# Factors Affecting Viscosity: Using Poiseuille's Equation to Approximate the Viscosity of Water

This investigation attempts to calculate the viscosity of water using Poiseuille's equation. With temperature kept relatively constant (10 $^\circ$), the factors length and radius of the capillary along with the pressure applied are varied to determine their relationships to viscosity.

## Underlying Physics

Viscosity is a measure of a substance's internal friction. The higher the viscosity, the higher the friction; the slower it flows. This property allows some specific materials classed as amorphous to seem solid while other materials to seemingly defy the laws of gravity and surface tension at extremely low temperatures; achieving superfluidity.

This investigation will focus on the flow of water through capillary tubes. Figure 1 illustrates the ideal laminar flow inside the tubes, where velocity is highest in the centre and decreases the closer it is to the stationary boundary; the capillary. The Hagen-Poiseuille equation will be used to calculate viscosity:

Where $$\Delta P$$ is the pressure loss.

$\mathrm{\mu}$ is the dynamic viscosity.

$\mathrm{L}$ is the length of the capillary.

$\mathrm{Q}$ is the volumetric flow rate.

$\mathrm{r}$ is the internal radius of the capillary.

To calculate the viscosity of water, the equation is rearranged to

Due to the nature of this experiment, $\mathrm{\Delta P}$ can be calculated by

Where $\mathrm{h}$ is the height difference.

$\mathrm{\rho}$ is the density of water.

$\mathrm{g}$ is the acceleration due to Earth's gravity.

This gives the final equation used:

Independently derived by Gotthilf Hagen and Jean Poiseuille, this equation is sufficient for all liquids flowing in pipes with lengths significantly larger than its diameter assuming that the fluid

- is Newtonian
- is incompressible
- does not accelerate within the pipe
- flows in a laminar fashion

## Procedure

### Apparatus

*fig.1*

In preparation for the experiment, 12 capillary tubes were acquired so that six were of different radii and another six of different lengths. The factors were measured using a travelling microscope and Vernier calliper respectively. The temperature of flowing tap water was then measured with a glass thermometer.

The experiment was then set up as shown in fig.2. Varying head height was achieved with a meter stick parallel to the clamp stand. The variables height, capillary length and radius each had six variations with 6 repetitions. Water exiting the capillary flowed directly down the burette except in the case of several experiments within the variations on radius (1.3075; 1.8350; 2.1525; 3.1125) where the water was collected first in a large beaker before being transferred to the burette in separate parts.

## Results

### Uncertainties

For uncertainties in the average of measured diameter, the following were used:

The calibration uncertainty for the travelling microscope was determined to be $\pm 0.005$mm. Substituting into

For the first entry, this gives

Using this method, uncertainties were obtained of the measurements and calculations presented.

Flow rate measured with Varying Height

Flow rate measured with Varying Radius

Flow rate measured with Varying Length

*fig.4: Varying Height*

*fig.5: Varying Length*

*fig.6: Varying Diameter*

### Example Calculation for Viscosity using Poiseuille's Equation

Substituting values:

### Final Uncertainty

The final uncertainty is calculated by

Where $\mathrm{\frac{\Delta U}{U}}$ is the final percentage uncertainty.

$\mathrm{\frac{\Delta h}{h}}$ is the percentage uncertainty in height.

$\mathrm{\frac{\Delta r}{r}}$ is the percentage uncertainty in radius.

$\mathrm{\frac{\Delta Q}{Q}}$ is the percentage uncertainty in flow rate.

$\mathrm{\frac{\Delta l}{l}}$ is the percentage uncertainty in length.

Substituting in the entries for the greatest error gives

Using the average value of $\mathrm{\mu}$ calculated:

This yields the final uncertainty for the viscosity of water calculated by this experiment

## Conclusion

The viscosity of water was calculated to be $3.04 \times 10^{-6} \pm 1.16 \times 10^{-6}$ Pa s. At $10^\circ$C, this is a lot smaller than it should be.

### Evaluation

Possible reasons for such a result would include many systematic uncertainties, as the data does seem consistent internally. Apparatus wise, height measurement could be improved with a more precise instrument. Length measurements could be improved vastly by using a mercury thread to ensure internal radii consistency.

Furthermore, Poiseuille's equation incorporates several simplifications that contradict the nature of water. eg. Water is compressible and has impurities that would alter the ideal result.

Another minor uncertainty stems from surface tension causing a small bulging in the constant head apparatus leading to a minute change in pressures.