A similarity solution of a three-dimensional boundary layer is investigated. The outer flow is given by U  = ( −  xz , −  yz , z 2 ), corresponding to an axisymmetric poloidal circulation with constant potential vorticity. This flow is an exact solution of the Navier–Stokes. A wall is introduced at y  = 0 along which a boundary layer develops towards x  = 0. We show that a similarity reduction to a system of ODEs is possible. Two distinct solutions are found, one of them through numerical path-continuation, and their stability is investigated. A second three-dimensional solution is also identified for two-dimensional outer flow. The solutions are generalized for outer flows scaling with different powers of z and similar results are found. This behaviour is related to the non-uniqueness of the Falkner–Skan flows in a three-dimensional sense, with a transverse wall-jet.

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 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.