Introducing an extension of a recently derived dimension-reduced model for an infinitely deep inviscid and irrotational layer, a two-layer system is examined in the present paper. A second thin viscous layer is added on top of the original one-layer system. The set-up is a combination of a long-wave approximation (upper layer) and a deep-water approximation (lower layer). Linear stability analysis shows the emergency of Rayleigh-Taylor and Kelvin-Helmholtz instabilities. Finally, numerical solutions of the model reveal spatial and temporal pattern formation in the weakly nonlinear regime of both instabilities. This article is part of the theme issue 'Stokes at 200 (Part 1)'.
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