We investigate a gravity-driven thin film flow of a non-Newtonian liquid over an inclined micro-patterned surface. We demonstrate the effect of micro-patterning on the film draining rate and the velocity profile by varying the relative slit width (Tr) and the length of the periodic irregularities (L). We unveil the interplay of the substrate structure and the fluid rheology by modeling the non-Newtonian thin film using the Carreau model, and the rheology of the film is varied for different values of power index n. Through numerical simulations, we delineate the effects of inertia, viscous, and capillary forces on the physics of thin film flow. We report a significant augmentation of flow velocity for both shear-thinning and shear-thickening fluids as a result of substrate micro-patterning, with the relative slit width playing a dominant role while the length of the periodic irregularities has only a minor influence on drainage characteristics. However, when the sole effect of fluid rheology is considered, flow velocity enhances for pseudoplastic fluid and decreases for dilatant fluid in comparison to Newtonian fluid. We examine the combined effect of rheology and substrate topography, revealing the dominating influence of micro-patterning at high slit-widths, while the fluid rheology has a greater role to play at lower slit-widths. We also demonstrate that the susceptibility of flow physics on varying rheology or topography is greatest for low viscosity liquids. Finally, we mark different regimes where the augmentation of average velocity and surface velocity are individually achieved. Hence, we propose a suitable combination of substrate structure and fluid rheology to engineer a flow characteristic. Based on the suitability for various applications, we provide the key to simultaneously optimizing the fluid rheology and substrate micro-patterning for precise engineering and controlling the draining characteristics of a thin film.

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