Extended series solutions and bifurcations of the Dean equations

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 $K$.