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In this paper we consider the effect of sinusoidal gravity modulation of size ε on a differentially heated infinite slot in which a vertical temperature stratification is imposed on the walls. The slot problem is characterized by a Rayleigh number, Prandtl number, and the imposed uniform stratification on the walls. When ε is small, we show by regular perturbation expansion in ε that the modulation interacts with the natural mode of the system to produce resonances, confirming the results of Farooq & Homsy (1994). For ε ∼ O(1) we show that the modulation can potentially destabilize the longwave eigenmodes of the slot problem. This is achieved by projecting the governing equations onto the least-damped eigenmode, and investigating the resulting Mathieu equation via Floquet theory. No instability was found at large values of the Prandtl number and also low stratification, when there are no travelling modes present.To understand the nonlinear saturation mechanisms of this growth, we consider a two-mode model of the slot problem with the primary mode being the least-damped travelling parallel-flow mode as before and a secondary mode of finite wavenumber. By projecting the governing equations onto these two modes we obtained the equations for temporal evolution of the two modes. For modulation amplitudes above critical, the growth of the primary mode is saturated resulting in a stable weak nonlinear synchronous oscillation of the primary mode. An unexpected and intriguing feature of the coupling is that the secondary mode exhibits very high-frequency bursts which appear once every cycle of the forcing frequency.

We investigate streaming in a square cavity where a lateral temperature gradient interacts with a constant gravity field modulated by small harmonic oscillations of order ε. The Boussinesq equations are expanded by regular perturbation in powers of ε, and the O(ε2) equations contain Reynolds-stress-type terms that cause streaming. The resulting hierarchy of equations is solved by finite differences to investigate the O(ε1) and O(ε2) fields and their parametric dependence on the Rayleigh number Ra, Prandtl number Pr, and forcing frequency ω. It has been found that the streaming flow is quite small at small values of Ra, but becomes appreciable at high Ra and starts to influence such flow properties as the strength of the circulation and the overall heat transfer. Under suitable parametric conditions of finite frequency and moderate Pr the periodic forcing motion interacts with the instabilities associated with the O(εo) base flow leading to resonances that become stronger as Ra increases. It is argued that these resonances will have their greatest effect on streaming for Pr ≈ 1. At low frequencies the streaming flow shows marked structural changes as Ra is increased leading to an interesting change in the sign of the O(ε2) contribution to the Nusselt number. Also, as the frequency is changed the O(ε2) Nusselt number again changes sign at approximately the resonant frequency.

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