Nonlinear dynamics and chaos of magnetized resonant surface waves in a rectangular container

In this paper the nonlinear evolution of the three-dimensional instability of resonated standing surface waves along the interface of a weakly viscous, incompressible magnetic liquid within a rectangular basin is investigated. A combination of the Rosensweig instability with Faraday instability is created, where the system is assumed to be stressed by a normal alternating magnetic field together with an external vertical oscillating force. First, it is assumed that the liquid is inviscid and thereby the motion is irrotational, a system of nonlinear coupled evolution equations governing the complex amplitudes of the different modes is derived. Second, a system of linear equations, derived via solving the linearized Navier–Stokes equations, is obtained. Consequently, the nonlinear equations of the complex amplitudes that correspond to the ideal fluid case are modified by adding the linear damping. This system is exploited to determine the steady-state solutions and then studying their stability both analytically and numerically. The results show that the liquid viscosity rather than the magnetic field affects the qualitative behavior of the wave motion and the system response alternates between the regular periodic and chaotic behavior depending on the specific values of some parameters.

► Weakly viscous magnetic liquid in a rectangular basin.
► Effects of normal alternating magnetic field.
► The container is stressed by a vertical oscillating force.
► Linear damping is obtained by means of boundary layer theory.
► Steady-state solutions are obtained and discussed numerically.