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.

中文翻译：

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带有和不带有气泡的粗糙疏水表面上的液体滑移

模拟了带有和不带有夹带气泡的粗糙壁疏水表面上的液体滑移过程。在这里，从 Navier-Stokes 方程开始，通过渐近分析构建了一组偏微分方程 (PDE) 和粗糙疏水表面的一般有效滑移张量的边界条件。固有滑移和表面粗糙度被认为是表面的特征。该解决方案基于完全恢复 PDE 和 Navier 滑移边界集的弱变化形式。对于夹带气泡的表面，建立了基于特征函数展开的半解析模型。除了表面特征外，半解析解中还考虑了气泡的大小和接触角。两种模型都使用已发布的数据以及直接数值模拟进行了验证。基于模型结果，我们提出了有效滑移长度与表面特征和夹带气泡的相关性。我们发现表面粗糙度减少了液体在表面上的滑动。然而，如果表面的凹凸不平处充满气泡，只要气泡接触角很小，有效滑移长度就会显着增加。有趣的是，具有较大接触角的气泡可能会起到相反的作用，并将具有较大固有滑移的疏水表面变为无滑移甚至粘性表面。这些结果阐明了关于碳基纳米管和纳米通道中水流实际滑移长度数量级的争议。