Complex solutions of the Dean equations and non-uniqueness at all Reynolds numbers

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.