Linear stability analysis in channel with slippery and rough boundaries

This study examines two-dimensional flow stability in a channel with a superhydrophobic wall and small amplitude roughness. We consider two primary cases: (1) the upper surface is smooth and allows slip, while the lower surface is rough and has no slip; and (2) both surfaces exhibit slip behavior. By using the orthogonal modal method for the Orr–Sommerfeld equation, we rigorously assess the flow's linear stability. The results demonstrate that the slip coefficient and rough wave number significantly influence stability characteristics. Variations in the slip coefficient lead to notable changes in stability thresholds, while the rough wave number alters the flow's response to perturbations. These findings enhance our understanding of the interactions between fluid dynamics, surface properties, and stability phenomena in channels, providing valuable insights for optimizing flow control strategies and designing surfaces in various engineering applications.