Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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



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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

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, Fourth Internat. Conf. on Synergetics, Ed. by H. Haken, Springer-Verlag, Berlin (1983)
  • [3] R. E. Zelazo and J. R. Melcher, Dynamics and stability of ferro-fluids, J. Fluid Mech. 39, 1 (1969)
  • [4] M. I. Shlimois and Yu. L. Raikher, Experimental investigations of magnetic fluids, I.E.E.E. Trans. Mag. 16, 237 (1980)
  • [5] 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)
  • [6] 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)
  • [7] 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)
  • [8] 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)
  • [9] J. T. Stuart, Bifurcation theory in non-linear hydrodynamic stability, Applications of bifurcation theory (Proc. Advanced Sem., Univ. Wisconsin, Madison, Wis., 1976) Academic Press, New York, 1977, pp. 127–147. Publ. Math. Res. Center, No. 38. MR 0495631
  • [10] A. C. Newell, Envelope equation, Lect. Appl. Math. 15, 157 (1974)
  • [11] W. Eckhaus, Studies in nonlinear stability theory, Springer (1965)
  • [12] C. G. Lang and A. C. Newell, The post-buckling problem for thin elastic shells, SIAM J. Appl. Math. 21, 605 (1971)
  • [13] J. Pedlosky, The finite Amplitude Dynamics of Baroclinic Waves, Application of Bifurcation Theory, Ed. by P. H. Rabinowitz, New York, Academic Press (1977)
  • [14] M. A. Weissman, Nonlinear wave packets in the kelvin-helmholtz instability, Phil. Trans. Roy. Soc. London 290, 58 (1979)
  • [15] S. K. Malik and M. Singh, Nonlinear instability in superposed magnetic fluids, Third Internat. Conf. on Magnetic Fluids (1983)
  • [16] G. B. Whitham, Linear and nonlinear waves, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Pure and Applied Mathematics. MR 0483954
  • [17] V. G. Makhan′kov, Dynamics of classical solitons (in nonintegrable systems), Phys. Rep. 35 (1978), no. 1, 1–128. MR 481361, https://doi.org/10.1016/0370-1573(78)90074-1
  • [18] 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)

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 76E30, 35Q20

Retrieve articles in all journals with MSC: 76E30, 35Q20

Additional Information

DOI: https://doi.org/10.1090/qam/757174
Article copyright: © Copyright 1984 American Mathematical Society

American Mathematical Society