This paper concerns a linear study of the convective parametric instability in the case of a Newtonian fluid confined in a Hele-Shaw cell and submitted to a vertical periodic motion. The gradient of temperature, applied to the fluid layer, is either in the same direction that gravity or in the opposite one. An asymptotic analysis shows that the Hele-Shaw approximation leads to two linear formulations depending on the order of magnitude of the Prandtl number. For these two asymptotic cases, the convective threshold is determined. It turns out that in the Hele-Shaw geometrical configuration, parametric oscillations have no influence on the criterion of stability when the Prandtl number is in the order of the unity or very superior to the unity. However, when the Prandtl number is small than unity, the parametric oscillations can affect the convective instability threshold.

The effect of a time-sinusoidal magnetic field on the onset of convection in a horizontal magnetic fluid layer heated from above and bounded by isothermal non magnetic boundaries is investigated. The analysis is restricted to static and linear laws of magnetization. A first order Galerkin method is performed to reduce the governing linear system to the Mathieu equation with damping term. Therefore, the Floquet theory is used to determine the convective threshold for the free-free and rigid-rigid cases. With an appropriate choice of the ratio of the magnetic and gravitational forces, we show the possibility to produce a competition between the harmonic and subharmonic modes at the onset of convection.

Thermal instability in a horizontal Newtonian liquid layer with rigid boundaries is investigated in the presence of vertical quasiperiodic forcing having two incommensurate frequencies omega1 and omega2. By means of a Galerkin projection truncated to the first order, the governing linear system corresponding to the onset of convection is reduced to a damped quasiperiodic Mathieu equation. The threshold of convection corresponding to quasiperiodic solutions is determined in the cases of heating from below and heating from above. We show that a modulation with two incommensurate frequencies has a stabilizing or a destabilizing effect depending on the frequencies ratio omega=omega2/omega1. The effect of the Prandtl number in a stabilizing zone is also examined for different frequency ratios.

The aim of this paper is the derivation of the Ginzburg–Landau equation [as introduced by A. C. Newell and J. A. Whitehead, J. Fluid Mech. 38, 279 (1969)] from the hydrodynamic equations for an infinite Hele–Shaw cell. The dimensional analysis and the asymptotic study allow one to distinguish two nonlinear formulations, each one depends on the order of magnitude of the Prandtl number. The first formulation corresponds to the case Pr=O(1) or Pr≫1, whereas the second corresponds to the case Pr=O(ε*2), where ε*≪1 denotes the aspect ratio of the cell. Here a weakly nonlinear analysis is performed for the two formulations.