458 10 Transport Processes
whereFfis the frictional force on the object andris the radius of the object. The negative
sign indicates that the frictional force is in the opposite direction to the velocity of the
object.EXAMPLE10.10
An iron sphere of density 7.874 g mL−^1 is falling at a steady speed in glycerol at 20◦C. The
density of glycerol is 1.2613 g mL−^1 and its viscosity at this temperature is 1.49 kg m−^1 s−^1
(1.49 Pa s). If the radius of the sphere is 5.00 mm, find the speed, assuming that Stokes’ law
is valid.
Solution
If the sphere has a constant speed the frictional force must be equal in magnitude to the
gravitational force, which (corrected for buoyancy) is equal to
∣∣
Fg∣∣
4 πr^3
3(
ρFe−ρgly)
g
4 π
3(
5. 00 × 10 −^3 m) 3 (
7874 kg m−^3 − 1261 .3kgm−^3)(
9 .80 m s−^2) 0 .0339 kg m s−^2 0 .0339 Nv∣∣
Fg∣∣6 πηr
0 .0339 kg m s−^2
6 π(
1 .49 kg m−^1 s−^1)(
5. 00 × 10 −^3 m) 0 .242 m s−^1The Reynolds number can be approximately applied to motion of a spherical object
through a liquid by replacing the radius of the tube by the radius of the object and
replacing the mean flow speed with the speed of the object through the liquid.EXAMPLE10.11
Obtain an approximate Reynolds number for the falling sphere in Example 10.10. Comment
on your result.
SolutionR
R〈u〉ρ
η
(
5. 00 × 10 −^3 m)(
0 .242 m s−^1)(
1261 .3kgm−^3)1 .49 kg m−^1 s−^1 1. 02The flow is almost certainly laminar.Exercise 10.9
a.Find the speed of the iron sphere in Example 10.10 if it is falling in water at 20◦C instead
of glycerol assuming that Stokes’ law is valid. The viscosity of water at 20◦C is 0.001002 kg
m−^1 s−^1 and its density is 998.2 kg m−^3. Find the value of the Reynolds number and decide
if Stokes’ law is valid.
b.Find the speed of an iron sphere falling in glycerol at 20◦C if the radius of the sphere is
5.00 cm. Calculate the Reynolds number and comment on your result.
c.Estimate the diameter of the largest iron sphere for which Stokes’ law is valid if it is falling
in glycerol at 20◦C.