Rayleigh–Taylor instabilities in miscible fluids with initially piecewise linear density profiles

The stability of some simple density profiles in a vertically orientated two-dimensional porous medium is considered. The quasi-steady-state approximation is made so that the stability of the system can be approximated. As the profiles diffuse in time, the instantaneous growth rates evolve in time. For an initial step function density profile, the instantaneous growth rate was numerically found to decay like $$T^{-1/2}$$ for large times T, and the corresponding eigenfunctions scale with $$\mathrm{e}^{\omega \sqrt{T}}$$ where $$\omega $$ is a constant. For density profiles initially corresponding to a finite layer, the instantaneous growth rate eventually decayed like $$T^{-1}$$. This corresponds to an instability with algebraic growth, and the eigenfunctions scale with $$T^p$$ (where p is a constant) for large time. For a species initially linearly distributed in a finite layer, when the concentration has an increasing gradient in the downwards direction, the stability of the system was similar to that found for a uniformly distributed finite layer. However, when the concentration had a decreasing gradient in the downwards direction, the growth rates remained constant for a long period time, but eventually decayed in the same way as found in a uniformly distributed finite layer, for very large times. Numerical simulations were performed to validate the predictions made by the linear stability analysis.