Slender-body theory
Encyclopedia
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
.
and typical diameter 
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 
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.
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...
and electrostatics
Electrostatics
Electrostatics is the branch of physics that deals with the phenomena and properties of stationary or slow-moving electric charges....
, 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
Electrostatics
Electrostatics is the branch of physics that deals with the phenomena and properties of stationary or slow-moving electric charges....
.
Theory for Stokes flow
Consider slender body of length and typical diameter with , surrounded by fluid of viscosityViscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher 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 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 OseenCarl Wilhelm Oseen
Carl Wilhelm Oseen was a theoretical physicist in Uppsala and Director of the Nobel Institute for Theoretical Physics in Stockholm....
), which acts as a Greens function for a single Stokeslet. We have
where
is the identity tensor.Asymptotic analysis
Asymptotic analysis
In mathematical analysis, asymptotic analysis is a method of describing limiting behavior. The methodology has applications across science. Examples are...
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.