In
fluid dynamics and
electrostatics, slender-body theory is a methodology that can be used to take advantage of the slenderness of a body to obtain an approximation to a field surrounding it and/or the net effect of the field on the body. Principal applications are to
Stokes flow — at very low
Reynolds numbers — and in
electrostatics.
Theory for Stokes flow
Consider slender body of length
and typical diameter
with
, surrounded by fluid of
viscosity
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inte ...
whose motion is governed by the
Stokes equations. Note that the
Stokes' paradox implies that the limit of infinite aspect ratio
is singular, as no Stokes flow can exist around an infinite cylinder.
Slender-body theory allows us to derive an approximate relationship between the velocity of the body at each point along its length and the force per unit length experienced by the body at that point.
Let the axis of the body be described by
, where
is an arc-length coordinate, and
is time. By virtue of the slenderness of the body, the force exerted on the fluid at the surface of the body may be approximated by a distribution of
Stokeslets along the axis with force density
per unit length.
is assumed to vary only over lengths much greater than
, and the fluid velocity at the surface adjacent to
is well-approximated by
.
The fluid velocity
at a general point
due to such a distribution can be written in terms of an integral of the
Oseen tensor (named after
Carl Wilhelm Oseen), which acts as a
Greens function for a single Stokeslet. We have
:
where
is the identity tensor.
Asymptotic analysis can then be used to show that the leading-order contribution to the integral for a point
on the surface of the body adjacent to position
comes from the force distribution at
. Since
, we approximate
. We then obtain
:
where
.
The expression may be inverted to give the force density in terms of the motion of the body:
:
Two canonical results that follow immediately are for the drag force
on a rigid
cylinder
A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
A cylinder may also be defined as an infin ...
(length
, radius
) moving a velocity
either parallel to its axis or perpendicular to it. The parallel case gives
:
while the perpendicular case gives
:
with only a factor of two difference.
Note that the dominant length scale in the above expressions is the longer length
; the shorter length has only a weak effect through the logarithm of the aspect ratio. In slender-body theory results, there are
corrections to the logarithm, so even for relatively large values of
the error terms will not be that small.
References
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* {{Citation , last = Hinch , first = E. J. , year = 1991 , title = Perturbation Methods , publisher = Cambridge University Press , isbn = 978-0-521-37897-0
Fluid dynamics