Vortex Shape and Geometry
Country: United States
Date: January 2009
When forming a vortex (with fog machine) forcing fog
through a round hole cut in the end of a trash can, the vortex is
round like a smoke ring. I understand this is due to the faster air
in the center drawing in the slower air near the edges forming the
doughnut. However, when I placed a triangular cut-out over the hole
the vortex still formed a doughnut shape. What property or
explanation is there for this happening?
I think your question is a lot more complicated than it appears at first look.
Without proof or justification, it seems "reasonable" that a closed loop -- from
a triangular (3 sides), a quadrilateral (4 sides), up to circular (an infinite
number of sides), will resolve into a torus spinning about the circular axis of
the "doughnut". The reasoning is that "sharp" edges will "soften" into a rotating
"smoke ring" -- the "doughnut". Here the rotation of the ring is in the direction
of the propagation of the ring. But there is another alternative, for which I do
not have a ready explanation.
Suppose that the direction of rotation is perpendicular to the direction of
translation of the vortex -- that is, the configuration is a "whirlpool",
spinning around the axis of translation? In this case, the speed of rotation
is faster the closer you get to the axis of rotation. This is similar to the
effect that occurs when ice skaters fold their arms. The conservation of momentum
makes the speed of rotation increase as their arms are pulled inward.
I do not have an answer, but maybe the question will raise some interesting
discussion for you, your friends and your teachers.
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Update: June 2012