Bifurcating Instability of the Free Surface of a Ferrofluid

Consider a slab of ferrofluid bounded below by a fixed boundary and above by a vacuum. If the fluid is subjected to a vertically directed magnetic field of sufficient strength, surface waves appear. The equations which describe this phenomenon are derived. In the physical space no natural Banach spa...

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Published in:SIAM journal on mathematical analysis Vol. 14; no. 4; pp. 736 - 766
Main Authors: Twombly, Evan Eugene, Thomas, J. W.
Format: Journal Article
Language:English
Published: Philadelphia Society for Industrial and Applied Mathematics 01.07.1983
Subjects:
Applied mathematics
Banach spaces
Boundary conditions
Eigenvalues
Magnetic fields
Permeability
ISSN:0036-1410, 1095-7154
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Abstract Consider a slab of ferrofluid bounded below by a fixed boundary and above by a vacuum. If the fluid is subjected to a vertically directed magnetic field of sufficient strength, surface waves appear. The equations which describe this phenomenon are derived. In the physical space no natural Banach space structure is available due to the free surface. In order to use the available bifurcation theory, a transformation of coordinates is made, mapping the surface flat. In the new coordinate system the equations define an operator between Banach spaces. The minimum eigenvalue of the linearized operator is the critical magnetic field strength where the planar surface loses stability. Using a generalized inverse of the Frechet derivative of the operator and the implicit function theorem, the existence of a nontrivial branch of solutions is proved. A local stability criterion is also obtained and applied to three periodic structures: rolls, squares and hexagons.
AbstractList Consider a slab of ferrofluid bounded below by a fixed boundary and above by a vacuum. If the fluid is subjected to a vertically directed magnetic field of sufficient strength, surface waves appear. The equations which describe this phenomenon are derived. In the physical space no natural Banach space structure is available due to the free surface. In order to use the available bifurcation theory, a transformation of coordinates is made, mapping the surface flat. In the new coordinate system the equations define an operator between Banach spaces. The minimum eigenvalue of the linearized operator is the critical magnetic field strength where the planar surface loses stability. Using a generalized inverse of the Frechet derivative of the operator and the implicit function theorem, the existence of a nontrivial branch of solutions is proved. A local stability criterion is also obtained and applied to three periodic structures: rolls, squares and hexagons.
Author Twombly, Evan Eugene
Thomas, J. W.
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Cites_doi 10.1017/S0022112077000731
10.1007/978-3-642-96379-7
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10.1109/TMAG.1975.1058798
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Copyright [Copyright] © 1983 Society for Industrial and Applied Mathematics
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Snippet Consider a slab of ferrofluid bounded below by a fixed boundary and above by a vacuum. If the fluid is subjected to a vertically directed magnetic field of...
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SubjectTerms Applied mathematics
Banach spaces
Boundary conditions
Eigenvalues
Magnetic fields
Permeability
Title Bifurcating Instability of the Free Surface of a Ferrofluid
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