NONLINEAR DISPERSIVE INSTABILITIES IN MAGNETIC FLUIDS* · 2018-11-16 · NONLINEAR DISPERSIVE INSTABILITIES IN MAGNETIC FLUIDS* By S. K. MALIK and M. SINGH Simon Fraser University
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QUARTERLY OF APPLIED MATHEMATICS 359OCTOBER 1984
NONLINEAR DISPERSIVE INSTABILITIES IN MAGNETIC FLUIDS*
By
S. K. MALIK and M. SINGH
Simon Fraser University
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.
1. Introduction. In recent years, the nonlinear instability of two superposed ferrofluids
has received considerable attention of various authors [ 1]—[4], The magnetic fluids are
synthesized in the laboratory and consist of ordinary nonconducting liquids in which very
fine small particles (mean diameter varying rom 30 to 150° A) of ferromagnetic material
are suspended freely. The very small size of these suspended particles, which are
distributed uniformly and homogeneously in the fluid, prevents coagulation. The introduc-
tion of an applied electric or a magnetic field does not cause the separation of the
magnetic particles from the liquid. Upon switching off the field, the fluid fully recovers its
original characteristics. The magnetic fluid is assumed to be non-conducting, and the only
forces involved are due to polarization. Cowley and Rosensweig [1] have demonstrated
that an instability sets in when the applied magnetic field H, which is normal to the fluid
surface, is slightly greater than the critical magnetic field Hc. It was reported in their
experiments that such an instability results in the appearance of the regular hexagonal
cells on the fluid surface. This experimental observation has been theoretically studied by
Gailitis [5], Kuznetsov and Spektor [6], and Brancher [7] with the use of the energy
method. Gailitis [5] has demonstrated the existence of hard excitation of the steady waves,
and shown that for certain values of the magnetic field strength, the hexagonal cell is
replaced by a square cell with possible hystersis behaviour for the subcritical values of the
field. Based on bifurcation analysis, Twombly and Thomas [8] have also obtained similar
results.
The aim of this presentation is to examine how a continuous band width of the modes
affects the description of the post critical instability. The method we employ was
developed by Stuart [9], and later modified by Newell [10] to consider the development of
the waves at the critical or the bifurcation point. In Sec. 2, we formulate the problem, give
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NONLINEAR DISPERSIVE INSTABILITIES 371
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