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Dropping the Ball (Slowly)
Michael Fowler, UVa
6/12/06
Stokes’ Law
We’ve seen how viscosity acts as a frictional brake on the rate at which water flows through a
pipe, let us now examine its frictional effect on an object falling through a viscous medium.
To
make it simple, we take a sphere.
If we use a very viscous liquid, such as glycerin, and a small
sphere, for example a ball bearing of radius a millimeter or so, it turns out experimentally that
the liquid flows smoothly around the ball as it falls, with a flow pattern like:
(The arrows show the fluid flow as seen by the ball.
This smooth flow
only takes place for fairly
slow motion
, as we shall see.)
If we knew mathematically precisely how the velocity in this flow pattern varied near the ball,
we could find the total viscous force on the ball by finding the velocity gradient near each little
area of the ball’s surface, and doing an integral.
But actually this is quite difficult.
It was done
in the 1840’s by Sir George Gabriel Stokes.
He found what has become known as Stokes’ Law:
the drag force
F
on a sphere of
radius
a
moving through a fluid of viscosity
η
at speed
v
is
given by:
6.
Fa
v
πη
=
Note that this drag force is
directly proportional to the radius
.
That’s not obvious—one might
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 Fall '07
 MichaelFowler
 Friction, Drag force, Sir George Gabriel Stokes, Michael Fowler

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