Nonlinear dispersive instabilities in magnetic fluids
Authors:
S. K. Malik and M. Singh
Journal:
Quart. Appl. Math. 42 (1984), 359-371
MSC:
Primary 76E30; Secondary 35Q20
DOI:
https://doi.org/10.1090/qam/757174
MathSciNet review:
757174
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Abstract: An asymptotic nonlinear theory of the two superposed magnetic fluids is presented taking into account the spatial as well as temporal effects. A generalized formulation of the evolution equation governing the amplitude is developed which leads to the nonlinear Klein-Gordon equation. The various stability criteria are derived from this equation. Obtained also are the bell shaped soliton and the kink solutions.
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M. D. Cowley and R. E. Rosensweig, The interfacial instability of a ferromagnetic fluid, J. Fluid Mech. 30, 671 (1967)
R. E. Rosensweig, Pattern formation in magnetic fluids, Fourth Internat. Conf. on Synergetics, Ed. by H. Haken, Springer-Verlag, Berlin (1983)
R. E. Zelazo and J. R. Melcher, Dynamics and stability of ferro-fluids, J. Fluid Mech. 39, 1 (1969)
M. I. Shlimois and Yu. L. Raikher, Experimental investigations of magnetic fluids, I.E.E.E. Trans. Mag. 16, 237 (1980)
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)
E. A. Kuznetsov and M. D. Spektor, Existence of a hexagonal relief on the surface of a dielectric fluid in an external electric field, Sov. Phys. JETP 44, 136 (1976)
J. P. Brancher, Waves and instabilities on a plane interface between ferrofluids and nonmagnetic fluids, Thermo-mechanics ${s^s}$ of Magnetic Fluids, Ed. by B. Berkovsky, Hemisphere Publishing Corp. (1977)
E. Twombly and J. W. Thomas, Theory of non-linear waves on the surface of a magnetic fluid, I.E.E.E. Trans. on Magnetics 16, 214 (1980)
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A. C. Newell, Envelope equation, Lect. Appl. Math. 15, 157 (1974)
W. Eckhaus, Studies in nonlinear stability theory, Springer (1965)
C. G. Lang 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, Ed. by P. H. Rabinowitz, New York, Academic Press (1977)
M. A. Weissman, Nonlinear wave packets in the kelvin-helmholtz instability, Phil. Trans. Roy. Soc. London 290, 58 (1979)
S. K. Malik and M. Singh, Nonlinear instability in superposed magnetic fluids, Third Internat. Conf. on Magnetic Fluids (1983)
G. B. Whitham, Linear and nonlinear waves, Wiley, New York (1974)
V. G. Makhankov, The dynamics of classical solitons, Phys. Rep. 35C, 1 (1978)
V. M. Zaitsev and M. I. Shliomis, The nature of the instability of the interface between two liquids in a constant field, Dokl. Akad. Nauk SSR 188, 1261 (1969)
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Article copyright:
© Copyright 1984
American Mathematical Society
