A hollow coffee stirrer, used as the capillary tube, was inserted into the side of a 710-mL plastic water bottle and fastened with plumber's putty. This was the equipment suggested at a workshop attended by one of the authors (TAW) at the Summer 2001 AAPT Meeting in Rochester, N.Y. However, instead of visual measurements of the height of the fluid reservoir as a function of time, we suspended the bottle on a force probe connected to a Vernier⁴ interface and accumulated the data with the Vernier Logger Pro software. Instead of height, we measured the corresponding weight as a function of time and fit the data to

where W_1BG is the weight of the bottle and fluid remaining after the flow stops. The subscript 1 anticipates use of this equation in a later section. The original bottom of the plastic bottle was cut off, the bottle was inverted, and the tube was placed near the bottom of the uniform diameter portion of this reservoir. Another piece of stirrer was used at the open end to maintain a constant geometry of the flexible plastic when supporting the amount of filled fluid. A data-sampling rate of one point/s was used for all experiments.

Equation (2) is exactly analogous to radioactive decay with a constant background added. Figure 1(a) presents both data and a graph of Eq. (2) using the best-fit values for the adjustable parameters of W₁₀, ₁, and W_1BG. For instructional and analysis purposes the data were imported into a spreadsheet (Microsoft Excel was used in this study) and sliders were constructed for each of the three parameters so students may study how a variation of each parameter affects the mathematical fit to the data. Students are asked to first find the best visual fit by manipulating the sliders, and finally to use Excel's Solver⁵ add-in tool to obtain the final best fit shown in Fig. 1. Solver is an optimization tool that we used to minimize the sum of squares of differences between the data and the corresponding values obtained from Eq. (2). Figure 1(b) is the same data and fit shown on a log-linear scale but with the background term subtracted. The overlap of the two graphs demonstrates excellent agreement except for small differences at large times.

Figure 1a.

Figure 1b.

Using the Solver best-fit value of ₁ = 0.00642, a tube length L = 13.9 cm, a diameter of the reservoir = 6.81 cm, and known properties of the fluid (water was used in all experiments), the diameter of the stirrer was found to be 1.9 mm from Eq. (1), which is close enough for our purposes to a direct measurement of about 2.1 mm by using a folding micrometer. The hollow cross section of the stirrer was not uniform, but more closely resembled that of a cylindrical cavity with a thin solid rod attached to the inner surface. We don't know what effect this has on Poiseuille's equation. For quantitative purposes it would be best to use a uniform capillary tube, but this was not the intent of our experiment. However, we now simply use this experiment in various lab courses as a measurement of the effective inside diameter of the stirrer. In lab courses where statistical treatment of data is emphasized, students routinely determine the tube diameter to be within 0.1 mm of the directly measured quantity. Historically this experiment was used to measure the viscosity of various fluids.

The exponential time dependence of column height exactly models radioactive decay⁶ where the number of nuclei present at time t is given by N = N₀exp(–t). The decay constant, , represents the probability per unit time for a nucleus to decay and is related to the half-life, T_1/2, the time for half of the original number of nuclei to decay, by T_1/2 = ln 2. In the fluid flow experiment the value for may then also be determined from this relationship by measuring the time for some reference height of fluid to fall by a factor of two.