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Vertical motion

Viscosity of water

The logarithm to base 10 of the terminal velocities in water vterm in cm/s versus the logarithm to base 10 of the radius r in cm of five metal bearing balls, different in size but with the same density. Data are taken from J.P. Owen and W.S. Ryu, Eur. J. Phys. 26, 1085 (2005). The straight line fit to the data is given by log(vterm) = 2.3 + 0.46log(r).

Contrary to the very viscous syrup which has been used in the previous experiment and for which the terminal velocities of the metal bearing balls were clearly in regime I, for water it appears that its viscosity is such that the terminal velocities of the metal bearing balls are in regime II. This is most clearly shown in an experiment elaborated by Julia P Owen and William S Ryu, for which the resulting dependence of the terminal velocity v_term on the radius r of the metal bearing balls, all with the same density of ρ_s=8.02×10³ kg/m³, is represented in the above figure.

Let us recal that in regime II the terminal velocity depends on the radius according to

v_term = β √r     where     β = √(4πρ_sg/3C₂)

When one takes the logarithm of the above expression, one obtains

log(v_term) = log(β) + 0.5×log(r)

Owen and Ryu find 0.46±0.07 for the coefficient in front of log(r). That result shows thus good agreement with the assumption that the experiment is performed under the conditions of regime II.
For log(β) they obtain the value of 2.3. And, since they work with the logarithm to base 10 and with units cm instead of with units m, the latter result implies β=200 √cm/s=20 √m/s.

Furthermore, at Princeton (NJ, USA), where the experiments were performed, one has for the gravitational acceleration g=9.8 m/s². Hence, we may determine C₂=4πρ_sg/3β²=823 kg/m³.
However, as we will see in the following, the latter formula, which is perfect for solid spheres in air, cannot be applied in water because of the buoyant force in water. As we will lateron see, the latter effect can be incorporated in the above formula on substituting ρ_s by ρ_s-ρ_water. In that case one finds C₂=4π(ρ_s-ρ_water)g/3β²=720 kg/m³, which value is of the same order of magnitude as the density of water.

water