The Geometry of Axisymmetric Ideal Fluid Flows with Swirl

• Published in: Arnold mathematical journal Vol. 3; no. 2; pp. 175 - 185
• Main Authors: Washabaugh, Pearce, Preston, Stephen C.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01-06-2017; Springer Nature B.V
• Subjects: Axisymmetric flow; Conjugate points; Criteria; Curvature; Flow control; Fluid dynamics; Fluid flow; Incompressible flow; Manifolds (mathematics); Mathematical analysis; Mathematics; Mathematics and Statistics; Research Contribution; Toruses; Euler-Arnold; Euler equation; Axisymmetric; Ideal fluid; Stability; Curvature
• ISSN: 2199-6792; 2199-6806
• DOI: 10.1007/s40598-016-0058-2

The sectional curvature of the volume preserving diffeomorphism group of a Riemannian manifold
M
can give information about the stability of inviscid, incompressible fluid flows on
M
. We demonstrate that the submanifold of the volumorphism group of the solid flat torus generated by axisymmetric fluid flows with swirl, denoted by
D
μ
,
E
(
M
)
, has positive sectional curvature in every section containing the field
X
=
u
(
r
)
∂
θ
iff
∂
r
(
r
u
2
)
>
0
. This is in sharp contrast to the situation on
D
μ
(
M
)
, where only Killing fields
X
have nonnegative sectional curvature in all sections containing it. We also show that this criterion guarantees the existence of conjugate points on
D
μ
,
E
(
M
)
along the geodesic defined by
X
.