This chapter extends the development of the previous chapter by applying Hamiltonâ€™s extended principle to a fluid surrounding a rigid structure. The energies are derived, and the control volume is examined in detail. Boundary conditions are derived and studied. 2D flows past a circular cylinder that is free to move transversely are formulated, and applied to reduced-order modeling. The derived general governing equations for the structure and the flow oscillator are compared with certain published models: Krenk and Nielsen, Hall, Berger, and Tamura and Matsui. It is concluded that the general formulation of this chapter is a good framework for the development of flow-oscillator models of vortex-induced oscillations.