Starting with the
theoretical model
After our initial observations it was made clear
that this was a case of laminar flow, as the orifice was significantly smaller
than the container and most containers were cylindrical. By taking these things
into consideration we concluded the following system of energy equations. However in fact due to our initial observations it was made clear that
eventually, the surface rests above the orifice at a certain height. This
variable is not to be confused with the thin layer ‘dy’. As the top layer reflects the thin
layer ‘dy’,
the bottom layer surrounding the orifice reflects a thin layer ‘dx’, as the flow rate is constant.
Notable, the fluid moves in
the negative direction of the y-axis!
~All
work done downward if flow direction of gravity is positive.
~All
work done upward against flow is negative.
~The
change in energy reflects to the work done on the liquid.
~As
the total work done by the
external pressure is zero so it is not included.
Where:
As the displacement is downwards the work
than is also downwards, but as the pressure differs with greater pressure above
than bellow, and it produces negative work. It
is not the difference in pressures that is relevant, it is the external forces
produced, which are opposing the difference in pressures. The internal pressure
does not play a part, and the difference in pressures in the equation arises
from the equilibrium conditions of the Young-Laplace equation. This gives us
the force due to surface tension.
This now yields:
Where:
Additional notable equations and
approximations:
, 
(this is confirmed
experimentally [9])
, 
, when 
,
This expression is true for contact
angles, as 
, 
- is represented in the upper sketch for the
top, same principles apply for the orifice, just with different curvatures and
contact angles. In reality the surface of the fluid is curved along the ‘xz’ plane (at the top) and along the ‘yz’ plane (at
the bottom). In most cases the effective radii at the bottom (at the orifice)
closely resembles the radii of the cross section, i.e. semisphere
– as long as we are talking about Newtonian
liquids, which are the only once we are considering.
Due to that we shall continue to formally
use this approximation 
, as it would not only be difficult to
estimate the curvature at the top experimentally, we lack the needed equipment
to do so. But from all of the initial observation it was quite clear that the
curvature at the top is barely noticeable under the naked eye, which means that
the maximum curvature (minimum radius) is half a sphere. So, we take that to be
our maximum force pulling the liquid back at the maximum height, and meat the
rest of the criteria in the “Error
analysis” section. Therefore as the only way to continue is to approximate
the curvature (at the top) as a maxima, and leave an error margin for it in the
error analysis, we do that. As for the curvature at the orifice, we do take it
as a semisphere as it falls under the met criteria
for forming a droplet. That is in no
shape or form the minimum nor maximum curvature, nor the shape of the droplet
(which depends upon the liquid), but it does fall under valid curvatures. i.e. 
Notable,
the curvature can be proven that is ‘positive curvature’. It is very much
dependent on the contact angle of the fluid to the surface of the container. [4]
This now yields:
The way we continue to expand our
designed model is, by first making an applicable prediction for the simples and
most common cases that may occur, and only once we are satisfied with them do
we make a general model.
A: Common cases
1) Cylinder:
At the point of 
, we see that we get a
stationary point – which remains static,
[7]
The system energy equation thus reduces
to the following solution for the velocity of the jet:
-This
is done just for fun, to show how our model
expanded Torricelli’s law to include stationary
states in case of orifices,
2) Cone:
As we continued investigating
we started using identical cups as containers, since this proved to be a much
better controlled environment. Here we encountered the problem that this no
longer resembled a container with a cylindrical shape. However, the main and
only difference is the radius at the end of the flow. By this, it is meant that
it is easier to measure the overall height at which the flow stops, and by
knowing the dimensions of the container we can get any radius at any height if
so needed.
, where 
and 
are the dominant radius and height of the
cone, respectfully.
At the point of , we see that we see
that we get a stationary point – which remains static,
As, 
, the negative solution
to the system hold’s no physical value.
The system energy equation thus reduces
to the following solution:
-This
is done just for fun, to show how our model
expanded Torricelli’s law to include stationary
states in case of orifices,
