Mf. Elsayed et Dk. Callebaut, NONLINEAR EHD STABILITY OF THE INTERFACIAL WAVES OF 2 SUPERPOSED DIELECTRIC FLUIDS, Journal of colloid and interface science, 200(2), 1998, pp. 203-219
The slow modulation of the interfacial capillary-gravity waves of two
superposed dielectric fluids with uniform depths and solid horizontal
boundaries, under the influence of a normal electric field and in the
absence of surface charges at their interface, is investigated by usin
g the multiple-time scales method. It is found that the complex amplit
ude of quasi-monochromatic traveling waves can be described by a nonli
near Schrodinger equation in a frame of reference moving with the grou
p velocity. The stability characteristics of a uniform wave train are
examined analytically and numerically on the basis of the nonlinear Sc
hrodinger equation, and some limiting cases are recovered. Three cases
appear, depending on whether the depth of the lower fluid is equal to
, greater than, or less than the depth of the upper fluid. The effect
of the normal electric field is determined for the three stability reg
ions of the pure hydrodynamic case. It is found that the normal electr
ic field has a destabilizing influence in the first stability region a
nd a stabilizing effect in the second and third stability regions. Mor
eover, one new unstable region or two new stable and unstable regions
appear, all of which increase when the electric field increases. On th
e other hand, the complex amplitude of quasi-monochromatic standing wa
ves near the cutoff wavenumber is governed by a similar type of nonlin
ear Schrodinger equation in which the roles of time and space are inte
rchanged. This equation makes it possible to estimate the nonlinear ef
fect on the cutoff wavenumber. (C) 1998 Academic Press.