A novel rapid methodology for generating bilayer base-flow solutions involving surface patterning

Two bilayer flow arrangements, involving either (i) shear- or (ii) gravity-driven fluids, are explored in the presence of a periodically repeating surface topography. Key simplifying assumptions are that the flow is considered two-dimensional and inertialess. The solutions are obtained utilising a novel first-integral representation of the Navier-Stokes equations, leading to a highly efficient methodology for generating the accompanying flow structure within the layers in tandem with the disturbance experienced by the internal interface separating them-for configuration (ii) this extends also to the upper layer's free-surface shape. Representative examples of each flow type, with a change of key parameters, are provided with particular emphasis on (ii).