Prediction of resistance induced by surface complexity in lubricating layers: Application to super-hydrophobic surfaces

Super Hydrophobic (SH) coatings are widely used to mitigate drag in various applications. Numerous studies have demonstrated that the beneficial wall-slip effect produced by these materials disappears in laminar flow regimes. The main mechanisms considered to be behind the decrease in performance are Marangoni-induced stresses and air/liquid interface deformation. In the present study, a new mechanism is proposed to explain the loss of performances of SH-surfaces in laminar flow regimes. Here we consider the flow of air inside the plastron and the associated momentum loses induced by roughness elements with different geometric characteristics. The effects of air motion within the plastron is coupled to the outer fluid with a homogenised boundary condition approach. To this end, numerical simulations at the scale of the roughness element were conducted as a function of the porosity and the tortuosity of the domain to determine the slip velocity at the air-liquid interface. The homogenised boundary condition is then implemented in a theoretical model for the outer flow to compute drag on SH-spheres at low $Re$ numbers. Experiments of laminar SH falling spheres indicate that high values of the tortuosity and low values of porosity lead to a loss of performances when considering drag reduction. As anticipated, a 3D printed sphere with low tortuosity and similar porosity demonstrated near-optimal drag reductions. A comparative study between the predicted values and experiments shows that the homogenised model is able to accurately predict the drag on SH surfaces for values of the porosity and tortuosity estimated from microscopy images of the SH textured surface.