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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| Vydané v: | SIAM journal on mathematical analysis Ročník 14; číslo 4; s. 736 - 766 |
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| Hlavní autori: | , |
| Médium: | Journal Article |
| Jazyk: | English |
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Philadelphia
Society for Industrial and Applied Mathematics
01.07.1983
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| 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. |
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| 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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| CitedBy_id | crossref_primary_10_1090_qam_1233527 crossref_primary_10_1007_s00332_021_09719_z crossref_primary_10_1016_j_physd_2017_03_004 crossref_primary_10_1007_s00033_018_0924_y crossref_primary_10_1016_j_jmmm_2010_11_045 crossref_primary_10_1090_qam_1193659 crossref_primary_10_1016_j_jmmm_2016_09_030 crossref_primary_10_1017_jfm_2015_565 crossref_primary_10_1137_23M1569125 crossref_primary_10_1007_s12043_019_1907_z crossref_primary_10_1016_j_jmmm_2020_167118 crossref_primary_10_1016_0167_2789_88_90099_1 crossref_primary_10_1134_S0012266111050144 crossref_primary_10_1016_0020_7225_94_00073_5 |
| Cites_doi | 10.1017/S0022112077000731 10.1007/978-3-642-96379-7 10.1017/S0022112069002011 10.1109/TMAG.1975.1058798 10.1016/0022-1236(78)90080-0 10.1007/BFb0060079 10.1007/978-3-642-45944-3_6 10.1109/TMAG.1980.1060591 10.1017/S0022112067001697 10.1007/BF00253044 10.1098/rspa.1976.0067 |
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| Copyright | [Copyright] © 1983 Society for Industrial and Applied Mathematics |
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| References | Sokolnikoff I. S. (R19) 1964 Sattinger David H. (R4) 1973 R8 Adams Robert A. (R22) 1975 Misner Charles W. (R20) 1973 Joseph Daniel D. (R12) 1972; 49 Jefimenko D. D. (R17) 1966 Barkov Yu. D. (R7) 1977; 3 Bashtovoi V. G. (R9) 1977; 3 R10 R21 R11 R14 Gailitis A. (R5) 1969; 5 R13 R15 Moskowitz R. (R3) 1975 R18 Zaitsev V. M. (R6) 1969; 188 Kuznetsov E. A. (R2) 1976; 71 R1 |
| References_xml | – volume: 5 start-page: 68 year: 1969 ident: R5 publication-title: Magnit. Gidrodinamika – volume: 71 start-page: 262 year: 1976 ident: R2 publication-title: Zh. Eksper. Teoret. Fiz. – volume-title: Gravitation year: 1973 ident: R20 – volume: 3 start-page: 137 year: 1977 ident: R7 publication-title: Magnit. Gidrodinamika – ident: R8 doi: 10.1017/S0022112077000731 – ident: R21 doi: 10.1007/978-3-642-96379-7 – volume: 188 start-page: 1261 year: 1969 ident: R6 publication-title: Dokl. Akad. Nauk SSSR – ident: R10 doi: 10.1017/S0022112069002011 – volume-title: Tensor Analysis, Theory and Applications to Geometry and Mechanics of Continua year: 1964 ident: R19 – volume: 3 start-page: 23 year: 1977 ident: R9 publication-title: Magnit. Gidrodinamika – ident: R15 doi: 10.1109/TMAG.1975.1058798 – ident: R14 doi: 10.1016/0022-1236(78)90080-0 – volume-title: Topics in stability and bifurcation theory year: 1973 ident: R4 doi: 10.1007/BFb0060079 – ident: R18 doi: 10.1007/978-3-642-45944-3_6 – volume-title: Electricity and Magnetism year: 1966 ident: R17 – volume-title: Sobolev spaces year: 1975 ident: R22 – ident: R11 doi: 10.1109/TMAG.1980.1060591 – ident: R1 doi: 10.1017/S0022112067001697 – year: 1975 ident: R3 publication-title: IEEE Spectrum – volume: 49 start-page: 321 year: 1972 ident: R12 publication-title: Arch. Rational Mech. Anal. doi: 10.1007/BF00253044 – ident: R13 doi: 10.1098/rspa.1976.0067 |
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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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| Title | Bifurcating Instability of the Free Surface of a Ferrofluid |
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