Liquid Flow and its Mathematical Representations
Sam Otten
MTH 202
Prof. Figa
Otten - 1
Liquid Flow and its Mathematical Representations
Consider a liquid flowing from its container.
You have most assuredly
encountered this phenomenon many times in your life: when filling your car with fuel,
when using the running water in your home, or when getting a refill of soda at your
favorite fast-food restaurant. The “liquid” could also be metaphorically, as in the cases of
heat flowing from a cooling pie, electric charge draining from a capacitor, or
appropriations flowing from a governmental body. These situations exist all around us.
With this paper I would like to demonstrate that liquid flow problems have real world
significance, as well as a strong basis in mathematics, particularly differential equations.
Let’s contemplate a very simple liquid flow situation, so that we can see the
essence of what’s happening. Suppose that water is in a cylindrical tank with a base area
of 1, and this tank has a hole in the bottom with an area of k. If we define the function
y(t) to be the height of the water in the tank (which also corresponds to its water volume)
and we choose the bottom of the tank to be the y-axis, then we can formulate a simple
differential equation for this situation where the flow rate is equal to water height and the
drain area: y’=-ky (the negative sign is included because the water level will be
decreasing). This differential equation can easily be solved, resulting in y(t)=A0e-kt,
where A0 is the beginning water level. This model of water flow seems to make sense,
because as k gets larger, the water height will reach zero (its steady-state) more rapidly.
The inverse of this statement is also true.
Two tanks can also be coupled together, as in the picture on the next page, to
create a slightly more interesting situation. Water from the first tank flows into the second
Otten - 2
tank, and water that flows from the second tank is
pumped back into the first tank. Therefore, this is a
closed environment, where no water is lost from the
system. We will still use tanks with a base area of 1
unit, for simplicity’s sake. Let y1 and k1 be the water
height and drain area respectively for the first tank.
Similarly, let y2 and k2 be the water height and drain
area for the second tank. We can devise a differential
equation for each tank, as we did with a single tank earlier, considering only its own
flow, which would result in y1’=-k1y1 and y2’=-k2y2. Progressing logically, we can
conclude that the first tank’s water level is affected negatively by its own water flow, but
positively by the flow from the second tank. Using the same reasoning for the second
tank we can generate a system of differential equations that represents the situation:
y1’=-k1y1+k2y2
y2’=k1y1 k2y2
Let’s consider the steady-states for different configurations of the two-tank system.
Theoretically, if the two drain areas are of equal size (k1=k2) and the tanks start with
equal water levels, there will be change in water level. If the drain areas are equal but the
tanks do not start with the same amount of water, their steady-states will still be with
equal amounts of water in each tank. However, if one of the tanks has a larger drain area
(k1<k2 or k1>k2) then the tank with the larger drain area will have a steady-state of zero,
while the tank with the smaller drain area will have a steady-state of all the water from the system.
Otten - 3
Of course we can add a third tank to our water
flow system. Using the same procedure as before to
devise a system of differential equations, we produce
the following:
y1’=-k1y1+0y2+k3y3
y2’=k1y1 k2y2+0y3
y3'=0y1+k2y2 k3y3
This system is much more complex than the one- or twotank models. In fact, under many circumstances the water
levels of the three-tank system will oscillate, never reaching
a steady-sate. However, the causes and implications of that fact will not be discussed in
this paper.
Of course, it is not always convenient for liquids to simply flow out of a hole in
the bottom of the container. Some situations may require it to be pumped out of the top,
and in these cases it is very important to know the amount of work being done so proper
arrangements can be made for electrical capacity, pump power, etc. The general formula
for 'work' is below:
Work=Force x Distance
This formula can be simplified, because when liquids are being pumped the force that
must be overcome is the force of gravity, so mass now becomes the determinant factor in
the amount of gravity needed to be overcome:
Work=mass x Distance
Otten - 4
Unfortunately, to measure mass in a complex system of liquids is not an easy task. A
more reasonable factor to measure would be volume. Then, using the density of the
liquids, the mass could be calculated:
Work=density x volume x Distance
Finally, the work done when pumping a
liquid may also be represented by an
integral. This integral includes the density,
the distance the liquid is moved, the crosssectional area of the tank and the thickness.
The integral is shown below:
The mathematical representations we have been discussing have legitimate
applications in the real world. An industry incorporating liquids can use models similar
to these to calculate the flow it needs or desires and how to achieve it. It can determine
what size of drainage area is needed, or how much energy will be needed. A city that is
setting up its water system can estimate the amount of water flow required, and then use
differential equations to determine how to achieve that flow. A company manufacturing
soda machines will want to control the flow of the drink, so that it fills the cup rapidly
enough to avoid impatience, but not so fast that it is hard to avoid spillage. The delicate
balance lies in the mathematics. As was stated earlier, the "liquid" could be metaphorical
Otten - 5
while using the same type of mathematical models. For instance, the flowing three-tank
system could represent the three socioeconomic classes in our country, which would give
new meaning to Reagan's "trickle-down" economics. I have mentioned only a few, but
there are countless examples in our world where the models discussed in this paper exist.
References
Anton, H., Bivens, I., et al. (2002). Calculus: 7th Edition, (p. 495-509). New York: John
Wiley & Sons, Inc.
Figa, J. (2003). Selected Lectures from Calculus 2, (February 21). Grand Valley State
University: Mathematics Department.
Hughes-Hallett, D., Gleason, A., et al. (2000). Calculus: 2nd Edition. New York: John
Wiley & Sons, Inc.
Kendig, K. (Feb. 1999). When and Why Do Water Levels Oscillate in Three Tanks?
Mathematics Magazine, (p. 22-31). Washington, D.C.: Mathematical Association
of America.