Engine Oil’s Viscosity
An experimental engine oil with density of 900 kg/m^3 is being tested in a laboratory to determine its viscosity. A 1-mm-diameter steel sphere is released into a much larger, transparent tank of the oil (Figure 6.24). After the sphere has fallen through the oil for a few seconds, it falls at a constant speed. A technician records that the sphere takes 9 s to pass marks on the container that are separated by 10 cm. Knowing that the density of steel is 7830 kg/m^3, what is the oil’s viscosity?
Approach
To calculate the oil’s viscosity, we will use an equilibrium-force balance involving the drag force to determine the speed at which the steel sphere falls through the oil. When the sphere is initially dropped into the tank, it will
accelerate downward with gravity. After a short distance, however, the sphere will reach a constant, or terminal, velocity. At that point, the drag F_{D} and buoyancyF_{B} forces that act upward in the free body diagram exactly balance the sphere’s weight ω (Figure 6.25). The viscosity can then be found from the drag force following Equation (6.16).
F_{D}\approx 3\pi \mu d v (Special case for a sphere : Re < 1) (6.16)
Finally, we will double-check the solution by verifying that the Reynolds number is less than one, a requirement
when Equation (6.16) is used.
