Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Nonlinear focussing in magnetic fluids

Authors: S. K. Malik and M. Singh
Journal: Quart. Appl. Math. 44 (1987), 629-637
MSC: Primary 76W05
DOI: https://doi.org/10.1090/qam/872815
MathSciNet review: 872815
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The phenomenon of nonlinear focussing or collapse is presented for two superposed magnetic fluids subjected to a normal magnetic field. We show that the focussing is direction-dependent and is more pronounced at shorter wavelengths as well as at higher values of density ratio. Nonlinear focussing occurs if the dimensions of the system are higher than one and the magnetic field is in the subcritical regime. Because of this nonlinear effect, the regular pattern formation may develop local spots of highly irregular behaviour.

References [Enhancements On Off] (What's this?)

  • [1] M. D. Cowley and R. E. Rosensweig, The interfacial instability of a ferromagnetic fluid, J. Fluid Mech. 30, 671 (1967)
  • [2] R. E. Rosensweig, Pattern formation in magnetic fluids, in Evolution of Order and Chaos, Springer-Verlag, Berlin, 1982
  • [3] S. K. Malik and M. Singh, Nonlinear dispersive instabilities in magnetic fluids, Quart. Appl. Math. 42, 359-371 (1984) MR 757174
  • [4] R. E. Zelazo and J. R. Melcher, Dynamics and stability of ferro-fluids, J. Fluid Mech. 39, 1 (1969)
  • [5] S. K. Malik and M. Singh, Modulational instability in magnetic fluids, Quart. Appl. Math. 43, 57-64 (1985) MR 782256
  • [6] V. E. Zakharov and V. S. Synakh, The nature of the self-focussing singularity, Sov. Phys. JETP 41, 465 (1976)
  • [7] V. E. Zakharov, Collapse of Langmuir waves, Sov. Phys. JETP 35, 908 (1972)
  • [8] J. D. Gibbon and M. J. McGuinness, Nonlinear focussing and the Kelvin-Helmholtz instability, Phys. Lett. 77A, 118 (1980) MR 586848
  • [9] A. C. Newell, Bifurcation and nonlinear focusing in Pattern Formation by Dynamic Systems and Pattern Recognition, Springer-Verlag, Berlin, pp. 244-265, 1979
  • [10] V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focussing and one-dimensional self-modulation of waves in nonlinear media, Sov. Phys. JETP 34, 62-69 (1972) MR 0406174
  • [11] S. K. Malik and M. Singh, Second harmonic resonance in a self-gravitating fluid column, J. Math. Anal. Appl. 83, 1 (1981) MR 632321
  • [12] M. A. Weissman, Nonlinear wave packets in the Kelvin-Helmholtz instability, Philos. Trans. Roy. Soc. London 290, 639 (1979)
  • [13] F. H. Berkshire and J. D. Gibbon, Collapse in the n-dimensional nonlinear Schrödinger equation, Stud. Appl. Math. 69, 229 (1983) MR 721132

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 76W05

Retrieve articles in all journals with MSC: 76W05

Additional Information

DOI: https://doi.org/10.1090/qam/872815
Article copyright: © Copyright 1987 American Mathematical Society

American Mathematical Society