The nonlinear analysis of the Rayleigh-Taylor
instability of two immiscible, viscous magnetic fluids in porous
media, is performed for two layers, each has a finite depth. The
system is subjected to both vertical vibrations and normal magnetic
fields. The influence of both surface tension and gravity
force is taken into account. Although the motions are assumed
to be irrotational in each fluid for small perturbations, weak
viscous effects are included in the boundary condition of the
normal stress balance. The method of multiple scale expansion
is used for the investigation. The evolution of the amplitude
is governed by a nonlinear Ginzburg-Landau equation which
gives the criterion for modulational instability. When the viscosity
and Darcy’s coefficients are neglected, the cubic nonlinear
Schr¨odinger equation is obtained. Further, it is shown
that, near the marginal state, a nonlinear diffusion equation is
obtained in the presence of both viscosity and Darcy’s coeffi-
cients. Stability analysis and numerical simulations are used
to describe linear and nonlinear stages of the interface evolution
and then the stability diagrams are obtained. Regions of
stability and instability are identified. |
