The combined effect of gravity modulation and temperature modulation in Rayleigh–Bénard convection with stress-free boundaries in the presence of a magnetic field in the vertical direction is investigated numerically. Gravity modulation is incorporated by placing a horizontal layer of fluid on a sinusoidally vibrating plate, while temperature modulation is added by assuming that the temperature gradient has a periodic component along with the steady part. The Floquet method is adopted to analyse the problem. When the temperature of the fluid layer surpasses a critical value, oscillatory convection starts. The excited wave at the onset of convection may oscillate synchronously or with half of the forcing frequency. The value of Rayleigh number ((textrm{Ra}_o)) on the threshold of convection is investigated varying various modulation parameters and fluid parameters. The (textrm{Ra}_o) increases with the increase in strength of magnetic field, whereas this value decreases with an increase in any one or both the modulation amplitudes. The union of two modulations leads to two types of bi-critical points, viz. SH and SS. The SH bi-critical point corresponds to the coexistence of a harmonic wave and a subharmonic wave at convection onset, while SS refers to the coexistence of two subharmonic waves with distinct wavenumbers. The second type of bi-critical point is new in a modulated system. Finally, the results of temperature modulation and gravity modulation are discussed separately as a particular case of the original system.
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