Slender-body theory
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Jump to: navigation, searchIn 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. Principle applications are to Stokes flow at very low
Reynolds numbers and in electrostatics.
[edit] Theory for Stokes flow
Consider slender body of length and typical diameter 2a with ,
surrounded by fluid of viscosity whose motion is governed by the
Stokes equations. Note that 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 s is an
arc-length coordinate, and t 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 a, 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 s0 comes from the force
distribution at | s s0 | = O(a). 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 F on a rigid cylinder (length , radius a) moving a velocity u
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 O(1) corrections to the logarithm, so even for
relatively large values of the error terms will not be that
small.
[edit] References
Batchelor, G. K. (1970), "Slender-body theory for particles of
arbitrary cross-section in Stokes flow", J. Fluid Mech. 44: 419440,
doi:10.1017/S002211207000191X Cox, R. G. (1970), "The motion of
long slender bodies in a viscous fluid. Part 1. General Theory", J.
Fluid Mech. 44: 791810, doi:10.1017/S002211207000215X Hinch, E. J.
(1991), Perturbation Methods, Cambridge University Press, ISBN
0521378974Retrieved from
"http://en.wikipedia.org/wiki/Slender-body_theory"
Categories: Fluid dynamics
