Re: [Phys-L] fluids

[John Denker <jsd@av8n.com>]

On 2/7/19 11:00 AM, Anthony Lapinski wrote:

> As a stream of water falls from a faucet, the stream narrows. I've heard
> this is due to continuity -- A1v1 = A2v2. The water accelerates as it
> falls, so the area decreases.

Yes.  That's in interesting topic.
 
> But isn't continuity for fluids in pipes/hoses, etc? As the area changes,
> the velocity changes (not the other way around).

An /equation/ says both things happen;  it does not say which causes which.
For equations, "the other way around" is not distinct from the original.
F=ma says exactly the same thing as a=F/m.
By way of contrast, causation has a direction, from cause to effect.
An equation is not a statement of causation.

Continuity is first cousins to conservation.  It's always true.
The full fancy statement of continuity takes density into account, but
for simplicity let's assume constant density.  Also let's assume the
stream does not break up into droplets.

Subject to those assumptions, continuity says that if the fluid speeds
up, the cross-sectional area must decrease.  Otherwise you would violate
the law of conservation of water.  This says what must happen.  It does
not explain the mechanics of how it happens.

====================

However, we can /also/ discuss the mechanics.  Let's do that now.

> I thought the stream narrows because the air pressure is slighter larger at
> the lower height (P2 = P1 + pgh). 

That's not a valid argument.  Continuity says the stream /must/ narrow
as it speeds up, no matter what the air is doing.  For example, imagine
a field that changes the speed of the fluid without changing the pressure
of the air ... perhaps a magnetic field acting on a magnetic fluid in
non-magnetic air.  The stream would still get narrower.

> But it seems the air pressure hardly
> changes in a few centimeters to cause the stream to narrow.

That's true.  So you are right to be suspicious of the pressure
gradient argument.

> Or does the surface tension in the water cause it to remain intact as it
> falls, acting like it's in a pipe?

As for the forces from the outside pushing in:
 -- Yes, surface tension contributes.
 -- Air pressure contributes also.
 -- You don't need a pressure gradient, as discussed below.
 -- There has to be some applied pressure and/or some surface tension;
  otherwise the fluid would instantly flash into vapor.
 ++ HOWEVER, none of that helps answer the question that was asked;
  forsooth, the sign of the effect is wrong.

There is a handy concept in fluid dynamics:  Whenever you see a
/curved/ streamline, there is lower pressure on the inside of the
curve and higher pressure on the outside.  This is a direct corollary
of Newton's 2nd law.

Remarkable fact:  In the accelerating fluid, the streamlines are
curved /outward/.  If the streamlines were moving inward in straight
lines, they would crash into each other.  You would get a cone, such
that the fluid reached zero cross section after a finite time.  So
the stream lines simply must curve outwards.  The concave side is
the outside.

That means that in any given slice transverse to the flow, the
pressure is /higher/ in the middle of the stream than it is at
the edges.  The force on the fluid is outward, so as to make it
less cone-like. So the role of external pressure in explaining
the qualitative behavior is much less important than you might
have guessed.

There is a high-school physics concept at work here:  explaining
the inward /motion/ is very different from explaining the inward
(or outward!) /force/.  The inward motion is simply inherited from
the upstream parts of the flow.

A constant transverse velocity plus an increasing longitudinal
velocity would give you "some" curvature, but the actual curvature
is more than that, so a transverse pressure gradient is needed.

This is a very condensed explanation of a complicated phenomenon;
feel free to ask follow-up questions.