We consider the situation of a heavier fluid placed above a lighter one in a vertically
arranged Hele-Shaw cell. The two fluids are miscible in all proportions. For this
configuration, experiments and nonlinear simulations recently reported by Fernandez
et al. (2002) indicate the existence of a low-Rayleigh-number (Ra) ‘Hele-Shaw’ instability
mode, along with a high-Ra ‘gap’ mode whose dominant wavelength is on
the order of five times the gap width. These findings are in disagreement with linear
stability results based on the gap-averaged Hele-Shaw approach, which predict much
smaller wavelengths. Similar observations have been made for immiscible flows as
well (Maxworthy 1989).In order to resolve the above discrepancy, we perform a linear stability analysis
based on the full three-dimensional Stokes equations. A generalized eigenvalue problem
is formulated, whose numerical solution yields both the growth rate and the
two-dimensional eigenfunctions in the cross-gap plane as functions of the spanwise
wavenumber, an ‘interface’ thickness parameter, and Ra. For large Ra, the dispersion
relations confirm that the optimally amplified wavelength is about five times the gap
width, with the exact value depending on the interface thickness. The corresponding
growth rate is in very good agreement with the experimental data as well. The eigenfunctions
indicate that the predominant fluid motion occurs within the plane of the
Hele-Shaw cell. However, for large Ra purely two-dimensional modes are also amplified,
for which there is no motion in the spanwise direction. Scaling laws are provided
for the dependence of the maximum growth rate, the corresponding wavenumber, and
the cutoff wavenumber on Ra and the interface thickness. Furthermore, the present
results are compared both with experimental data, as well as with linear stability
results obtained from the Hele-Shaw equations and a modified Brinkman equation.
A note on the linear stability of a two-dimensional Stokes layer