Figure 8-12. Comparison of two dimensional velocity fields.
In comparison, consider the two-dimensional velocity field V =yˆe 1 , 0 ≤y ≤h,
which is illustrated in figure 8-12(b). Here the velocity field may be thought of as
representing the flow of fluid in a river. The curlof this velocity field is
curl V =∇× V =
∣∣
∣∣
∣∣
ˆe 1 ˆe 2 ˆe 3
∂
∂x
∂
∂y
∂
∂z
y 0 0
∣∣
∣∣
∣∣=−ˆe^3.
In this example, the velocity field is rotational. Consider a spherical ball dropped
into this velocity field. The curl V tells us that the ball rotates in a clockwise direction
about an axis normal to the xy plane. Observe the difference in velocities of the water
particles acting upon the upper and lower surfaces of the sphere which cause the
clockwise rotation.
Using the right-hand rule, let the fingers of the right hand move in the direction
of the rotation. The thumb then points in the −ˆe 3 direction.
The curltells us the direction of rotation, but it does not tell us the angular
velocity associated with a point as the following example illustrates. Consider a basin
of water in which the water is rotating with a constant angular velocity ω=ω 0 ˆe 3.
The velocity of a particle of fluid at a position vector r =xˆe 1 +yˆe 2 is given by
V =ω×r =
∣∣
∣∣
∣∣
ˆe 1 ˆe 2 ˆe 3
0 0 ω 0
x y 0
∣∣
∣∣
∣∣=−ω^0 yˆe^1 +ω^0 xˆe^2.
