Nonlinear field instability and chaos in magnetic fluids

Malik, S. K.; Singh, M.

doi:10.1090/qam/1233527

Authors: S. K. Malik and M. Singh
Journal: Quart. Appl. Math. 51 (1993), 519-534
MSC: Primary 76E25; Secondary 34C37, 76W05
DOI: https://doi.org/10.1090/qam/1233527
MathSciNet review: MR1233527
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Abstract: We study the nonlinear dynamics of normal field instability in a ferrofluid under the influence of a uniform magnetic field. In addition, a small normal sinusoidal magnetic field is superimposed on the system. An equation governing the evolution of small but finite amplitude is obtained. Applying the Melnikov method, it is shown that there exist transverse homoclinic orbits leading to chaotic motions.

R. E. Rosensweig, Ferrohydrodynamics, Cambridge Univ. Press, Cambridge, 1985

M. D. Cowley and R. E. Rosensweig, The interfacial stability of a ferromagnetic fluid, J. Fluid Mech. 30, 671 (1967)

S. K. Malik and M. Singh, Nonlinear dispersive instabilities in magnetic fluids, Quart. Appl. Math. 42 (1984), no. 3, 359–371. MR 757174, DOI https://doi.org/10.1090/S0033-569X-1984-0757174-8

A. Gailitis, Formation of the hexagonal pattern on the surface of a ferromagnetic fluid in an applied magnetic field, J. Fluid Mech. 82, 401 (1977)

Evan Eugene Twombly and J. W. Thomas, Bifurcating instability of the free surface of a ferrofluid, SIAM J. Math. Anal. 14 (1983), no. 4, 736–766. MR 704488, DOI https://doi.org/10.1137/0514056

J. P. Brancher, Waves and instabilities on a plane interface between ferrofluids and nonmagnetic fluids, Thermomechanics of Magnetic Fluids, B. Berkovsky, ed., Hemisphere, Washington, D.C., 1978

A. G. Boudouvis, J. L. Puchalla, L. E. Scriven, and R. E. Rosensweig, Normal field instability and pattern in pool of ferrofluid, J. Magn. Magn. Mat. 65, 307 (1987)

Mary Silber and Edgar Knobloch, Pattern selection in ferrofluids, Phys. D 30 (1988), no. 1-2, 83–98. MR 939268, DOI https://doi.org/10.1016/0167-2789%2888%2990099-1

S. K. Malik and M. Singh, Nonlinear waves in magnetic fluids, Continuum Mechanics and its Applications, G. A. C. Graham and S. K. Malik, eds., Hemisphere, Washington, D.C., 1989

J. C. Bacri, U. d’Ortona, and D. Salin, Magnetic-fluid oscillator: Observation of nonlinear period doubling, Phys. Rev. Lett. 67, 50 (1991)

V. K. Melnikov, On the stability of the centre for time periodic perturbations, Trans. Moscow Math. Soc. 12, 1 (1963)

John Guckenheimer and Philip Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied Mathematical Sciences, vol. 42, Springer-Verlag, New York, 1983. MR 709768
Francis C. Moon, Chaotic vibrations, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1987. An introduction for applied scientists and engineers. MR 914494
Stephen Wiggins, Introduction to applied nonlinear dynamical systems and chaos, Texts in Applied Mathematics, vol. 2, Springer-Verlag, New York, 1990. MR 1056699
S. K. Malik and M. Singh, An explosive instability in magnetic fluids, Quart. Appl. Math. 50 (1992), no. 4, 613–626. MR 1193659, DOI https://doi.org/10.1090/qam/1193659

C. G. Lange and A. C. Newell, The post-buckling problem for thin elastic shells, SIAM J. Appl. Math. 21, 605 (1971)

J. Pedlosky, The finite amplitude dynamics of baroclinic waves, Application of Bifurcation Theory, P. H. Robinowitz, ed., Academic Press, New York, 1971

M. A. Weissman, Nonlinear wave packets in the Kelvin-Helmholtz instability, Philos. Trans. Roy. Soc. London Ser. A 290, 639 (1979)

Paul K. Newton, Chaos in Rayleigh-Bénard convection with external driving, Phys. Rev. A (3) 37 (1988), no. 3, 932–934. MR 927227, DOI https://doi.org/10.1103/PhysRevA.37.932
S. K. Malik and M. Singh, Chaos in Kelvin-Helmholtz instability in magnetic fluids, Phys. Fluids A 4 (1992), no. 12, 2915–2922. MR 1192761, DOI https://doi.org/10.1063/1.858518