the elastic deformations of the matrix, where a memory effect of the orientations is included. Although the particles are displaced along one spatial direction only, the system already shows rich behavior: as a function of the magnetic dipole moment, we find a phase transition between "soft-elastic" states with finite interparticle separation and finite compressive elastic modulus on the one hand, and "hardened" states with touching particles and therefore diverging compressive elastic modulus on the other hand. Corresponding phase diagrams are derived neglecting thermal fluctuations of the magnetic particles. In addition, we consider a situation in which a spatially homogeneous magnetization is initially imprinted into the material. Depending on the strength of the magneto-mechanical coupling between the dipole orientations and the elastic deformations, the system then relaxes to a uniaxially ferromagnetic, an antiferromagnetic, or a spiral state of magnetization to minimize its energy. One purpose of our work is to provide a largely analytically solvable approach that can provide a benchmark to test future descriptions of higher complexity. From an applied point of view, our results could be exploited, for example, for the construction of novel damping devices of tunable shock absorbance.

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Section: Introductionmentioning

confidence: 99%

Hardening transition in a one-dimensional model for ferrogels

Annunziata

Menzel²,

Löwen³

2013

The Journal of Chemical Physics

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“…b Localised axisymmetric peaks, termed 'spot A' solutions, have been observed experimentally (see Richter 2011, reproduced with permission) of Zaitsev and Shliomis (1970) to a finite-depth ferrofluid. These results, close to the bifurcation point, were supplemented by other studies of two-dimensional periodic free surfaces; see Silber and Knobloch (1988) for normal-form analysis, Bohlius et al (2011Bohlius et al ( , 2007 for deriving amplitude equations near onset and Horn (2015), Groves and Horn (2018) for a Dirichlet-Neumann formulation and local bifurcation theory. With the exception of the Groves & Horn work, all these studies included a linear magnetisation law.…”

Section: Theoretical Approaches To Ferrofluidsmentioning

confidence: 87%

Localised Radial Patterns on the Free Surface of a Ferrofluid

2021

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This paper investigates the existence of localised axisymmetric (radial) patterns on the surface of a ferrofluid in the presence of a uniform vertical magnetic field. We formally investigate all possible small-amplitude solutions which remain bounded close to the pattern’s centre (the core region) and decay exponentially away from the pattern’s centre (the far-field region). The results are presented for a finite-depth, infinite expanse of ferrofluid equipped with a linear magnetisation law. These patterns bifurcate at the Rosensweig instability, where the applied magnetic field strength reaches a critical threshold. Techniques for finding localised solutions to a non-autonomous PDE system are established; solutions are decomposed onto a basis which is independent of the radius, reducing the problem to an infinite set of nonlinear, non-autonomous ODEs. Using radial centre manifold theory, local manifolds of small-amplitude solutions are constructed in the core and far-field regions, respectively. Finally, using geometric blow-up coordinates, we match the core and far-field manifolds; any solution that lies on this intersection is a localised radial pattern. Three distinct classes of stationary radial solutions are found: spot A and spot B solutions, which are equipped with two different amplitude scaling laws and achieve their maximum amplitudes at the core, and ring solutions, which achieve their maximum amplitudes away from the core. These solutions correspond exactly to the classes of localised radial solutions found for the Swift–Hohenberg equation. Different values of the linear magnetisation and depth of the ferrofluid are investigated and parameter regions in which the various localised radial solutions emerge are identified. The approach taken in this paper outlines a route to rigorously establish the existence of axisymmetric localised patterns in the future.

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“…5 In addition, the dynamics of the crests has also been intensively studied both experimentally [6][7][8][9][10][11][12] and theoretically. [13][14][15][16] However, these researches focus mainly on the magnetic fluid layer at a relatively larger scale. With advances in microtechnology, it is interesting to further investigate the instability phenomena of an extremely thin layer, in which stronger effects by surface tension might lead to interesting behaviors.…”

Section: Introduction and Experimental Setupmentioning

confidence: 99%

Ordered microdroplet formations of thin ferrofluid layer breakups

Chen

2010

Physics of Fluids

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The ordered breakup pattern of a thin layer of ferrofluid drop subjected to a uniform perpendicular field is experimentally investigated. The results confirm a universal pattern formation containing numerous breaking droplets of a uniform size, which is independent of the initial area of ferrofluid drop and the propagating directions of the formation waves. Two quantitative observations regarding the size and number of breaking droplets are concluded. Both the experiments and theoretical analysis show the correlation between the diameter of breaking droplets ͑d͒ and magnetization strength ͑M͒ can be characterized as d ϰ 1 / M 2 . The uniform size of breaking droplets under a constant field strength results in a linear proportionality between the number of breaking droplets ͑N͒ and the initial area of ferrofluid drop ͑A͒ as N ϰ A, which is verified by the experiments.

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Solution of the adjoint problem for instabilities with a deformable surface: Rosensweig and Marangoni instability

Cited by 13 publications

References 30 publications

Hardening transition in a one-dimensional model for ferrogels

Hardening transition in a one-dimensional model for ferrogels

Localised Radial Patterns on the Free Surface of a Ferrofluid

Ordered microdroplet formations of thin ferrofluid layer breakups

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