The Viscosity of Water Using Poiseuille’s Law

The viscosity of a fluid is a measure of the internal friction and resistance to force, known by experimentation to vary with temperature. This investigation sought to test the viscosity of water at different temperatures, finding values close to, and generally within experimental uncertainty of, values from an empirical model. At 16.2 ◦C, the theoretical model suggests η = 1.10 mPa·s, and this investigation achieved η = 1.2± 0.1 mPa·s.

Introduction

Viscosity governs the resistance to motion of a substance under an applied force. In 1838, French physicist Jean Poiseuille derived an equation for resistance to flow, finding it to be directly proportional to the length of pipe L and viscosity, and inversely proportional to its radius a.

He used this to relate the volumetric flow rate Q to the pressure difference ∆p through a horizontal pipe of uniform cross- section [1], giving Poiseuille’s law for laminar flow, ∆p = 8ηL πa4 Q (1) for some measurable η, the viscosity of a fluid.

This requires laminar flow, which is defined as the smooth sliding of infinitesimal parallel non-mixing layers [2].

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This equation was later discovered independently by Gotthilf Hagen in 1839, and has since been derived from the Navier- Stokes equations, first by Stokes and Hagenbach. The pressure difference between the ends of the pipe is ∆p = ρgh for density ρ (value not used in calculation) and height of fluid above the non-open end h. Using this and dmdt = ρQ, η can be calculated experimentally for a measured rate of change of mass as, η = πgha4 8L 1 dm dt (2) with dimensions Nm−2s = Pa·s (pascal-second) and using constant g = 9.

81ms−2, the acceleration due to gravity [5].

A fluid for which the viscous stress varies linearly with the strain rate can be described as Newtonian. Many common liquids and gases, such as air and water, are assumed to be Newtonian, which means η is constant for a fluid at constant temperature, with several empirical models based on experimental values. One of the simplest is Vogel’s three-parameter exponential [3], η = Ae B T−C (3) with experimentally calculated fitting parameters A = 2.41× 10−5 Pa·s, B = 247.8 K and C = 140 K [6].

Method

As shown in Fig. 1, a cubic water container of volume 8×10−3 m3 was filled with water to h = 10.00± 0.07 cm, FIG. 1: The experimental set-up of the equipment, showing (a) the container with water of height h above the centre of a capillary, (b), of diameter 2a, vertically above a beaker (c) positioned on a digital mass balance (d). measured by a ruler, at different temperatures around 16 ◦C, measured by a digital thermometer.

The container was fitted with one of three capillaries, and we opted to use the middling width capillary (’blue’) with diameter 0.75 ± 0.02 mm, measured with a travelling microscope three times prior to use. The length of each capillary was measured by a ruler and all were found to be L = 15.00± 0.07 cm. An automated program on the connected computer recorded 180 measurements of the mass from the balance, taking approximately 0.3 s to take each measurement and with a 1.0 s pause before beginning the next measurement.

This was completed three times for the blue and white capillaries, but only once for the red (most narrow). The temperature and height of the water in the container were monitored during measurements, but not controlled. For each dataset, a least squares fitting was completed to plot a linear trendline with the data, with the slope and its error also calculated. This slope could be used along with measured constants h, a and L in Equation (2) to obtain a value of the viscosity of water, η.

Results

Table 1 shows the values of η for three different temperatures of water using the blue capillary, along with theoretical values calculated from Equation (3), with uncertainties as calculated in the Appendix. Fig. 2 shows the rate of change of mass for three measuring periods using the blue capillary, with residuals for a linear and quadratic model to the 16.2 ± 0.1 ◦C data set shown (similar is found for other temperatures). 1 S. E. Shackleton The Viscosity of Water using Poiseuille’s Law A χ2 evaluation was performed on each dataset. Taking the blue capillary at 16.2 ± 0.1 ◦C data as an example, with an error on mass measurements as 0.01 g, a linear fit gives reduced value χ2ν = 66.5, and a quadratic fit has χ2ν = 10.5. The normalised residuals in Fig. 2 show a pattern for the linear fit, but for the quadratic model they seem symmetrically distributed without a clear pattern.

Discussion

The values of η measured using the blue capillary, shown in Table 1, are all close to the empirical model values, although not quite accurate to within experimental uncertainty. However, those measured with the red and white capillaries, mean 0.8±1.5 mPa·s and 1.4±0.1 mPa·s respectively for 16.5 ◦C, were less accurate. Equations (1,2), governing viscosity, assume laminar flow through the capillary [4]. For the wider capillaries, this assumption may not be valid, as if the speed of the water is too great it will have turbulent flow.

This had an effect on the accuracy of the white and potentially blue capillaries. The red capillary is the most narrow, and had an extremely slow flow rate. Thus the uncertainty in the mass balance readings had a larger effect. Further, the automated sample rate was too long, causing step increases which meant that the gradient could not be reliably calculated.

This meant the error on the slope dmdt dominated the uncertainty on η (see Appendix), making it larger than the value of η itself, meaning this value is highly unreliable. Another loss of accuracy in measurements for η arises FIG. 2: The rate of change of mass over the measuring period, using the blue capillary (similar was produced for white and red). Each repeat was at a different water temperature. Residuals are shown for the 16.2± 0.1 ◦C line, for a linear and quadratic χ2 model. Error bars on mass are too small to be seen. from the assumption of negligible change of height, h, during the measurements.

Due to the small volume expended during measurements and the large cross- sectional area of the water container, this height difference was generally of millimetre order and thus was ignored, however for the widest capillary (white), which had the highest volumetric flow rate, this is less valid. The χ2 evaluation previously detailed shows a reasonable fit of the rate of change of mass to a quadratic model. While not a perfect model, the value of χ2ν is dependent on the choice of uncertainty of mass measurements. 0.01 g was chosen because there was frequent fluctuation on the scale from the hundredths of grams, however this could be larger.

We opted to take measurements over a long time period and using digital data acquisition to reduce the effects of the error on mass measurements. However, calculations of η from Equation (2) were assuming a linear rate of change of mass, for its slope to be calculated. As the quadratic model fits better, Equation (2) could be altered to include a time derivative of this model, to achieve a more accurate value of viscosity. The theoretical prediction of water viscosity can also be made more accurate using higher parameter exponentials, although little documentation exists regarding their parameters or uncertainties.

Conclusions

In this investigation, the viscosity of water was measured for different temperatures using capillaries of different diameters using Poiseuille’s law for laminar flow. Those measured with the medium width capillary were most accurate to Vogel’s empirical model, which states that viscosity varies as a reciprocal exponential of temperature.

For this capillary, a mean value of η = 1.2 ± 0.3 mPa·s is within experimental uncertainty of ηV ogel = 1.10 mPa·s. For the wider and narrower capillaries, the calculated values of η were further from the empirical model. The main sources of uncertainty were the error on measurements of diameter using the travelling microscope and the error on the slope. This could be improved by using more precise digital instruments and modifying Equation (2) to include a quadratic slope.

References

1. Pfitzner, J., ”Poiseuille and his law”, Anaesthesia, 1976, Vol. 31, pp 273-275.
2. Sutera, S. P. and Skalak, R., The ”History of Poiseuille’s Law”, Annual Review of Fluid Mechanics, Jan. 1993, Vol. 25:1-20.
3. Andrade, E. C., ”Viscosity of Liquids”, Nature, Jan. 23, 1932, Vol. 129.3247, pp 125-126.
4. Young, H. D. and Freedman, R. A., University Physics, 13th Ed., Pearson Addison-Wesley, San Francisco (2012), pp. 389- 90.
5. CODATA Value: Standard Acceleration of Gravity, 2018. NIST Reference on Constants, Units, and Uncertainty https://physics.nist.gov/cgi-bin/cuu/Value?gn As of Nov 2019.
6. Engineering Properties of Water, 2019. Engineers Edge https://www.engineersedge.com/physics/water density viscosity specific weight 13146.htm As of Nov 2019.