This research builds upon Çelik's earlier study \cite{z-talip}, which explored various flow patterns and constructed a bifurcation diagram for the flow with Stokes approximation in a Z-shaped domain driven by dual lids. In this study, enhanced solutions near the re-entrant corner are initially derived using the asymptotic matching technique. These solutions direct attention towards a comprehensive analysis of flow structures and behaviors at the re-entrant corner point. The formation and bifurcation of the separation bubble that appears near the corner is investigated in detail. Subsequently, the $(h_1,h_2)$ control space diagram, which was previously developed on the basis of cavity height parameters, is recreated for a Reynolds number of $50$. Additionally, the impact of the Reynolds number ($Re$) on vortex formation is analyzed. The findings indicate that at $Re=50$, distinct flow patterns occur, and an increase in the Reynolds number accelerates vortex formation, leading to earlier transitions in flow topology and more complex bifurcation scenarios.