Normal Field Instability Research Articles

Magnetic field induced instability pattern evolution in an immiscible alloy

The magnetic field induced instability patterns have been observed in an undercooled immiscible Co–Cu alloy by an in situ magnetization measurement of the supercooled alloy melt. With the increase in magnetic field intensity and gradient, the undercooled immiscible melt experienced a transition from a core-shell structure to a layered structure at a lower field intensity and then a typical normal field instability pattern with the applied higher magnetic field gradient. Due to the different magnetic response ability of the separated phases in the presence of a magnetic field gradient, the transition of the morphology was complex, and its detailed investigations can provide important insight for better understanding of the ferrofluid and the creation of functional material. Furthermore, under an appropriate field gradient condition, it can achieve the subtle transitions between the diverse morphologies in an immiscible alloy.

A simplified phase-field lattice Boltzmann method with a self-corrected magnetic field for the evolution of spike structures in ferrofluids

This research presents a numerical analysis of the normal field instability for an initially flat layer of ferrofluid under the influence of magnetic field. A coupling between the simplified lattice Boltzmann method and the self-correcting procedure is developed to capture the velocity field and magnetic field. The proposed method has the ability to simulate complex hedgehog and comb-like spike structures without using an additional magnetization equation. A single dipole permanent magnet is defined instead of multiple point dipoles which makes this method much simpler and more efficient compared to the numerical approaches available in the literature. A comparison between the simulation results and experimental findings is provided to verify the validity of our method. A criterion for the prediction of spikes is presented for uniform magnetic fields. This study also investigates the effects of different types of magnetic fields, their strengths, and the effect of surrounding non-magnetic fluid on the spike structures. Moreover, the description of magnetic field lines, distribution of magnetic flux density, and energy estimation are also provided in this work which gives a useful insight into the hydrodynamic as well as the magnetostatic behavior of ferrofluids.

Reversible transition from ferrofluid drop to spikes due to an approaching magnet

Remarkably, we experimentally evidenced normal field instability-driven reversible transition from ferrofluid droplet to spikes, when a sessile ferrofluid drop is exposed to a varying magnetic field gradient due to an approaching magnet, without flipping its direction of motion. We find the reversible transition is attributed to the critical magnetization of the ferrofluid and a characteristic wavelength that control normal-field instability. We extend the theory of magnetic instability by including non-uniformity in the magnetic field in all directions to predict the critical condition for transition, which is in good agreement with experiments. The spacing between spikes and spike height, and the number of spikes measured from experiments are in agreement with existing theory.

The 2-3-4 spike competition in the Rosensweig instability

The horizontal free surface of a magnetic liquid (ferrofluid) pool turns unstable when the strength of a vertically applied uniform magnetic field exceeds a threshold. The instability, known as normal field instability or Rosensweig’s instability, is accompanied by the formation of liquid spikes either few, in small diameter pools, or many, in large diameter pools; in the latter case, the spikes are arranged in hexagonal or square patterns. In small pools where only few spikes – 2, 3 or 4 in this work – can be accommodated, their appearance/disappearance/re-appearance observed in experiments, as applied field strength varies, is investigated by computer-aided bifurcation and linear stability analysis. The equations of three-dimensional capillary magneto-hydrostatics give rise to a three-dimensional free boundary problem which is discretized by the Galerkin/finite element method and solved for multi-spike surface deformation coupled with magnetic field distribution simultaneously with a compact numerical scheme based on Newton iteration. Standard eigenvalue problems are solved in the course of parameter continuation to reveal the multiplicity and the stability of the emerging deformations. The computational predictions reveal selection mechanisms among equilibrium states and explain the corresponding experimental observations and measurements.

Magnetic field distribution in a magnetic liquid spike

Abstract The Rosensweig, or normal field instability, is a well known phenomenon where formation of a spike like shape appears on a magnetic liquid surface when it is exposed to a sufficiently large external magnetic field. While the surface shape of magnetic liquid has an impact on the magnetic field distribution and vice versa, it is somehow essential to know the magnetic field distribution within the magnetic liquid. The aim of the magnetic field calculation is to show that the magnetic field density is distributed similarly as the shape of the free surface. The calculation is placed in the 3D space and performed at the equilibrium state of magnetic liquid using the Finite Element Method based software Opera Field. Only a single spike of the magnetic liquid is included in the computational model, while the pattern of the surface deformation is assumed to be hexagonal and distributed periodically over the surface.

A free boundary approach to the Rosensweig instability of ferrofluids

We establish the existence of saddle points for a free boundary problem describing the two-dimensional free surface of a ferrofluid which undergoes normal field instability (also known as Rosensweig instability). The starting point consists in the ferro-hydrostatic equations for the magnetic potentials in the ferrofluid and air, and the function describing their interface. The former constitute the strong form for the Euler-Lagrange equations of a convex-concave functional. We extend this functional in order to include interfaces that are not necessarily graphs of functions. Saddle points are then found by iterating the direct method of the calculus of variations and by applying classical results of convex analysis. For the existence part we assume a general (arbitrary) non linear magnetization law. We also treat the case of a linear law: we show, via convex duality arguments, that the saddle point is a constrained minimizer of the relevant energy functional of the physical problem.

Shape and fission instabilities of ferrofluids in non-uniform magnetic fields

We study static distributions of ferrofluid submitted to non-uniform magnetic fields. We show how the normal-field instability is modified in the presence of a weak magnetic field gradient. Then we consider a ferrofluid droplet and show how the gradient affects its shape. A rich phase transition phenomenology is found. We also investigate the creation of droplets by successive splits when a magnet is vertically approached from below and derive theoretical expressions which are solved numerically to obtain the number of droplets and their aspect ratio as a function of the field configuration. A quantitative comparison is performed with previous experimental results, as well as with our own experiments, and yields good agreement with the theoretical modelling.

Instability Pattern Formation in a Liquid Metal under High Magnetic Fields

Magnetic field can generate interface instability when some liquids are put close to magnetic field. A well-known interface instability is called Rosensweig instability or normal field instability. Here we report that pure liquid Co can be highly undercooled close to its Curie temperature in strong magnetic field with very high magnetization and exhibiting unique morphology instability called the normal field instability. To obtain such unique instability pattern, the sample size, undercooling and magnetic field intensity need fulfill certain condition. In the present study, we have studied the required condition for obtaining normal field instability. The magnetization of the undercooled liquid Co is measured in a wide temperature range with different magnetic field intensities and calculated as a function of undercooling and field intensity. The critical size and critical magnetization for the normal field instability are calculated with the changing temperature and field intensity. Then the required conditions including the critical size, the minimum undercooling and field intensity for the existence of the instability pattern formation are determined.

Onset of Normal Field Instability in a Ferrofluid in Microgravity

A ferrofluid is a magnetic liquid composed of nanoscale ferrous particles sus- pended in a low-viscosity carrier fluid. When subjected to a magnetic field, the surface of a ferrofluid deforms into peaks and valleys along the magnetic field lines. The onset of surface deformation is called the Normal Field Instability, and the theory describing the NFI identifies a critical magnetic field below which no magnetically driven surface deformations occur. This critical field depends on the gravitational acceleration and, according to the theory, should disappear as local gravitational acceleration approaches zero. Previous studies have been inconclusive on the existence of a critical magnetic field in reduced gravity. For our work, we designed a payload for a suborbital rocket mission which launched through the RockSat-C program. Our experiment incorporated a ferrofluid sam- ple and a uniform magnetic field which could be varied across a discrete range of value. During the microgravity portion of the rocket’s flight, we obtained video of the ferrofluid’s behavior to compare it to data taken in Earth’s gravity.Â

The Onset of Normal Field Instability in a Ferrofluid in a Reduced Gravity Environment

A ferrofluid is a magnetic liquid comprised of nanoscale ferrous particles suspended in a low-viscosity carrier fluid. When subjected to a magnetic field, the surface of a ferrofluid deforms into peaks and valleys along the magnetic field lines. The onset of surface deformation is called the normal field instability. The theory describing the NFI identifies a critical magnetic field below which no magnetically driven surface deformations occur. The critical field depends on the gravitational acceleration and, according to the theory, should disappear as local gravitational acceleration approaches zero. While there have been previous demonstrations of the normal field instability in a reduced gravity environment, data has been inconclusive on the existence of a critical magnetic field in reduced gravity. For our work, we designed a sounding rocket payload for a suborbital rocket mission. Our experiment incorporates a ferrofluid sample and a uniform magnetic field which can be varied across a discrete range of values to observe surface deformations at different applied fields. During the microgravity portion of the rocket’s flight, we obtain video of the ferrofluid’s behavior and compare it to data taken in Earth’s gravity. A team of five undergraduate students designed and built the payload that will characterize the role of gravity in setting the critical magnetic field strength for the onset of the NFI. Data will be used to further understanding of ferrofluid behavior in a microgravity environment to advance its use in various space-based applications.

Anomalous magnetism and normal field instability in supercooled liquid cobalt

Supercooled liquids may offer fascinating phenomena as compared with the normal state. In the case of supercooled paramagnetic liquids, completely different phenomena in high magnetic fields should be observed thanks to the high undercooling leading to higher magnetization and very strong magnetic coupling in the liquids. Here, we report the measurement of an undercooled pure liquid metal with a higher magnetization than the solid at the same temperature. We also observe the normal field instability pattern formation in the undercooled melt. Although the origin of the high magnetism of the liquid remains unclear, we discuss the possibility of a short range ordering effect.

Weakly nonlinear study of normal-field instability in confined ferrofluids

Similar to the classic three-dimensional Rosensweig instability, a ferrofluid confined in a vertical Hele-Shaw cell subjected to an in-plane normal magnetic field develops a periodic array of peaked interfacial structures. We perform a weakly nonlinear analysis that is able to reproduce the morphology of such pattern formation phenomenon at lowest nonlinear order. A mode-coupling theory is applied to compare the early nonlinear evolution of the interface with static shapes obtained when relevant forces equilibrate. Our nonlinear results indicate that the time-evolving shapes tend to approach stable stationary solutions.

Experimental characterization of thermal conductance switching in magnetorheological fluids

We experimentally investigate thermal conductance switching in Fe-based magnetorheological fluids (MRFs). The transient hot-square technique is employed to directly measure enhancement in the thermal conductivity of bulk samples with volume concentrations up to 33% along the field direction. The ratio of the thermal conductivities of bulk MRFs under no and strong (∼290 kA/m) field is approximately 1.3, nearly independent of particle concentration. Significantly higher on-off conductance ratios can be achieved at a device level by exploiting the normal field instability to form columns of MRFs across an air gap. We experimentally demonstrate reversible switching in one implementation of this device concept.

The normal field instability under side-wall effects: comparison of experiments and computations

We consider a single spike of ferrofluid, arising in a small cylindrical container, when a vertically oriented magnetic field is applied. The height of the spike as well as the surface topography is measured experimentally by two different technologies and calculated numerically using the finite element method. As a consequence of the finite size of the container, the numerics uncovers an imperfect bifurcation to a single spike solution, which is forward. This is in contrast to the standard transcritical bifurcation to hexagons, common for rotational symmetric systems with broken up-down symmetry. The numerical findings are corroborated in the experiments. The small hysteresis observed is explained in terms of a hysteretic wetting of the side wall.

Amplitude Equation for the Rosensweig Instability

Since its discovery in 1967 [1], the normal field or Rosensweig instability attracted the attention of experimentalists and theoreticians, alike. The phenomenon describes the transition of an initially flat ferrofluid surface to hexagonally ordered surface spikes as soon as an applied magnetic field exceeds a certain critical value. Ferrofluids are suspensions of magnetic nanoparticles in a suitable carrier liquid. One of the most prominent property is the superparamagnetic behavior in external magnetic fields, which accounts for the large magnetic susceptibility and the high saturation magnetization in rather low magnetic fields. If one starts to crosslink a mixture of a ferrofluid and a polymer solution with cross-linking agents, a superparamagnetic elastic medium, called ferrogel, is obtained [2]. As in usual ferrofluids, the initially flat surface of ferrogels becomes linearly unstable beyond a critical magnetic field [3]. A nonlinear analysis of the Rosensweig instability, however, turned out to be very complicated mainly due to the fact that the instability necessarily involves a deformable surface. In addition, the driving force is provided by the boundary conditions at the deformed surface and not by the bulk equations. A second problem arises from the fact that this instability is intrinsically dynamic in nature, although the linear threshold and the critical wavelength can be obtained in a static description (as an energetic minimum neglecting viscous effects). A nonlinear description, however, has to treat this instability as a breakdown of traveling surface waves.

A note on the similarity between the normal-field instability in ferrofluids and the thermocapillary instability

A striking resemblance between the normal-field instability in ferromagnetic fluids and the interfacial mode of the thermocapillary instability in viscous fluids is presented. A nonlinear evolution equation describing the dynamics of the free surface for a ferrofluid layer subject to a uniform normal magnetic field is derived, and compared to that for a thin viscous layer heated from below. Their similarity predicts the possibility of mutual nonlinear stability control.

The surface topography of a magnetic fluid: a quantitative comparison between experiment and numerical simulation

The normal field instability in magnetic liquids is investigated experimentally by means of a radioscopic technique which allows a precise measurement of the surface topography. The dependence of the topography on the magnetic field is compared to results obtained by numerical simulations via the finite-element method. Quantitative agreement has been found for the critical field of the instability, the scaling of the pattern amplitude and the detailed shape of the magnetic spikes. The fundamental Fourier mode approximates the shape to within 10% accuracy for a range of up to 40% of the bifurcation parameter of this subcritical bifurcation. The measured control parameter dependence of the wavenumber differs qualitatively from analytical predictions obtained by minimization of the free energy.

Numerical treatment of free surface problems in ferrohydrodynamics

The numerical treatment of free surface problems in ferrohydrodynamics is considered.Starting from the general model, special attention is paid to field–surface and flow–surfaceinteractions. Since in some situations these feedback interactions can be partly or even fullyneglected, simpler models can be derived. The application of such models to the numericalsimulation of dissipative systems, rotary shaft seals, equilibrium shapes of ferrofluid drops,and pattern formation in the normal-field instability of ferrofluid layers is given. Ournumerical strategy is able to recover solitary surface patterns which were discoveredrecently in experiments.

A simple optical device for measuring free surface deformations of nontransparent liquids

In this paper, a novel optical device for measuring the deformation of liquid free surfaces is presented. The device employs a laser beam, which can be focused on any chosen location on the free surface. The key measurement is of the intensity of the beam reflected from a location on the free surface where the deformation exhibits a local extremum. The optics of the device is so designed as to measure a maximum intensity when the distance between the focusing lens and the selected point on the free surface is equal to the focal length, thus enabling a height measurement. The device is tested in ferrofluid pools where the height of the spikes of the normal field instability is measured. The simplicity of the suggested technique enables the fabrication of a quite cheap device for measuring surface deformation of nontransparent liquids, which provides good accuracy and reproducibility.

Numerical simulation of normal-field instability in the static and dynamic case

The normal-field instability is studied numerically by a finite element approach based on the 3D equations for the magnetostatic potential and the 3D Navier–Stokes equations for the velocity and pressure fields of the magnetic fluid. The finite element method is able to determine the critical value of the magnetic field intensity for the onset of the instability and to describe the transition from the undisturbed surface to the fully developed pattern in the dynamic case.