An improved formalism for a flowing two-fluid equilibrium with constant density is developed. This extends the usual single-fluid model. In this generalization, the magnetic field is replaced by two quantities, the generalized vorticities of each species. Criteria are found for when the single-fluid model is adequate and when the more general two-fluid model is necessary. The two-dimensional equilibria with purely azimuthal ion flow are studied analytically and numerically. Spherical torus and compact toroid equilibria are found that are relevant to the current experiment. The ion flow and plasma beta as well as the size parameter are found to play a major role in the question of whether two-fluid corrections are needed.