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DRIP-DROP, drip-drop, drip-drop-drip. Dripping taps can be unbearably
unpredictable. If you鈥檝e ever spent a sleepless night listening to leaky
plumbing, you鈥檒l know how annoying it can be. Just as your brain thinks it鈥檚
worked out the pattern, the tap will fox you. The suspense of waiting for the
next drop of cold water to hit a victim鈥檚 forehead was once used in American
prisons to elicit confessions, and drove some prisoners insane.

The same unpredictability has tortured scientists attempting to understand
dripping taps. 鈥淚t鈥檚 a very old problem,鈥� says Jack Swift at the Center for
Nonlinear Dynamics, part of the University of Texas at Austin. Now one American
scientist can put all us out of our misery. Last year, Osman Basaran of Purdue
University in Indiana solved the key equations governing fluid drips and learned
to play on the edge of chaos. He hopes his results will be used in all kinds of
ways, from helping to build better ink-jet printers to making whole new
technologies possible.

It isn鈥檛 hard to see how drops form. To begin with, fluid forms a bulge at
the mouth of a tap or nozzle, which becomes a pear shape as more fluid enters.
The fluid at the bottom begins to fall faster than it is replenished from above,
so the pear develops a neck. The neck thins and pinches off, and the drop falls
(see Diagram).
But why do taps sometimes drip and sometimes produce a
stream of water? Why are drips the size they are? How does one drip affect the
next?

Whether in taps, rivers or ink-jet printers, fluids are devilishly hard to
understand. Physicists know that fluid motion is affected by gravity, pressure,
viscosity and surface tension. The Navier-Stokes equation, which describes how
all these forces behave, was first written down in 1827. The trouble is, for any
situation the equation has an infinite number of possible solutions. How do you
work out which one is right?

One place to start is to see how the flow is constrained: in a pipe, for
example, water can鈥檛 flow through the walls. Given a boundary condition like
this, the infinite set of solutions can often be trimmed down to a single,
correct flow pattern.FIG-mg22795001.JPG

But a dripping tap has no boundaries. Once water leaves, it can go anywhere.
People were so intimidated by this lack of a fixed boundary that for the best
part of a century they didn鈥檛 even try to solve the Navier-Stokes equation for
taps. It wasn鈥檛 until 1994 that Jen Eggers at the University of Chicago had a
go.

Eggers noticed that after the drop pinches off, a thin thread is left behind
which boings back into the next burgeoning drop. This gave him the idea that the
water emerging from the tap could be thought of as a bouncing spring that
produces a drop whenever part of the spring snaps off. So Eggers simplified the
Navier-Stokes equation to make it like that of a bouncing spring, which is easy
to solve. He then set up a computer program which ran the simplified solution
repeatedly to simulate sequences of hundreds of drops dripping from a tap.

Other research groups began to run their own programs based on the spring
idea. Swift and his colleagues at the University of Texas used it to simulate
drips from a ceiling, and found that heating air near the ceiling stops the
drips forming. 鈥淚t鈥檚 amazing what you could do with these algorithms,鈥� says
Swift.

Predicting sequences of drops can mean big money if simulations help to
improve commercially important processes. For example, when the neck of a drop
boings back into the next drop in line, it can shatter, creating what are called
鈥渟atellite鈥� droplets that splash around the main drops. In ink-jet printing,
satellites create nasty fuzz on the page. Satellites can also be a waste of
material in any type of spraying process, from painting cars to coating crisps
with flavour. A new method of printing transistors involves dripping
nanometre-sized drops of semiconductor onto a plastic base. If the transistors
are to work, the drops must be dripped just right.

But Eggers鈥檚 spring model is only an approximation. 鈥淒ripping is not really a
mass-spring system,鈥� says Basaran. As he points out, it鈥檚 hard to tell how
accurate the spring approximation is if you can鈥檛 solve the more precise
Navier-Stokes equations. And when you are trying to predict how a long sequence
of drops will behave, even small errors can be crucial because they accumulate
from drip to drip.

To create accurate simulations of a sequence of drips, Basaran reckoned he
had to understand the flow of fluid everywhere inside a droplet鈥攁n
ambitious goal that most physicists in the field had abandoned. To do this, he
divided the forming drop into a mesh of tiny segments that ended at the boundary
of the drop. He then set his students to work designing a computer program to
calculate the velocity of fluid in each segment, with the aim of working out
where the segment should move next. As the drop changes shape, the computer must
constantly redefine the mesh.

The model confirms physicists鈥� assumptions about how and why drops form.
Without surface tension, there would be no such thing as drips. Surface tension
holds the emerging liquid in a sac dangling from the tap, which then breaks
away. It also accelerates the break-up: once a neck forms, surface tension
squeezes fluid out of the neck. This effect can be so strong that the pressure
in the neck of a falling drop can be hundreds of times that in the tap nozzle,
Basaran has calculated. He also found that viscosity works against the effects
of surface tension, which explains why you can pour sticky liquids like honey or
syrup in a long, thin stream without it breaking into drips. The flow rate and
the diameter of the tap also influence when you get a stream of water and when a
series of drips.

So much for the basics. What Basaran really wanted to look at was sequences
of drops鈥攁t how each drop affects its successor. His software took up to
two days over each drop, and he didn鈥檛 want to wait years for a result, so he
returned to spring-like algorithms. But unlike Eggers, he had a good model of
how drops form, which he could use to check the spring approximation. His new
approximate algorithm predicted drop times that agreed to within a few per cent
of the exact solution, even after hundreds of drops.

Some of his results confirmed what technologists in industry already knew. At
very low flow rates, satellite drops are more likely to form. With higher rates
you get a regular sequence鈥攅very drop the same. Higher rates still produce
drip-drop, drip-drop鈥攅very other drop the same. Increasing the flow rate
further makes the period double again, so there is a repeating pattern of four
drops, all of different sizes and coming out at different intervals. This period
doubling continues until every eighth drop is the same, then every sixteenth,
and before long only every infinitieth drop is the same鈥攁nd that鈥檚 chaos.
Eventually, the drops merge and you get a stream of liquid.

Physicists already knew that dripping taps could go chaotic. Now Basaran鈥檚
algorithm can predict the route to chaos for real taps. His simulations match
photographs of drop sequences perfectly. By calculating just how the system
slides towards chaos, his team has taken prediction to its limits.

But there were some surprises in store. Basaran wondered what would happen if
the flow rate fluctuated, rather than increasing uniformly until the system goes
chaotic. He knew this was the sort of thing likely to happen in industry, when
old or imperfect equipment is used. So starting with a flow rate at which every
drop is the same, he increased the flow till the drip-drop pattern appeared,
then reduced it again to the original value, where the uniform drip-drip-drip
should have reappeared. But it didn鈥檛. The dripping stayed stuck in the
drip-drop, drip-drop phase. Basaran had discovered that
a tap鈥檚 behaviour depends on its history.

This capacity of systems to remember their past is called hysteresis, and
it鈥檚 familiar to scientists who study other systems, such as the way a piece of
iron responds to a changing magnetic field. But no one had thought it might
happen to dripping taps. Imperceptible fluctuations in flow rate or viscosity
can lead to radical change in a tap鈥檚 dripping pattern. The period could be
doubled and redoubled by moving up and down in flow rate, eventually even
becoming chaotic at much lower flows than might have been expected.

Basaran realised that in industrial processes hysteresis could pose a
problem. 鈥淪omething might seem to drip as they want, and then it stops doing
that,鈥� he says. Not what you need if you鈥檝e designed equipment to make evenly
sized chocolate drops, or ink spots of a certain size for printing. Worst of
all, the simulations showed that hysteresis tended to happen for exactly the
flow rates manufacturers are most likely to be using鈥攂etween the slow
flows that lead to satellite drops and the fast ones that cause jetting.

Armed with the new simulations, manufacturers can now at least predict when
hysteresis will occur, and act accordingly. For example, an ink-jet manufacturer
can ensure that the ink droplets aren鈥檛 going to ruin a print run by failing to
dry and smudging the image, or clogging up part of the printer. In coating
things like photographic film and sticky tape, they will want to avoid
fluctuating thickness.

Manufacturers may even be able to put the effect to good use. 鈥淧eople could
get to a pattern of drips that鈥檚 not accessible without manipulating the flow
rate to get there,鈥� says Basaran. This could help some delicate modern
technologies such as DNA chips, designed to perform instant gene profiling,
where enzymes must be deposited in precise amounts. Basaran is particularly
excited about the production of electronic circuits by depositing nano-sized
droplets of semiconductor in particular patterns. 鈥淚t鈥檚 partly the dripping
problem that is holding this back,鈥� he says.

So the next time you鈥檙e stuck at home with a dripping tap, console yourself
with the thought that at least it鈥檚 useful to someone. It may make you curse the
landlord, but it鈥檚 also one of the few chaotic systems that we have actually got
sorted. In theory, at least.