Appl Math Optim 44:245–272 (2001)
2001 Springer-Verlag New York Inc.
The Effects of Flexion and Torsion on a Fluid Flow
Through a Curved Pipe
Department of Mathematics, University of Zagreb,
Bijeniˇcka 30, 10000 Zagreb, Croatia
Abstract. We consider an injection of incompressible viscous ﬂuid in a curved
pipe with a smooth central curve γ . The one-dimensional model is obtained via
singular perturbation of the Navier–Stokes system as ε, the ratio between the cross-
section area and the length of the pipe, tends to zero. An asymptotic expansion of
the ﬂow in powers of ε is computed. The ﬁrst term in the expansion depends only
on the tangential injection along the central curve γ of the pipe and the velocity as
well as the pressure drop are in the tangential direction. The second term contains
the effects of the curvature (ﬂexion) of γ in the direction of the tangent while the
effects of torsion appear in the direction of the normal and the binormal to γ . The
boundary layers at the ends of the pipe are studied. The error estimate is proved.
Key Words. Curved pipe, Asymptotic analysis, Navier–Stokes equations.
AMS Classiﬁcation. 35B40, 35B25, 76D05, 76D10.
1.1. The Geometry
Let γ be a smooth curve of class C
(the central curve of the pipe) parameterized by its
arc length y
∈ [0,]. Let ϕ:[0,] → R
be its natural parameterization, i.e.,
At each point ϕ(y
∈ [0,], of the curve γ we deﬁne the curvature (ﬂexion) as
)| and Frenet’s basis:
) = ϕ