Localised Radial Patterns on the Free Surface of a Ferrofluid

被引:0
|
作者
Dan J. Hill
David J. B. Lloyd
Matthew R. Turner
机构
[1] University of Surrey,Department of Mathematics
来源
关键词
Localised patterns; Ferrofluid; Quasilinear PDE; Asymptotic methods; Radial solutions; Primary classification: 35B36; Secondary classification: 76M45 and 35Q35;
D O I
暂无
中图分类号
学科分类号
摘要
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.
引用
收藏
相关论文
共 50 条
  • [1] Localised Radial Patterns on the Free Surface of a Ferrofluid
    Hill, Dan J.
    Lloyd, David J. B.
    Turner, Matthew R.
    JOURNAL OF NONLINEAR SCIENCE, 2021, 31 (05)
  • [2] Miscible ferrofluid patterns in a radial magnetic field
    Chen, Ching-Yao
    Yang, Y. -S.
    Miranda, Jose A.
    PHYSICAL REVIEW E, 2009, 80 (01):
  • [3] Thermooptical mirror on the free ferrofluid surface
    Zauls, V
    Liberts, G
    Shakars, J
    Cebers, A
    OPTICAL INORGANIC DIELECTRIC MATERIALS AND DEVICES, 1997, 2967 : 260 - 265
  • [4] Fully nonlinear simulations of ferrofluid patterns in a radial magnetic field
    Oliveira, Rafael M.
    Miranda, Jose A.
    PHYSICAL REVIEW FLUIDS, 2020, 5 (12):
  • [5] Stability Properties of Non-Radial Steady Ferrofluid Patterns
    Escher, Joachim
    Matioc, Bogdan-Vasile
    COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS, 2011, 36 (03) : 363 - 379
  • [6] BIFURCATING INSTABILITY OF THE FREE-SURFACE OF A FERROFLUID
    TWOMBLY, EE
    THOMAS, JW
    SIAM JOURNAL ON MATHEMATICAL ANALYSIS, 1983, 14 (04) : 736 - 766
  • [7] Modelling of a free-surface ferrofluid flow
    Habera, M.
    Hron, J.
    JOURNAL OF MAGNETISM AND MAGNETIC MATERIALS, 2017, 431 : 157 - 160
  • [8] THE INFLUENCE OF MAGNETIC FIELD ON FREE SURFACE FERROFLUID FLOW
    Habera, M.
    Fabian, M.
    Svikova, M.
    Timko, M.
    MAGNETOHYDRODYNAMICS, 2013, 49 (3-4): : 402 - 406
  • [9] Radial Stiffness of a Ferrofluid Seal
    Ravaud, R.
    Pinho, M.
    Lemarquand, G.
    Dauchez, N.
    Genevaux, J. M.
    Lemarquand, V.
    Brouard, B.
    IEEE TRANSACTIONS ON MAGNETICS, 2009, 45 (10) : 4388 - 4390
  • [10] On the critical free-surface over localised topography
    Keeler, J. S.
    Binder, B. J.
    Blyth, M. G.
    JOURNAL OF FLUID MECHANICS, 2017, 832 : 73 - 96