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reaches a terminal velocity v at which the viscous retarding force F
T
plus the buoyant(Archimedes ) force F equals the weight mg of the
A
sphere. Let  be the density of the sphere and  the density of the

4
fluid. The weight of the sphere is then   R  3 g and the buoyant
3
4
force is  R  3 g  ; when the terminal velocity is reached, the total
3
force is zero and

4 3 4 3
 R  g  6  Rv   R  g . (6.30)
T
3 3
When the terminal velocity of a sphere of known radius and
density is measured, the viscosity of the fluid in which it is falling
can be found from the equation above

2
2 R g
  (   )  . (6.31)
9 v T

Conversely, if the viscosity is known, the radius of the sphere
can be determined by measuring the terminal velocity. This method
was used by Millikan to determine the radius of very small

electrically charged oil drops (used to measure the eleetrical charge
of the individual electron) by observing free fall in air.

7.8 Reynolds Number

When the velocity of a fluid flowing in a tube exceeds a

certain critical value (which depends on the properties of the fluid and
the diameter of the tube),
the nature of the flow
becomes extremely

complicated. Within a
very thin layer adjacent to
the tube walls, called the

boundary layer, the flow
is still laminar(fig.7.10a).
The flow velocity in the
boundary layer is zero at

the tube walls and
increases uniformly
throughout the layer. The

properties of the boundary
Figure 7.11

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