The falling-sphere method is used to determine the viscosity of a high viscosity fluid, namely glycerol. The effect of temperature on the viscosity of glycerol is also determined. Stokes's law is used as the basis for calculations with corrections applied for experimental procedures such as finite falling distances and edge effects.

The falling-sphere method is a simple way to study high-viscosity fluids at varying temperatures. It can be used to find the viscosity of a fluid using the effects of gravity on a smooth sphere falling through a standing fluid. Stokes's law provides the basic necessary theory behind this method. Because Stokes's law requires use of a fluid of infinite volume, it must be modified through the work of Gibson and Jacobs to account for the effects of limitations on the fluid caused by experimental procedure [1]. The viscosity of a fluid is a useful quantity to have available as a reference to determine if a fluid would be good in a certain capacity, for example, as a lubricator alternative to oil for use in vehicles. It would be unwise to use a very viscous substance in this capacity because it would create too much friction, so fluids comparable to oil in their viscosity should be investigated first.

The force needed to separate molecules of the fluid according to Stokes is

where R is the radius of the sphere, n is the viscosity of the fluid, and vc is the velocity of the sphere through the infinite fluid. This force can be set equal to the gravitational force modified to account for the buoyant effect as follows

where pS is equal to the density of the sphere, pL is equal to the density of the liquid, and g is the acceleration due to gravity. Constant velocity can be set equal to L/t (distance/time) and the equation solved for n as

The velocity must be modified for the compression of the fluid by the cylinder walls (known as edge effects) by

where x is the ratio of sphere diameter to cylinder diameter. The velocity must also be modified for the finite falling distance by

where y is the ratio of sphere diameter to total liquid height.

Continuous velocity can now be described as

The new values for vc can now be substituted into equation (3) to yield

In this format, all variables necessary to calculate n can be measured directly except for density, which can be easily calculated from measurements [1].

The viscosity of glycerol is determined by the falling-sphere method. A cylinder is filled with glycerol and two marks placed at the ends of the distance the sphere will be timed over. The cylinder is placed in a water bath to maintain a constant temperature through each trial.

A tube is held perpendicular to the center of the surface of the glycerol so the end is a few centimeters below the surface of the fluid. This guides the sphere in dropping through the center of the cylinder and helps to keep air bubbles from adhering to the sphere. It also slows the decent of the sphere to the fluid surface [1].

The mass of each sphere, the identity of its makeup, and its diameter is determined and recorded, as well as the temperature of the glycerol. The sphere is dropped down the tube, and the time it takes to travel between marks recorded. If the speed is too fast to record, a sphere of lower density should be to make the time measurable with some degree of accuracy. The temperature of the glycerol or the properties of the sphere are varied and the trials repeated until enough data are gathered to have a good representation for calculations.

The density of glycerol at varying temperatures is found using its coefficient of expansion. The density of the spheres is determined by finding the average volume and mass of each if using a fairly consistent source, such as steel bb's, or using individual measurements if there is a significant amount of variation, such as in glass beads.

[1] A. M. Halpern, Experimental Physical Chemistry:
A Laboratory Text. 2nd ed. (Prentice Hall, Upper Saddle River, NJ 1997).

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