Comparison between the homogenization and the multiscale methods for the analysis of very thin compressible flow between rough surfaces

The present work presents a comparison between the homogenization and the multiscale methods applied to the compressible Reynolds equation with irregular coefficients. The equation models a very thin compressible flow between rough surfaces. If the use of the homogenization method for the Reynolds equation with irregular coefficients is not new, it is for the multiscale method. Indeed, this last approach is borrowed from the flows in porous media (where only flows due to the pressure gradients are present) and is here extended to also take into account the Couette terms. The paper presents the detailed development of both methods and underlines similitudes and differences. Illustrative results obtained for a realistic geometry show the impact of the coarse mesh, the precision of the solution on the fine mesh and the computational effort of both methods compared to the original compressible Reynolds equation. Both methods worked well and the results show that they are reliable and efficient tools for the compressible Reynolds equation with irregular coefficients.