Steady, incompressible flow down a slowly curving circular pipe is considered. Both real and complex solutions of the Dean equations are found by analytic continuation of a series expansion in the Dean number, $K$. Higher-order Hermite–Padé approximants are used and the results compared with direct computations using a spectral method. The two techniques agree for large, real $K$, indicating that previously reported asymptotic behaviour of the series solution is incorrect, and thus resolving a long-standing paradox. It is further found that a second solution branch, known to exist at high Dean number, does not appear to merge with the main branch at any finite $K$, but appears rather to bifurcate from infinity. The convergence of the series is limited by a square-root singularity on the imaginary $K$-axis. Four complex solutions merge at this point. One corresponds to an extension of the real solution, while the other three are previously unreported. This bifurcation is found to coincide with the breaking of a symmetry property of the flow. On one of the new branches the velocity is unbounded as $K\rightarrow 0$. It follows that the zero-Dean-number flow is formally non-unique, in that there is a second complex solution as $K\rightarrow 0$ for any non-zero $\vert K\vert $. This behaviour is manifested in other flows at zero Reynolds number. The other two complex solutions bear some resemblance to the two solution branches for large real $K$.

Dynamo action is considered in a conducting cylindrical annulus surrounded by an insulator. The driving velocity field is assumed to be linear in the axial coordinate and to satisfy the incompressible Navier-Stokes equations. Such flows have recently been shown to exist with no forcing other than the similarity structure. Magnetic field instabilities with the same spatial structure are sought. The kinematic eigenvalue problem is found to have two growing modes for moderate values of the magnetic Reynolds number, Rm. As Rm → ∞ it is shown that the modes are governed by layers on the outer wall. The growing field saturates in a solution to the nonlinear dynamo problem. Three distinct steady solution families are found and the complicated bifurcation structure is investigated.

Steady, incompressible flow down a slowly-curving circular pipe is considered, analyti- cally and numerically. Both real and complex solutions are investigated. Using high-order Hermite–Pad ́e approximants, the Dean series solution is analytically continued outside its circle of convergence where it predicts a complex solution branch for real, positive Dean number, K . This is confirmed by numerical solution. It is shown that other previously unknown solution branches exist for all K > 0, which are related to an unforced com- plex eigensolution. This non-uniqueness is believed to be generic to the Navier–Stokes equations in most geometries. By means of path continuation, numerical solutions are followed around the complex K -plane. The standard Dean two-vortex solution is shown to lie on the same hypersurface as the eigensolution and the four-vortex solutions found in the literature. Elliptic pipes are considered and shown to exhibit similar behaviour to the circular case. There is an imaginary singularity limiting convergence of the Dean series, an unforced solution at K = 0 and nonuniqueness for K > 0, culminating in a real bifurcation

Steady Boussinesq flow in a weakly curved channel driven by a horizontal temperature gradient is considered. Linear variation in the transverse direction is assumed so that the problem reduces to a system of ordinary differential equations. A series expansion in $G$, a parameter proportional to the Grashof number and the square root of the curvature, reveals a real singularity and anticipates hysteresis. Numerical solutions are found using path continuation and the bifurcation diagrams for different parameter values are obtained. Multivalued solutions are observed as $G$ and the Prandtl number vary. Often fields with the imposed structure that satisfy all the governing equations are insensitive to the boundary conditions and can be regarded as perturbations of the homogeneous (or ‘unforced’) problem. Four such unforced solutions are found. In two of these the velocity remains coupled with temperature which, formally, scales as $1/G$ as $G\rightarrow 0$. The other two are purely hydrodynamic. The existence of such solutions is due to the unbounded nature of the domain. It is shown that these occur not only for the Dean equations, but constitute previously unreported solutions of the full Navier–Stokes equations in an annulus of arbitrary curvature. Two additional unforced solutions are found for large curvature.