Complex Solutions of the Navier-Stokes Equations

30 April 2015
Jonathan Mestel

It is well known that low-Reynolds-number flows ($R_e\ll1$) have unique solutions, but this statement may not be true if complex solutions are permitted.

We begin by considering Stokes series, where a general steady velocity field is expanded as a power series in the Reynolds number. At each order, a linear problem determines the coefficient functions, providing an exact closed form representation of the solution for all Reynolds numbers. However, typically the convergence of this series is limited by singularities in the complex $R_e$ plane. 

We employ a generalised Pade approximant technique to continue analytically the solution outside the circle of convergence of the series. This identifies other solutions branches, some of them complex. These new solution branches can be followed as they boldly go where no flow has gone before. Sometimes these complex solution branches coalesce giving rise to real solution branches. It is shown that often, an unforced, nonlinear complex "eigensolution" exists, which implies a formal nonuniqueness, even for small and positive $R_e$.

Extensive reference will be made to Dean flow in a slowly curved pipe, but also to flows between concentric, differentially rotating spheres, and to convection in a slot. In addition, certain fundamental exact solutions are shown to possess extra complex solutions.

by Jonathan Mestel and Florencia Boshier


  • Industrial and Applied Mathematics Seminar