Bifurcations of magnetized surface waves between two immiscible fluids inside a vertical cylindrical container

The nonlinear stability of magnetized standing waves on the plane interface separating two immiscible inviscid magnetic fluids in a cylindrical container is investigated by means of the method of the multiple scales. The fluid system is stressed by both a uniform magnetic field and a periodic acceleration normal to the free surface, with the effect of the surface tension between the two fluids taken into account. The modulation of the amplitudes and phases of the tow modes is represented by a system of nonlinear first-order differential equations. The different cases of resonance are studied. Numerical applications are achieved to indicate the stability of the different types of the fixed points. The trivial motion, a periodically modulated sinusoid (limit cycle), and a chaotically modulated sinusoid are discussed.