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

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Consider slender body of length   and typical diameter   with  , surrounded by fluid of viscosity   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 (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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  • Batchelor, G. K. (1970), "Slender-body theory for particles of arbitrary cross-section in Stokes flow", J. Fluid Mech., 44 (3): 419–440, Bibcode:1970JFM....44..419B, doi:10.1017/S002211207000191X, S2CID 121986116
  • Cox, R. G. (1970), "The motion of long slender bodies in a viscous fluid. Part 1. General Theory", J. Fluid Mech., 44 (4): 791–810, Bibcode:1970JFM....44..791C, doi:10.1017/S002211207000215X, S2CID 118908560
  • Hinch, E. J. (1991), Perturbation Methods, Cambridge University Press, ISBN 978-0-521-37897-0