Torricelli's Law and Fluid Dynamics
MATH 2413 Research Project 2002

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Representation

The apparatus used is a pipe, closed at the bottom and open at the top, with a hole near the bottom on the side of the pipe. Water is poured into the pipe from the open end while the hole is covered. Then, when the hole is opened, water escapes from the container until the water level is at the bottom of the hole.

We seek to fully describe this process, by examining the different characteristics of this system as they change over time.

Several properties remain constant:
>> R, the radius of the pipe.
>> r, the radius of the hole through which water escapes.
>> A, the cross-sectional area of the pipe.
>> a, the cross-sectional area of the hole.

Note that and , so only two variables are necessary. These essentially define the entire system.

Three properties will vary with time, and thus are functions of time:
>> h(t), the height of the column of water. This is measured from the bottom of the hole to the top of the column of water.
>> v(t), the horizontal velocity of water as it escapes the pipe through the hole.
>> V(t), the volume of water remaining in the pipe, disregarding water below the bottom of the hole.

Note that there are three initial conditions: h(0) = h₀, v(0) = v₀, and V(0) = V₀.

Time is to be measured such that t = 0 is when the hole is first opened and water is first released. It can then extend to infinity.

Through the initial experiment we seek to fully understand how the three functions of t as the system progresses.