Re: The Value of Science: Explanation vs Prediction

J. J. Lodder wrote:
David Ewan Kahana <dek@xxxxxxx> wrote:

Robert Carnegie wrote:
David Ewan Kahana wrote:
David Ewan Kahana wrote: (what?)
Strato (died circa 270 BCE), and third director of the Lycaeum,
criticized Aristotle's views on motion. He noticed that bodies
tend to accelerate when they fall. He cited rainwater falling
off of the corner of a roof as an example ... which often starts
as a continuous stream that breaks up into single drops by
the time it hits the ground, due as he suggested to the water
attaining a higher speed than when it left the roof.

Together with surface tension, I suppose - consider a stream of falling
water being "stretched" by gravity, like pulling taffy; with water,
when the stream gets narrow, it tends to "snap" into drops.


Well, I was cheating a bit ;->

The proper explanation has to do with hydrodynamics
of course, and Strato didn't know it, for sure: the
key point is that the flow is laminar at first, while the
velocity is low. This keeps the Reynolds number small.

Once the fluid falls far enough it speeds up, the Reynolds
number becomes high, and the flow becomes turbulent.
Turbulence causes the stream to break up.

No.


OK. Thanks for the correction, and apologies
to Robert Carnegie, who had the right idea.

I had always thought that the breakup was a
bulk phenomenon, so was related to turbulence.

Surface tension comes in on the level of the behaviour
of small droplets that break away ... they have so much
surface relative to volume, that surface tension becomes
very significant in such cases.

As the jet speeds up it's surface area decreases,
by conservation of mass.

That's certainly true. It's clearly a requirement
of the equation of continuity given an incompressible
fluid and an increasing velocity. In fact the narrowing
of the cross-sectional area is a famous demonstration
of the equation of continuity in the case of a cylindrically
symmetric free-falling water jet.

But the narrowing ceases at the point where the
stream breaks up into drops, of course.

When the jet becomes too narrow
it breaks up by whatshisname's instability,
since surface tension favours droplets.

Apparently Rayleigh was the first to analyze
the phenomenon in the non-viscous case,
and others extended the stability analysis to
include viscosity and other non-linear effects.

I note that surface tension alone would tend to
favour the formation of larger radius drops.

In that way the surface to volume ratio would be
made smallest.

But it would require a big horizontal mass flow of the
water to make a large spherical drop.

So there is clearly a competition between
inertia, which tends to inhibit sideways
and reverse flows of the water necessary to
form drops, and surface tension, which favours
the drops.

The driving force that produces the horizontal
flow must be a pressure difference between
the inside of the water jet (higher pressure) and
the air pressure outside the jet, which is balanced
by surface tension in the stable region of the flow.

But at some critical velocity and cross-sectional
area, surface tension wins out, and small perturbations
in the surface shape begin to grow.


But the Greeks couldn't know that.

But the breakup is irrelevant:
the right point is that the jet narrows down,
hence accerates, and Strato had that right.

Best,

Jan

Best regards,

David

.

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