The process of liquid slip on rough-walled hydrophobic surfaces with and without entrapped gas bubbles is modeled. Here, starting with the Navier–Stokes equations, a set of partial differential equations (PDE) and boundary conditions for the general effective slip tensor of a rough hydrophobic surface are constructed by an asymptotic analysis. The intrinsic slip and surface roughness are considered as the characteristics of the surface. The solution is based on a weak variation form that fully recovers the set of PDE and Navier slip boundary. For the surface with entrapped bubbles, a semi-analytical model based on eigenfunction expansion is developed. In addition to the surface characteristics, the size and contact angle of the bubbles are considered in the semi-analytical solution. Both models are validated with the published data as well as direct numerical simulation. Based on the model results, we present correlations of effective slip length with surface characteristics and entrapped bubbles. We found that surface roughness reduces liquid slippage on a surface. However, if the asperities on a surface are filled with gas bubbles, the effective slip length can significantly increase as long as the bubble contact angle is small. Interestingly, bubbles with a larger contact angle could act inversely and change a hydrophobic surface with a large intrinsic slip to a no-slip or even a sticky surface. These results shed light on the controversy over the order of magnitude of the actual slip length of water flow in carbon-based nanotubes and nanochannels. The model results also help understand the anomalies of high water production and high amounts of hydraulic fracturing fluid leak-off observed in tight oil and shale gas reservoirs.
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