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## : re: dripping spouts: capillaries, meniscus and venturi again

### Gavin Stairs on thu 8 aug 02

At 04:15 PM 07/08/2002 +0930, Ivor wrote to Snail Scott:
>Capillary activity is a curious phenomenon.
>
>A glass tube drawn out to give a bore of somewhat less that 1/10 and =
>1/100 of a millimetre can give a capillary lift of about ten centimetres =
>or more. So we would expect water to flow out of a tube of similar =
>diameter but shorter length. It does not and I cannot explain that. But =
>a Physicist might and also provide the mathematical calculations to =
>validate the proposition.

Hi Snail and Ivor,

Capillary action is another manifestation of adhesion and cohesion of the
surface layer of a liquid. Not only will water be drawn into and up a
tube, but mercury will be driven down and out of a similar
tube. Why? Because water adheres to a clean glass surface, while mercury
does not. If the glass is not clean, but has a waxy film on it, the water
column will act more like the mercury, because the water will not adhere to
the wax. Like water beading up on a car's waxed paint job, but sheeting on
a carefully cleaned window.

This makes the miniscus, or the curved surface of the liquid. In water and
glass, the edge goes up, while with mercury and glass it goes down, leaving
a rounded, convex surface. You can see the meniscus in an ordinary
drinking glass. It is the up-turned edge on the otherwise flat surface of
the water.

It is the meniscus which drives the capillary. Adhesion at the wall of the
tube draws the water up, while surface tension pulls the rest of the
surface up to a curve which is a function of the diameter of the tube. If
the angle of this curve at the glass surface is greater than a critical
angle, adhesion will pull it still higher. The surface curve follows,
while getting deeper as it goes. At some level, the angle will reach the
critical angle, and the capillary will be at balance. At this point the
pressure in the water at the capillary surface will be below atmospheric
pressure. The water is held up by the tension in the surface and the
tension at the adherent edge. You can calculate these forces from the
height of the column, the diameter of the capillary tube, and the angle of
contact.

Now notice one thing: the capillary depends on the geometry of the
tube. If that tube widens, as it does at the top of the tube, the liquid
will not rise past the point in the widening where the critical parameters
are met. Water will not flow out of the upper end of the tube. To do so,
it would have to get energy from some source. At equilibrium, there is no
such source. If there were such a source, you could pump water from low to
high, and you would have a potential perpetual motion machine, which is
prohibited in nature. However, water can evaporate from the upper surface,
which will cool the remaining water, but also draw more water up. This
works because the heat in the water is the energy source. Keep supplying
heat and the water transport will continue. This is how a water jar made
of earthenware cools water.

In the case of the teapot spout, I think something else is at work. It is
our old friend Venturi. While you are pouring tea from the spout, tea is
running past the upper surface of the hole in the spout. That means the
dynamic pressure in the tea is lower than the static pressure in the same
stream, while the dribble, which is not moving, will be at atmospheric
pressure, higher than in the moving stream. So tea will be drawn up the
hole from high pressure to low pressure. The hole geometry is critical,
because this phenomenon is dependant on the momentum balance in a moving
stream of liquid. If the surface curves such that the tea is drawn around
the curve and into the hole, the thing will work in reverse, and the
dribble will get worse, because the pressure at the entry to the hole will
be closer to the static pressure in the stream, which is atmospheric
pressure. So gravity may draw a bit of water down the tube, rather than
back up. The groove is presumably to help define a reliable
geometry. What is needed is a hole which meets the surface with a sharp
edge on the upstream side, and an axis which is perpendicular to the flow
stream, or even angled in the direction of the stream. Also no bumps on the
downstream side. The glaze surface needs to be flat and smooth. If there
is too much of a rounded depression at the mouth of the hole, an eddy may
develop there, and that will alter the pressure balance so that the pump
won't work. The downstream groove makes it easier to get a surface which
points downstream at the hole mouth, thus enhancing the pump. You can see
the same shape in various vent openings in racing (and street) cars, and
airplanes. The reverse scoop is used to draw air out of an opening and
into the passing stream. It is used for inducing cooling air flow and
similar purposes.

Dynamic and static pressure are used in a device called a Pitot tube to
measure the airspeed of an airplane. It is mounted on a probe in the air
stream in front of the airplane, with one opening to the front to measure
the static pressure and one opening to the side to measure the dynamic
pressure. The difference is a measure of the air speed.

Gavin