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 formula_1 and typical diameter formula_2 with formula_3, surrounded by fluid of viscosity formula_4 whose motion is governed by the Stokes equations. Note that the Stokes' paradox implies that the limit of infinite aspect ratio formula_5 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 formula_6, where formula_7 is an arc-length coordinate, and formula_8 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 formula_9 per unit length. formula_10 is assumed to vary only over lengths much greater than formula_11, and the fluid velocity at the surface adjacent to formula_6 is well-approximated by formula_13.
The fluid velocity formula_14 at a general point formula_15 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
 formula_16
where formula_17 is the identity tensor.
Asymptotic analysis can then be used to show that the leading-order contribution to the integral for a point formula_15 on the surface of the body adjacent to position formula_19 comes from the force distribution at formula_20. Since formula_21, we approximate formula_22. We then obtain
 formula_23
where formula_24.
The expression may be inverted to give the force density in terms of the motion of the body:
 formula_25
Two canonical results that follow immediately are for the drag force formula_26 on a rigid cylinder (length formula_1, radius formula_11) moving a velocity formula_29 either parallel to its axis or perpendicular to it. The parallel case gives
 formula_30
while the perpendicular case gives
 formula_31
with only a factor of two difference.
Note that the dominant length scale in the above expressions is the longer length formula_1; the shorter length has only a weak effect through the logarithm of the aspect ratio. In slender-body theory results, there are formula_33 corrections to the logarithm, so even for relatively large values of formula_34 the error terms will not be that small.