Multiscale interplay of curvature and hydrodynamic slippage in flow over a patterned topography

Abstract

Reduction of fluid-solid friction utilizing the phenomenon of hydrodynamic slippage underlies a variety of energy-saving innovations. For simple demonstration of such ideas, the flow is often assumed to take place over a planar substrate. On the other hand, both roughness and intentionally textured topographies are ubiquitous in nature and engineering. Accordingly, a shear flow intrinsically slipping over a topography with small-amplitude corrugations is considered. Deriving scaling insights from a kinematic interpretation of the boundary conditions, a singular perturbation theory is developed for the multiscale interplay of the degree of intrinsic slip with the curvature of the surface along the slip velocity. Four distinct scaling regimes are identified. Especially for strongly (but not necessarily perfectly) slipping sinusoidal surfaces, unique analytical predictions with a wide intrinsic slip and amplitude range of numerical accuracy vis-à-vis finite element simulations are obtained, whereas the predictions on the small-slip scaling regimes corroborate findings from the literature. If the corrugation amplitude increases beyond a critical value, then strongly slipping surfaces suffer a slip-to-stick transition losing their lubricating properties. Unlike existing analytical predictions, the developed large-slip model can represent the slip-to-stick transition with reasonable accuracy, despite their rather simple analytical form. For extra accuracy, the predictions are readily extensible through computer algebra.