Spouting and planform selection in the Rayleigh–Taylor instability of miscible viscous fluids

Abstract

A weakly nonlinear analysis is used to study the initial evolution of the Rayleigh–Taylor instability of two superposed miscible layers of viscous fluid between impermeable and traction-free planes in a field of gravity. Analytical solutions are obtained to second order in the small amplitude of the initial perturbation of the interface, which consists of either rolls or squares or hexagons with a horizontal wavenumber k. The solutions are valid for arbitrary values of k, the viscosity ratio (upper/lower) γ, and the depth ratio r, but are presented assuming that k=k_max(γ, r), where k_max is the most unstable wavenumber predicted by the linear theory. For all planforms, the direction of spouting (superexponential growth of interfacial extrema) is determined by the balance between the tendency of the spouts to penetrate the less viscous layer, and a much stronger tendency to penetrate the thicker layer. When these tendencies are opposed (i.e. when γ>1 with r>1), the spouts change direction at a critical value of r=r_c(γ). Hexagons with spouts at their centres are the preferred planform for nearly all values of γ and r, followed closely by squares; the most slowly growing planform is hexagons with spouts at corners. Planform selectivity is strongest when γ[ges ]10 and r[ges ]γ^1/3. Application of the results to salt domes in Germany and Iran show that these correspond to points (γ, r) below the critical curve r=r_c(γ), indicating that the domes developed from interfacial extrema having subexponential growth rates.