Rayleigh–Taylor instability of classical diffusive density profiles for miscible fluids in porous media: a linear stability analysis

Abstract

A Rayleigh–Taylor instability typically develops when a denser layer overlies a less dense one in the gravity field. In that case, the initial base state density profile is a step function for which linear stability analysis results are well known. We investigate here analytically the linear stability analysis of other classical diffusive density profiles for porous media flows. We find that, for a species A initially distributed in the upper half of the domain with an initial concentration profile of \((-X)^m\) for \(0<m<1\) where X is the vertical coordinate, and absent from the bottom half of the domain, for large times the eigenfunctions grow like \(\exp \left( \omega _0 T^{(m+1)/2} + \omega _1 \ln (T) \right) \) where \(\omega _0\) and \(\omega _1\) are constants and T is time. Thus, the growth rate defined by \((1/A) (\mathrm{{d}}A/\mathrm{{d}}T)\) decays like \(c_1 T^{(m-1)/2} +c_2 T^{(m-2)/3}\) whilst the maximum growing wavenumber scales with \(T^{(m-2)/6}\). These results are compared to the growth rates obtained using numerical linear stability analysis. Our analytical predictions provide a set of generalised results that pave the way to the analysis of Rayleigh–Taylor instabilities of nontrivial density profiles.