Re: No torque on an a-chrial body in Stokes flow?
- From: Uncle Al <UncleAl0@xxxxxxxxxxxxx>
- Date: Thu, 03 May 2007 12:05:22 -0700
Edward Green wrote:
[snip]
Simple counter-argument to the claimed theorem:
Consider an otherwise arbitrary body possessing a plane of mirror
symmetry. Form a compound object by joining the body to a long thin
strut of negligible cross section lying in an extension of the mirror
plane, terminated by a spherical mass of negligible diameter.
Given a statement, its opposite is not the negative it is the
contrapositive. The proof was that a chiral solid body translated
through a uniform medium must experience a net torque. If the body
has three equal moments of inertia its orientation is irrelevant.
Demonstrating that translating achiral bodies can experience net
torque is not a disproof of the chiral case statement.
http://www.igf.fuw.edu.pl/KB/HKM/PDF/HKM_027_s.pdf
pp. 25-27, the chiral case.
Solid equi-inertial bodies such as platonic polyhedra, a sphere, a
solid right cylinder of revolution with length = [sqrt(3)](radius)...
cannot be chiral. OTOH, open-cell foam bodies with those identical
shapes can trivially be chiral - simply position their internal struts
oriented as bonds are in the alpha-quartz lattice.
An equi-inertial hollow cylinder with inner radius R1 and outer radius
R2 has length = sqrt[3(R1^2 + R2^2)]. Are ya gonna whine about its
foam analogue being off-kilter when the hollow is aligned parallel or
normal to translation? It spins in either case.
If the drag force on the body does have a component out of the flow
direction, the compound body can be arranged so that this component
generates an uncompensated torque about an axis parallel to the flow
field. So, if no body or compound body possessing a symmetry plane
can experience torque in a Stokes flow, it must be that any object
possessing a symmetry plane is fully equivalent to a cube under these
flows.
"All white swans have high albedos" does not exclude "some dark swans
have high albedos." Have you ever seen a bright reflection come off a
jet black surface?
--
Uncle Al
http://www.mazepath.com/uncleal/
(Toxic URL! Unsafe for children and most mammals)
http://www.mazepath.com/uncleal/lajos.htm#a2
.
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