Thermo-magnetic convection and convective magneto-gravitational parametric resonance in a horizontal ferrofluid layer under time quasi-periodic vibrations: A balance harmonic analysis

Abstract

Convective instability in a horizontal ferromagnetic liquid layer with rigid boundaries is investigated in the presence of both an uniform magnetic field and a vertical quasi-periodic forcing having two incommensurate frequencies ω1 and ω2. A linearized convective instability analysis predicting the onset of convection is performed and by means of a Galerkin projection truncated to the first order, the linear dynamic of the system is reduced to a damped quasi-periodic Mathieu oscillator. Using a balance harmonic analysis, it turns out that at sufficiently low vibration frequencies, we recover the critical instability parameters related to the unmodulated case where the convection is mainly due to the magnetic mechanism. This feature means that the buoyancy mechanism does not contribute into the convective cells formation and consequently heating from below or from above does not alter the onset of convection. However, at sufficiently high vibration frequencies, both the magnetic and buoyancy mechanisms are operative and here we are dealing with a magneto-gravitational parametric resonance convection. The transition between these convective modes is occurred via bicritical states giving rise to cusp points in the magnetic Rayleigh number besides discontinuities in its corresponding critical wave number. In addition, it is shown that a proper tuning of the irrational frequency ratio ω=ω2/ω1 provides a control of the convection onset as well as the nature of its mechanism. Moreover, special attention is devoted to highlight the effect of the Prandtl number on the dynamic of the system at different values of the frequency ratio ω.