scholarly journals Localised Radial Patterns on the Free Surface of a Ferrofluid

2021 ◽  
Vol 31 (5) ◽  
Author(s):  
Dan J. Hill ◽  
David J. B. Lloyd ◽  
Matthew R. Turner
Keyword(s):  
Magnetic Field ◽  
Small Amplitude ◽  
Scaling Laws ◽  
Blow Up ◽  
Finite Depth ◽  
Critical Threshold ◽  
Far Field ◽  
Radial Solutions ◽  
The Core ◽  
Manifold Theory

AbstractThis 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.

Investigation of the Structure and Magnetic Properties of Bulk Amorphous FeCoYB Alloy

2017 ◽  
Vol 68 (9) ◽  
pp. 2162-2165 ◽  
Author(s):  
Katarzyna Bloch ◽  
Mihail Aurel Titu ◽  
Andrei Victor Sandu
Keyword(s):  
Magnetic Field ◽  
Magnetic Properties ◽  
Hyperfine Field ◽  
Irreversible Processes ◽  
Magnetization Process ◽  
Casting Method ◽  
The Core ◽  
Injection Casting ◽  
Core Losses ◽  
Microstructural Studies

The paper presents the results of structural and microstructural studies for the bulk Fe65Co10Y5B20 and Fe63Co10Y7B20 alloys. All the rods obtained by the injection casting method were fully amorphous. It was found on the basis of analysis of distribution of hyperfine field induction that the samples of Fe65Co10Y5B20 alloy are characterised with greater atomic packing density. Addition of Y to the bulk amorphous Fe65Co10Y5B20 alloy leads to the decrease of the average induction of hyperfine field value. In a strong magnetic field (i.e. greater than 0.4HC), during the magnetization process of the alloys, where irreversible processes take place, the core losses associated with magnetization and de-magnetization were investigated.

scholarly journals Planetary Magnetic Fields and Habitability in Super Earths

2018 ◽  
Vol 27 (1) ◽  
pp. 183-231 ◽  
Author(s):  
Pablo Cuartas-Restrepo
Keyword(s):  
Magnetic Field ◽  
Magnetic Fields ◽  
Scaling Laws ◽  
Thermal Evolution ◽  
Solid Core ◽  
Direct Influence ◽  
Physical Processes ◽  
Planetary Magnetic Fields ◽  
Planetary Magnetic Field ◽  
Planetary Dynamos

Abstract This work seeks to summarize some special aspects of a type of exoplanets known as super-Earths (SE), and the direct influence of these aspects in their habitability. Physical processes like the internal thermal evolution and the generation of a protective Planetary Magnetic Field (PMF) are directly related with habitability. Other aspects such as rotation and the formation of a solid core are fundamental when analyzing the possibilities that a SE would have to be habitable. This work analyzes the fundamental theoretical aspects on which the models of thermal evolution and the scaling laws of the planetary dynamos are based. These theoretical aspects allow to develop models of the magnetic evolution of the planets and the role played by the PMF in the protection of the atmosphere and the habitability of the planet.

scholarly journals Flux expulsion with dynamics

2016 ◽  
Vol 791 ◽  
pp. 568-588 ◽  
Author(s):  
Andrew D. Gilbert ◽  
Joanne Mason ◽  
Steven M. Tobias
Keyword(s):  
Magnetic Field ◽  
Magnetic Fields ◽  
Lorentz Force ◽  
Differential Rotation ◽  
Time Scale ◽  
Scaling Laws ◽  
Dynamical Regime ◽  
Kinematic Evolution ◽  
Flux Expulsion ◽  
The Magnetic Field

In the process of flux expulsion, a magnetic field is expelled from a region of closed streamlines on a $TR_{m}^{1/3}$ time scale, for magnetic Reynolds number $R_{m}\gg 1$ ($T$ being the turnover time of the flow). This classic result applies in the kinematic regime where the flow field is specified independently of the magnetic field. A weak magnetic ‘core’ is left at the centre of a closed region of streamlines, and this decays exponentially on the $TR_{m}^{1/2}$ time scale. The present paper extends these results to the dynamical regime, where there is competition between the process of flux expulsion and the Lorentz force, which suppresses the differential rotation. This competition is studied using a quasi-linear model in which the flow is constrained to be axisymmetric. The magnetic Prandtl number $R_{m}/R_{e}$ is taken to be small, with $R_{m}$ large, and a range of initial field strengths $b_{0}$ is considered. Two scaling laws are proposed and confirmed numerically. For initial magnetic fields below the threshold $b_{core}=O(UR_{m}^{-1/3})$, flux expulsion operates despite the Lorentz force, cutting through field lines to result in the formation of a central core of magnetic field. Here $U$ is a velocity scale of the flow and magnetic fields are measured in Alfvén units. For larger initial fields the Lorentz force is dominant and the flow creates Alfvén waves that propagate away. The second threshold is $b_{dynam}=O(UR_{m}^{-3/4})$, below which the field follows the kinematic evolution and decays rapidly. Between these two thresholds the magnetic field is strong enough to suppress differential rotation, leaving a magnetically controlled core spinning in solid body motion, which then decays slowly on a time scale of order $TR_{m}$.

scholarly journals Blow-up criteria and instability of normalized standing waves for the fractional Schrödinger-Choquard equation

2020 ◽  
Vol 10 (1) ◽  
pp. 311-330 ◽  
Author(s):  
Feng Binhua ◽  
Ruipeng Chen ◽  
Jiayin Liu
Keyword(s):  
Blow Up ◽  
Standing Waves ◽  
Radial Solutions ◽  
Choquard Equation ◽  
Decomposition Theory ◽  
Profile Decomposition ◽  
Strong Instability

Abstract In this paper, we study blow-up criteria and instability of normalized standing waves for the fractional Schrödinger-Choquard equation $$\begin{array}{} \displaystyle i\partial_t\psi- (-{\it\Delta})^s \psi+(I_\alpha \ast |\psi|^{p})|\psi|^{p-2}\psi=0. \end{array}$$ By using localized virial estimates, we firstly establish general blow-up criteria for non-radial solutions in both L2-critical and L2-supercritical cases. Then, we show existence of normalized standing waves by using the profile decomposition theory in Hs. Combining these results, we study the strong instability of normalized standing waves. Our obtained results greatly improve earlier results.

On the classical solutions for a Rosenau–Korteweg-deVries–Kawahara type equation

2021 ◽  
pp. 1-23
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo
Keyword(s):  
Cauchy Problem ◽  
Small Amplitude ◽  
Type Equation ◽  
Discrete Systems ◽  
Finite Depth ◽  
Capillary Waves ◽  
Classical Solutions ◽  
Well Posedness ◽  
Kawahara Equation ◽  
The Cauchy Problem

The Rosenau–Korteweg-deVries–Kawahara equation describes the dynamics of dense discrete systems or small-amplitude gravity capillary waves on water of a finite depth. In this paper, we prove the well-posedness of the classical solutions for the Cauchy problem.

Morse index and symmetry breaking for an elliptic equation with negative exponent in expanding annuli

2017 ◽  
Vol 147 (6) ◽  
pp. 1215-1232
Author(s):  
Zongming Guo ◽  
Linfeng Mei ◽  
Zhitao Zhang
Keyword(s):  
Elliptic Equation ◽  
Symmetry Breaking ◽  
Morse Index ◽  
Blow Up ◽  
Semilinear Elliptic Equation ◽  
Radial Solutions ◽  
Entire Solutions ◽  
Semilinear Elliptic ◽  
Image Position

Bifurcation of non-radial solutions from radial solutions of a semilinear elliptic equation with negative exponent in expanding annuli of ℝ2 is studied. To obtain the main results, we use a blow-up argument via the Morse index of the regular entire solutions of the equationThe main results of this paper can be seen as applications of the results obtained recently for finite Morse index solutions of the equationwith N ⩾ 2 and p > 0.

scholarly journals Development of an improved geomagnetic reference field of Antarctica

1999 ◽  
Vol 42 (2) ◽  
Author(s):  
A. De Santis ◽  
M. Chiappini ◽  
J. M. Torta ◽  
R. R. B. von Frese
Keyword(s):  
Magnetic Field ◽  
Secular Variation ◽  
Reference Model ◽  
Reference Field ◽  
Spherical Cap ◽  
Core Field ◽  
The Core ◽  
Spherical Cap Harmonic Analysis ◽  
The Antarctic ◽  
Magnetic Observatories

The properties of the Earth's core magnetic field and its secular variation are poorly known for the Antarctic. The increasing availability of magnetic observations from airborne and satellite surveys, as well as the existence of several magnetic observatories and repeat stations in this region, offer the promise of greatly improving our understanding of the Antarctic core field. We investigate the possible development of a Laplacian reference model of the core field from these observations using spherical cap harmonic analysis. Possible uses and advantages of this approach relative to the implementations of the standard global reference field are also considered.

scholarly journals ARBTools: A Tricubic Spline Interpolator for Three-Dimensional Scalar or Vector Fields

2019 ◽  
Author(s):  
Paul Walker ◽  
Ulrich Krohn ◽  
Carty David
Keyword(s):  
Magnetic Field ◽  
Vector Field ◽  
Vector Fields ◽  
Three Dimensional ◽  
Spline Method ◽  
Particle Trajectories ◽  
The Core ◽  
Numerical Integrators ◽  
Data Points ◽  
System Requirements

ARBTools is a Python library containing a Lekien-Marsden type tricubic spline method for interpolating three-dimensional scalar or vector fields presented as a set of discrete data points on a regular cuboid grid. ARBTools was developed for simulations of magnetic molecular traps, in which the magnitude, gradient and vector components of a magnetic field are required. Numerical integrators for solving particle trajectories are included, but the core interpolator can be used for any scalar or vector field. The only additional system requirements are NumPy.

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