Oscillatory Hele-Shaw convection

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Abstract
Time-dependent convection motions in the form of rolls in a thin vertical fluid layer (Hele-Shaw cell) heated from below are investigated numerically. Perpendicular to the convection-roll axis the fluid is bounded by parallel adiabatic rigid sidewalls. Stress-free top, bottom and end boundaries are assumed. The horizontal extension of the convection rolls is described by the wavenumber α. Solutions for the time-dependent behaviour of the convective motion are presented for a range of wavenumbers between α = ½π and 2π. The onset of the oscillation is shifted to higher Rayleigh numbers with increasing wavenumber. The oscillatory Hele-Shaw convection is caused by an instability of the thermal boundary layer, as is evident from the plotted temperature field and streamlines. From the variation of the Nusselt number with time it is found that the oscillatory motion starts with a sinusoidal time dependence and passes into a periodic state with several frequencies as the Rayleigh number is increased. Quantitative and qualitative agreement with previous experimental and numerical results is found.