Why does the stream of water from a faucet become narrower as it falls?

Asked by: Mike

Answer

This question refers to a rather well-known phenomenon
that everyone can observe. Just turn on the water in your
bathroom, and turn it down so it flows in a steady,
non-turbulent manner. You can observe that the stream
gets narrower as towards the sink, as it falls.

Why does this happen? The answer is really simple,
only very simple principles need to be known to understand
the behavior of the stream.

First, we assume 'steady' flow. By that, we mean that
the flow does not change in time, the stream stays and
behaves the same from one moment to the next. In this
case, there is one simple observation:

The amount of water in any section of the stream
stays the same.

This leads us to the next step:

The amount of water entering any section of the stream
must equal the amount of water leaving that section at
any time.

Since these observations are true for _any_ section of
the stream, it can be generalized to the following statement:

The amount of water running through the cross section
of the stream per unit time at any point is the same.

Now, how can we write the amount of water flowing through
the cross section of the stream in terms of other quantities?
It requires a little thought, but it is not too difficult.

First, it obviously depends on the cross section. The more
cross section there is, the more flow there will be per
unit time. (In analogy with a highway, the wider the highway,
the more traffic it can accommodate.)

Second, it also depends on the _speed_ of the flow.
The more speed, the more flow. (Again, in analogy with
a highway, the faster the cars are traveling, the more
traffic can run through.)

Lastly, the last dependence is on the density of the
fluid -- in general it can change from point to point
within the fluid (although it is constant to a very
good approximation for your tap water).

So, we arrive at the following equation:

_{1} * v_{1} * A_{1} = _{2} * v_{2} * A_{2}

Where _{1} and _{2} are densities at the two
cross sections in question, v_{1} and v_{2} are the
velocities, and A_{1} and A_{2} are the areas.

As mentioned before, the density of water is pretty
much a constant. So we have _{1} = _{2}, which leaves
us with:

v_{1} * A_{1} = v_{2} * A_{2}

Now, our observation suggests that, at lower points
in the stream, the cross section is smaller (i.e. the
stream is narrower). For the last equation to hold,
it must be true that the velocity at the lower part
of the stream must be _higher_ (so that the product
remains the same). How can that be? The answer is simple:
Gravity. Like all objects, a stream of water also
accelerates as it falls. Since the density of water is
a constant, and water tends to hold together (cohesion),
the only thing that can happen is the narrowing of the
stream -- which is exactly what happens.
Answered by: Yasar Safkan, Ph.D. M.I.T., Software Engineer, Istanbul, Turkey

'Physicists like to think that all you have to do is say, these are the conditions, now what happens next?'