The steady axisymmetric Euler flow of an inviscid incompressible swirling fluid is described exactly by the Squire-Long equation. This equation is studied numerically for the case of diverging flow to investigate the dependence of solutions on upstream, or inlet, and downstream, or outlet, boundary conditions and flow geometry. The work is performed with a view to understanding how the phenomenon of vortex breakdown occurs. It is shown that solutions fail to exist or, alternatively, that the axial flow ceases to be unidirectional, so that breakdown can be inferred, when a parameter measuring the relative magnitude of rotation and axial flow (the Squire number) exceeds critical values depending upon the geometry and inlet profiles. A `quasi-cylindrical' simplification of the Squire-Long equation is compared with the more complete Euler model and shown to be able to account for most of the latter's behaviour. The relationship is examined between `failure' of the quasi-cylindrical model and the occurrence of a `critical' flow state in which disturbances can stand in the flow.