Abstract
The stability of several density profiles constructed from a linear combination of step functions in a vertically orientated two-dimensional porous medium are considered. A quasi-steady-state approximation is made so that the initial stability of the system can be approximated. Using a similar approach to that of Tan and Homsy (Phys Fluids 29:3549–3556, 1986) a dispersion equation for each density profile is obtained analytically at time zero. Neutral stability curves are then obtained to allow the regions of the parameter space to be divided into stable and unstable regions.
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Acknowledgments
PMJT would like to thank Prof. Anne De Wit for numerous fruitful discussions. Both authors would like to thank all three anonymous referees for their constructive comments and suggestions, which helped to improve this paper.
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Appendix
Appendix
The dispersion equation for two separated layers is given by
where \(P_{nm}=R_1 k(s+(-1)^n k)+(-1)^m 2\sigma s\) and \(Q_{nm}=R_2 k(s+(-1)^n k)+(-1)^m 2\sigma s\) such that \(n\) and \(m\) are either 0 or 1.
The cut-off wavenumber for two separated layers satisfies the following equation:
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Gandhi, J., Trevelyan, P.M.J. Onset conditions for a Rayleigh–Taylor instability with step function density profiles. J Eng Math 86, 31–48 (2014). https://doi.org/10.1007/s10665-013-9649-2
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DOI: https://doi.org/10.1007/s10665-013-9649-2