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lower than the pressure acting on the leading surface of the sphere, and this pressure
differential gives rise to the pressure drag component. When the sphere is spinning—
say, clockwise—about a horizontal axis passing through its center, as shown in
Figure 6-12 , the fluid passing over the top of the sphere will be sped up while the fluid
passing under the sphere will be retarded.
Figure 6-12. Spinning sphere
Remember, because of friction, there is a thin boundary layer of fluid that attaches to
the sphere's surface. At the sphere's surface, the velocity of the fluid in the boundary
layer is 0 relative to the sphere. The velocity increases within the boundary layer as you
move further away from the sphere's surface. In the case of the spinning sphere, there
is now a difference in fluid pressure above and below the sphere due to the increase in
velocity above the sphere and the decrease in velocity below the sphere. Further, the
separation point on the top side of the sphere will be pushed further back along the
sphere. The end result is an asymmetric flow pattern around the sphere with a net lift
force (due to the pressure differential) perpendicular to the direction of flow. If the
surface of the sphere is roughened a little, not only will frictional drag increase, but this
lift effect will increase as well.
Don't let the term lift confuse you into thinking that this force always acts to lift, or
elevate, the sphere. The effect of this lift force on the sphere's trajectory is very much
tied to the axis of rotation about which the sphere is spinning as related to the direction
in which the sphere is traveling (that is, its angular velocity).
The magnitude of the Magnus force is proportional to the speed of travel, rate of spin,
density of fluid, size of the object, and nature of the fluid flow. This force is not easy to
calculate analytically, and as with many problems in fluid dynamics, you must rely on
experimental data to accurately estimate it for a specific object under specific conditions.
There are, however, some analytical techniques that will allow you to approximate the
Magnus force. Without going into the theoretical details, you can apply the Kutta-
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