Wavelength selection of fingering instability inside Hele-Shaw cells

Fingering instabilities involving fluids confined between two plates someti mes give rise to a typical wavelength lambda proportional to the gap h. Thi s unexplained behavior is investigated for the case of the Rayleigh-Taylor instability between two liquids of the same viscosity. Using qualitative sc aling arguments and linear stability analysis for a simplified model of hyd rodynamics, we show that, in the miscible case, h becomes a natural cut-off when diffusion is negligible, i.e., when the Peclet number Pe=h(3)Delta rh og/(etaD) is large (eta viscosity, g gravitational acceleration, D diffusiv ity, Delta rho density difference). The same result holds in the immiscible case for large capillary number Ca=h(2)Delta rhog/(12 gamma) (gamma surfac e tension). In this saturation regime, the dominant wavelength is given by lambda approximate to2.3h, while in the opposite limit (low Pe or low Ca) l ambda scales, respectively, as h/Pe or h/Ca-1/2. These results are in agree ment with a recent experimental study. (C) 2001 American Institute of Physi cs.