Rayleigh-Taylor instability in a finite cylinder: linear stability analysis and long-time fingering solutions

We consider the Rayleigh-Taylor instability problem of two initially stationary immiscible viscous fluids positioned with the denser above the less dense in a finite circular cylinder, such that their starting fluid-fluid interface is the horizontal midplane of the cylinder. The ensuing linear instability problem has a five-dimensional parameter space - defined by the density ratio, the viscosity ratio, the cylinder aspect ratio, the surface tension between the fluids and the ratio of viscous to gravitational time scales - of which we explore only part, motivated by recent experiments where viscous fluids exchange in vertical tubes (Beckett et al., J. Fluid Mech., 2011, vol. 682, pp. 652-670). We find that for these experiments, the instability is invariably 'side-by-side' (of azimuthal wavenumber 1 type) but we also uncover parameter regions where the preferred instability is axisymmetric. The fact that both 'core-annular' (axisymmetric) and 'side-by-side' (asymmetric) long-time flows are seen experimentally highlights the fact that the initial Rayleigh-Taylor instability of the interface does not determine the long-time flow configuration in these situations. Finally, long-time flow solutions are presented on the basis that they will be slowly varying fingering solutions.