Explanation
The project we
have been observing included, but was not limited to the following: “On a
plastic glass (container) create a small orifice and pour water in it. Wait
till the water drips out of the container. You will notice that the water level
will remain above the orifice. Investigate how the height h, between the center
of the orifice and the steady water level is dependent on the relevant
parameters.”
The reasoning
behind why we opted to choose the ‘change in energy’ approach to the first
approach with ‘force/pressure’, is so that we may further understand all of the
shortcomings of the model, compared to the real world. We wanted to understand
the entire process, from the moment we fill the cup to the moment it’s last
drop leaks out, not just how the phenomenon is possible, but also why it is. Whilst
for simpler case the initial approach is clearer, it would be rather unnatural
to tackle the more complicated questions with it.
Once we pour a
‘sufficient’ amount of water into the container, due to the presence of gravity
the water will drip out or form a small jet of water. The flow is highly
obstructed by friction forces such as viscosity and other losses. The velocity
of the water jet is an attainable form of Torricelli’s law and we have expanded it to include stationary states in
case of orifices. However, the
actual phenomenon is not dependent on viscosity, as it is only visible once an
equilibrium position has been reached – all dynamic terms are irrelevant, it is
not only stationary, but also static. The main reason this phenomenon is solely
dependent on the Laplace forces acting on the surface – pulling the water
molecules upward due to surface pressure. Once there isn’t enough water to
actively push out the water i.e. all the static forces cancel out, the water
levels. In an ideal case where the gravity is the only observed force no other
forces oppose it and the water levels at the height of the orifice. At an
attainable case, however the Young-Laplace pressure is greater than the
gravitational pressure exerted upon the orifice, resulting in a stable water
level above the orifice.
The height is
linearly dependent to the surface tension, inversely dependent on the density
of the fluid and gravitational acceleration. In the case of a cylinder
container we can easily see that the height is also inversely dependent on the
radii of the orifice and container. In the other case whilst not clear at first
the same is applicable – however with some minor changes to the equation.
A general solution to the phenomenon is: 
The
solution used by us, which best described the containers we were using is the
following one:
and its
theoretical error estimate is,
