Complex solutions of the Dean equations and non-uniqueness at all Reynolds numbers
File(s)
Accepted version
Author(s)
Boshier, FAT
Mestel, AJ
Type
Journal Article
Abstract
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.
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.
Date Issued
2017-03-29
Date Acceptance
2017-02-24
Citation
Journal of Fluid Mechanics, 2017, 818, pp.241-259
ISSN
1469-7645
Publisher
Cambridge University Press (CUP)
Start Page
241
End Page
259
Journal / Book Title
Journal of Fluid Mechanics
Volume
818
Copyright Statement
© 2017 Cambridge University Press. This paper has been accepted for publication and will appear in a revised form, subsequent to peer-review and/or editorial input by Cambridge University Press.
Subjects
Science & Technology
Technology
Physical Sciences
Mechanics
Physics, Fluids & Plasmas
Physics
bifurcation
mathematical foundations
Navier-Stokes equations
CURVED DUCTS
BIFURCATION
SERIES
MOTION
FLUID
PIPES
FLOW
Fluids & Plasmas
01 Mathematical Sciences
09 Engineering
Publication Status
Published