Fluid lubrication model over sinusoidal roughness with streamline-based approach

A method is proposed to deterministically obtain steady lubrication pressure for the Stokes flow in a channel bounded by a flat wall and a surface with roughness represented by sinusoidal waves. A streamline sufficiently far away from the rough surface is used to formulate a streamline-based lubrication equation with the velocity on the streamline, and the velocities on the streamline is imposed as a boundary condition. In the solution of the lubrication equation, by virtually moving the streamline towards the flat wall, the pressure on the flat wall is obtained, and then the wall-normal variation of the pressure is recovered from the wall pressure by a lubrication model that considers higher order terms. The proposed method is applied to lubrication flows in channels with roughness represented by a single sinusoidal wave and a superposition of several sinusoidal waves. Through comparison with analytical solutions, the validity of the proposed method is established, and the applicable range of superposition of waves is explained that lowest-wavenumber component in surface profile is sufficiently isolated from higher-wavenumber components. Although the problem setting intrinsically prohibits the application of the conventional Reynolds lubrication equation, this study provides new understandings for the pressure obeying the Reynolds lubrication equation and the role of the higher-order terms.