  We determine the flow structure in an axisymmetric diffuser or expansion
region connecting two cylindrical pipes when the inlet flow is a solid body
rotation with a uniform axial flow of speeds Omega and U, respectively. A
quasi-cylindrical approximation is made in order to solve the steady Euler
equation, mainly the Bragg-Hawthorne equation. As in our previous work on the
cylindrical region downstream [R Gonz\'alez et al., Phys. Fluids 20, 24106
(2008); R. Gonz\'alez et al., Phys. Fluids 22, 74102 (2010), R Gonz\'alez et
al., J. Phys.: Conf. Ser. 296, 012024 (2011)], the steady flow in the
transition region shows a Beltrami flow structure. The Beltrami flow is defined
as a field v_B that satisfies omega_B=nabla v_B= gamma v_B, with gamma =
constant. We say that the flow has a Beltrami flow structure when it can be put
in the form v = U e_z + Omega r e_theta + v_B, being U and Omega constants, i.e
it is the superposition of a solid body rotation and translation with a
Beltrami one. Therefore, those findings about flow stability hold. The
quasi-cylindrical solutions do not branch off and the results do not depend on
the chosen transition profile in view of the boundary conditions considered. By
comparing this with our earliest work, we relate the critical Rossby number
vartheta_cs (stagnation) to the corresponding one at the fold vartheta_cf [J.
D. Buntine et al., Proc. R. Soc. Lond. A 449, 139 (1995)].

Fluid Dynamics