Slender-body theory

Slender-body theory

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 and in electrostatics.

Theory for Stokes flow

Consider slender body of length ell and typical diameter 2a with ell gg a, surrounded by fluid of viscosity mu whose motion is governed by the Stokes equations. 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 oldsymbol{X}(s,t), 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 oldsymbol{f}(s) per unit length. oldsymbol{f} is assumed to vary only over lengths much greater than a, and the fluid velocity at the surface adjacent to oldsymbol{X}(s,t) is well-approximated by partialoldsymbol{X}/partial t.

The fluid velocity oldsymbol{u}(oldsymbol{x}) at a general point oldsymbol{x} due to such a distribution can be written in terms of an integral of the Oseen tensor, which acts as a Greens function for a single Stokeslet. We have: oldsymbol{u}(oldsymbol{x}) = int_0^ell frac{oldsymbol{f}(s)}{8pimu} cdot left( frac{mathbf{I + frac{(oldsymbol{x} - oldsymbol{X})(oldsymbol{x} - oldsymbol{X})}{|oldsymbol{x} - oldsymbol{X}|^3} ight) , mathrm{d}s where mathbf{I} is the identity tensor.

Asymptotic analysis can then be used to show that the leading-order contribution to the integral for a point oldsymbol{x} on the surface of the body adjacent to position s_0 comes from the force distribution at |s- s_0| = O(a). Since a ll ell, we approximate oldsymbol{f}(s) approx oldsymbol{f}(s_0). We then obtain: frac{partial oldsymbol{X{partial t} sim frac{ln(ell/a)}{4pimu} oldsymbol{f}(s) cdot Bigl( mathbf{I} + oldsymbol{X}'oldsymbol{X}' Bigr)where oldsymbol{X}' = partial oldsymbol{X}/partial s.

The expression may be inverted to give the force density in terms of the motion of the body:: oldsymbol{f}(s) sim frac{4pimu}{ln(ell/a)} frac{partial oldsymbol{X{partial t} cdot Bigl( mathbf{I} - extstylefrac{1}{2} oldsymbol{X}'oldsymbol{X}' Bigr)

Two canonical results that follow immediately are for the drag force F on a rigid cylinder (length ell, radius a) moving a velocity u either parallel to its axis or perpendicular to it. The parallel case gives: F sim frac{2pimuell u}{ln(ell/a)}while the perpendicular case gives: F sim frac{4pimuell u}{ln(ell/a)}with only a factor of two difference.

Note that the dominant length scale in the above expressions is the longer length ell; 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 ell/a the error terms will not be that small.

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