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

A weakly nonlinear analysis is used to study the initial evolution of the R ayleigh-Taylor instability of two superposed miscible layers of viscous flu id between impermeable and traction-free planes in a field of gravity. Anal ytical solutions are obtained to second order in the small amplitude of the initial perturbation of the interface, which consists of either rolls or s quares or hexagons with a horizontal wavenumber k. The solutions are valid for arbitrary values of k, the viscosity ratio (upper/lower) gamma, and the depth ratio r, but are presented assuming that k = k(max)(gamma, r), where k(max) is the most unstable wavenumber predicted by the linear theory. For all planforms, the direction of spouting (superexponential growth of inter facial extrema) is determined by the balance between the tendency of the sp outs to penetrate the less viscous layer, and a much stronger tendency to p enetrate the thicker layer. When these tendencies are opposed (i.e, when ga mma > 1 with r > 1), the spouts change direction at a critical value of r = r(c)(gamma). Hexagons with spouts at their centres are the preferred planf orm for nearly all values of gamma and r, followed closely by squares; the most slowly growing planform is hexagons with spouts at corners. Planform s electivity is strongest when gamma greater than or equal to 10 and r greate r than or equal to gamma(1/3). Application of the results to salt domes in Germany and Iran show that these correspond to points (gamma, r) below the critical curve r = r(c)(gamma), indicating that the domes developed from in terfacial extrema having subexponential growth rates.