<p>This study provides a comprehensive examination of the dynamic behavior of fluid-conveying pipes through a three-dimensional, geometrically nonlinear theoretical framework. The study concentrates on discerning dynamic instability phenomena affected by diverse boundary conditions and internal fluid flow velocities. We use Hamilton’s principle and the calculus of variations to find the governing equations of motion. These equations take into account Coriolis forces and the effects of fluid flow on structural stiffness. Three types of boundaries are studied: clamped-free, pinned-pinned, and clamped-clamped. The research investigates both divergence (buckling) and coupled-mode flutter instabilities, predicated on the assumptions of pipe inextensibility and nonlinear kinematics. Eigenvalue analysis is utilized to ascertain critical instability thresholds, assessing the influences of fluid velocity and geometric characteristics. The results indicate that instabilities are substantially influenced by flow velocity and support conditions, with Coriolis forces serving as primary contributors. A non-dimensional approach underscores the impact of mass and geometry, providing insights for the design of safer and more efficient piping systems. This study elucidates the substantial impact of Coriolis forces on flutter-type instabilities under particular boundary conditions, offering an innovative energy-centric viewpoint on fluid-structure interactions in fluid-conveying pipes.</p>