Nonlinear capillary-gravity waves in magnetic fluids
Authors:
S. K. Malik and M. Singh
Journal:
Quart. Appl. Math. 47 (1989), 59-70
MSC:
Primary 76W05; Secondary 76B15, 76E25
DOI:
https://doi.org/10.1090/qam/987895
MathSciNet review:
987895
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Abstract: The nonlinear wave propagation of capillary-gravity waves on the surface of a ferrofluid of finite depth is investigated by employing the method of multiple scales. The stability analysis reveals the existence of different regions of instability. We show that the nonlinear modulational instability cannot be suppressed by the application of a strong magnetic field, however strong it may be. The influence of the magnetic field is not only quite signficant but its effect is also different for different regions of stability.
R. E. Zelazo and J. R. Melcher, Dynamics and stability of ferrofluids: surface interactions, J. Fluid Mech. 39, 1 (1969)
R. E. Rosensweig, Ferrohydrodynamics, Cambridge, 1985
R. E. Rosensweig, Magnetic fluids, Ann. Rev. Fluid Mech. 19, 437 (1987)
- Rama Kant and S. K. Malik, Nonlinear waves in superposed magnetic fluids, Phys. Fluids 28 (1985), no. 12, 3534–3537. MR 815463, DOI https://doi.org/10.1063/1.865307
S. K. Malik and M. Singh, Finite amplitude Kelvin-Helmholtz instability in magnetic fluids, Phys. Fluids 29, 2853 (1986)
- S. K. Malik and M. Singh, Modulational instability in magnetic fluids, Quart. Appl. Math. 43 (1985), no. 1, 57–64. MR 782256, DOI https://doi.org/10.1090/S0033-569X-1985-0782256-4
M. J. Lighthill, Contribution to the theory of waves in nonlinear dispersive systems, J. Inst. Math. Appl. 1, 269 (1965)
T. B. Benjamin and J. E. Feir, The disintegration of wave train in deep water, Part I. Theory, J. Fluid Mech. 27, 417 (1967)
H. Hasimoto and H. Ono, Nonlinear modulation of gravity waves, J. Phys. Soc. Japan 33, 805 (1972)
T. Kawahara, Nonlinear self-modulation of capillary-gravity waves on liquid layer, J. Phys. Soc. Japan 38, 265 (1975)
- G. B. Whitham, Linear and nonlinear waves, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Pure and Applied Mathematics. MR 0483954
- R. W. Lardner, Uniformly valid expansion for capillary-gravity waves on a fluid of infinite depth, J. Phys. Soc. Japan 51 (1982), no. 12, 4079–4083. MR 688269, DOI https://doi.org/10.1143/JPSJ.51.4079
R. Kant and S. K. Malik, Nonlinear internal resonance in magnetic fluids, J. Mag. Mat. 65, 347 (1987)
- D. J. Benney, Significant interactions between small and large scale surface waves, Studies in Appl. Math. 55 (1976), no. 2, 93–106. MR 452076, DOI https://doi.org/10.1002/sapm197655293
- D. J. Benney, A general theory for interactions between short and long waves, Studies in Appl. Math. 56 (1976/77), no. 1, 81–94. MR 463715, DOI https://doi.org/10.1002/sapm197756181
M. I. Shliomis, Magnetic fluids, Soviet Phys. Uspekhi 17, 153 (1975)
R. E. Zelazo and J. R. Melcher, Dynamics and stability of ferrofluids: surface interactions, J. Fluid Mech. 39, 1 (1969)
R. E. Rosensweig, Ferrohydrodynamics, Cambridge, 1985
R. E. Rosensweig, Magnetic fluids, Ann. Rev. Fluid Mech. 19, 437 (1987)
R. Kant and S. K. Malik, Nonlinear waves in superposed magnetic fluids, Phys. Fluids 28, 3534–3537 (1985)
S. K. Malik and M. Singh, Finite amplitude Kelvin-Helmholtz instability in magnetic fluids, Phys. Fluids 29, 2853 (1986)
S. K. Malik and M. Singh, Modulational instability in magnetic fluids, Quart. Appl. Math. 43, 57–64 (1985)
M. J. Lighthill, Contribution to the theory of waves in nonlinear dispersive systems, J. Inst. Math. Appl. 1, 269 (1965)
T. B. Benjamin and J. E. Feir, The disintegration of wave train in deep water, Part I. Theory, J. Fluid Mech. 27, 417 (1967)
H. Hasimoto and H. Ono, Nonlinear modulation of gravity waves, J. Phys. Soc. Japan 33, 805 (1972)
T. Kawahara, Nonlinear self-modulation of capillary-gravity waves on liquid layer, J. Phys. Soc. Japan 38, 265 (1975)
G. B. Whitham, Linear and nonlinear waves, Wiley, New York, 1974
R. W. Lardner, Uniformly valid expansion for capillary-gravity waves on a fluid of infinite depth, J. Phys. Soc. Japan 51, 4079–4083 (1982)
R. Kant and S. K. Malik, Nonlinear internal resonance in magnetic fluids, J. Mag. Mat. 65, 347 (1987)
D. J. Benney, Significant interaction between small and large scale surface waves, Studies in Appl. Math. 55, 93–106 (1976)
D. J. Benney, A general theory of interactions between short and long waves, Studies in Appl. Math. 56, 81–94 (1977)
M. I. Shliomis, Magnetic fluids, Soviet Phys. Uspekhi 17, 153 (1975)
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Article copyright:
© Copyright 1989
American Mathematical Society