Pattern selection in ferrofluids

Mary Silber; Edgar Knobloch

doi:10.1016/0167-2789(88)90099-1

Pattern selection in ferrofluids

Mary Silber^*, Edgar Knobloch

^*Corresponding author for this work

Engineering Sciences and Applied Mathematics

Research output: Contribution to journal › Article › peer-review

34 Scopus citations

Abstract

Pattern formation by surface instabilities in a ferrofluid is studied as a function of an applied vertical magnetic field H. The effects of sidewalls are neglected. The problem is formulated as a bifurcation problem on an appropriately chosen doubly-periodic lattice. Normal forms for both square and hexagonal lattices are given and the necessary coefficients computed from the partial differential equations. On the square lattice solutions in the form of parallel ridges are never stable near onset. For a relative magnetic permeability μ<1.4 stable squares are formed. On the hexagonal lattice a finite amplitude instability produces hexagons. Analysis of a degenerate bifurcation predicts a hysteretic transition to ridges with increasing H.

Original language	English (US)
Pages (from-to)	83-98
Number of pages	16
Journal	Physica D: Nonlinear Phenomena
Volume	30
Issue number	1-2
DOIs	https://doi.org/10.1016/0167-2789(88)90099-1
State	Published - 1988

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Mathematical Physics
Condensed Matter Physics
Applied Mathematics

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10.1016/0167-2789(88)90099-1

Cite this

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title = "Pattern selection in ferrofluids",

abstract = "Pattern formation by surface instabilities in a ferrofluid is studied as a function of an applied vertical magnetic field H. The effects of sidewalls are neglected. The problem is formulated as a bifurcation problem on an appropriately chosen doubly-periodic lattice. Normal forms for both square and hexagonal lattices are given and the necessary coefficients computed from the partial differential equations. On the square lattice solutions in the form of parallel ridges are never stable near onset. For a relative magnetic permeability μ<1.4 stable squares are formed. On the hexagonal lattice a finite amplitude instability produces hexagons. Analysis of a degenerate bifurcation predicts a hysteretic transition to ridges with increasing H.",

author = "Mary Silber and Edgar Knobloch",

note = "Funding Information: We would like to thank J.P. Crutchiield for bringing this problem to our attention, and J.P. Crutchfield, C.D. Jeffries and Q.H. Lain for discussing their experiments with us. We also acknowledge informative correspondence with R.E. Rosensweig and J.W. Thomas. This work was supported by the ACM program of DARPA.",

year = "1988",

doi = "10.1016/0167-2789(88)90099-1",

language = "English (US)",

volume = "30",

pages = "83--98",

journal = "Physica D: Nonlinear Phenomena",

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T1 - Pattern selection in ferrofluids

AU - Silber, Mary

AU - Knobloch, Edgar

N1 - Funding Information: We would like to thank J.P. Crutchiield for bringing this problem to our attention, and J.P. Crutchfield, C.D. Jeffries and Q.H. Lain for discussing their experiments with us. We also acknowledge informative correspondence with R.E. Rosensweig and J.W. Thomas. This work was supported by the ACM program of DARPA.

PY - 1988

Y1 - 1988

N2 - Pattern formation by surface instabilities in a ferrofluid is studied as a function of an applied vertical magnetic field H. The effects of sidewalls are neglected. The problem is formulated as a bifurcation problem on an appropriately chosen doubly-periodic lattice. Normal forms for both square and hexagonal lattices are given and the necessary coefficients computed from the partial differential equations. On the square lattice solutions in the form of parallel ridges are never stable near onset. For a relative magnetic permeability μ<1.4 stable squares are formed. On the hexagonal lattice a finite amplitude instability produces hexagons. Analysis of a degenerate bifurcation predicts a hysteretic transition to ridges with increasing H.

AB - Pattern formation by surface instabilities in a ferrofluid is studied as a function of an applied vertical magnetic field H. The effects of sidewalls are neglected. The problem is formulated as a bifurcation problem on an appropriately chosen doubly-periodic lattice. Normal forms for both square and hexagonal lattices are given and the necessary coefficients computed from the partial differential equations. On the square lattice solutions in the form of parallel ridges are never stable near onset. For a relative magnetic permeability μ<1.4 stable squares are formed. On the hexagonal lattice a finite amplitude instability produces hexagons. Analysis of a degenerate bifurcation predicts a hysteretic transition to ridges with increasing H.

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Pattern selection in ferrofluids

Abstract

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