Archimedes to Hawking: Laws of Science and the Great Minds behind Them

By Clifford A. Pickover | Go to book overview

STOKES’S LAW OF VISCOSITY

Ireland, 1851. The frictional force exerted on a sphere moving in a fluid is proportional to the fluid viscosity and the radius and speed of the sphere.

CROSS REFERENCE: STOKES-EINSTEIN RELATION, NAVIER-STOKES EQUATIONS, STOKES’s LAW OF FLUORESCENCE, CLAUDE NAVIER, SIMEON POISSON, AND ADHÉMAR BARRÉ DE SAINT-VENANT.

In 1851, Herman Melville published Moby Dick. The New
York Times
and Reuters news service were founded. British
astronomer William Lassell discovered Ariel and Umbriel,
moons of Uranus. The first YMCA opened in the United States.
Rowland Hussey Macy founded Macy’s department store.

Consider a solid sphere of radius r moving with a velocity v through a fluid of viscosity μ. George Stokes determined that the frictional force F that resists the motion of the sphere can be stated as

Note that this drag force F is directly proportional to the sphere radius. This was not intuitively obvious because some researchers supposed that the frictional force would be proportional to the cross-section area, which erroneously suggest an r2 dependence. When dealing with tumbling molecules, the radius r is usually considered to be the Stokes radius, that is, the radius of a sphere that diffuses at the same rate as the molecule. The behavior of this imaginary sphere accounts for hydration and shape effects. Stokes’s Law tends to be most accurate for small, slowly moving particles in viscous liquids.

Table 8 should be useful in order to get a feel for typical viscosity values μ that are used in Stokes Law. Note that 1 pascal equals 1 kg/(m·s2), and the viscosity of water is, conveniently, about 1 mPa·s or 1 g/(m·s). [In actuality, the viscosity of water depends on temperature. At 293°K (20°C), the viscosity of water is 1.002 cP, where cP stands for centipoise. A centipoise is 1 millipascal second (mPa·s).]

Stokes’s Law has many practical applications. For example, the law is considered in industry when studying sedimentation that occurs during the separation of a suspension of solid particles in a liquid. In these applications, scientists are often interested in the resistance exerted by the liquid to the motion of the descending particle.

Consider a scenario in which a particle in a fluid is subject to the forces of gravity. For example, some older readers may recall the popular Prell

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