ñòð. 14 |

v = 'v(r)

^ (5.6)

see gure (5.2) for a sketch of this ow eld. The problem we now face is that

^

Ï•

Figure 5.2: Sketch of an axi-symmetric source-free ow in the x,y-plane.

de nition (5.1) is expressed in Cartesian coordinates while the velocity in equation

(5.6) is expressed in cylinder coordinates. In section (5.6) an expression for the curl

in cylinder coordinates will be derived. As an alternative, one can express the unit

^

vector ' in Cartesian coordinates.

Problem c: Verify that: 0 1

' = B x=r C :

;y=r

^@ A (5.7)

0

Hints, make a gure of this vector in the x y-plane, verify that this vector is

perpendicular to the position vector r and that it is of unit length. Alternatively

you can use expression (3.36) of section (3.5).

CHAPTER 5. THE CURL OF A VECTOR FIELD

44

Problem d: Use the expressions (5.5), (5.7) and the chain rule for di erentiation

to show that for the ow eld (5.6):

(r v)z = @v + v : (5.8)

@r r

Hint, you have to use the derivatives @r=@x and @r=@y again. You have learned

this in section (4.2).

5.3 The rst source of vorticity rigid rotation

In general, a nonzero curl of a vector eld can have two origins, in this section we

will treat the e ect of rigid rotation. Because we will use uid ow as an example we

will speak about the vorticity, but keep in mind that the results of this section (and

the next) apply to any vector eld. We will consider a velocity eld that describes

a rigid rotation with the z-axis as rotation axis and angular velocity .

Problem a: Show that the associated velocity eld is of the form (5.6) with v(r) =

r. Verify explicitly that every particle in the ow makes one revolution in

a time T = 2 = and that this time does not depend on the position of the

particle.

Problem b: Show that for this velocity eld: v = 2 ^.

z

r

This means that the vorticity is twice the rotation vector ^. This result is derived

z

here for the special case that the z-axis is the axis of rotation. (This can always

be achieved because one is free in the choice of the orientation of the coordinate

system.) In section (6.11) of Boas 11] it is shown with a very di erent derivation

that the vorticity for rigid rotation is given by ! =r v = 2 , where is the

rotation vector. (Beware, the notation used by Boas is di erent from ours in a

deceptive way!)

We see that rigid rotation leads to a vorticity that is twice the rotation rate. Imagine

we place a paddle-wheel in the ow eld that is associated with the rigid rotation, see

gure (5.3). This paddle-wheel moves with the ow and makes one revolution along

its axis in a time 2 = . Note also that for the sense of rotation shown in gure (5.3)

the paddle wheel moves in the counterclockwise direction and that the curl points

along the positive z-axis. This implies that the rotation of the paddle-wheel not only

denotes that the curl is nonzero, the rotation vector of the paddle is directed along

the curl! This actually explains the origin of the word vorticity. In a vortex, the

ow rotates around a rotation axis. The curl increases with the rotation rate, hence

it increases with the strength of the vortex. This strength of the vortex has been

dubbed vorticity, and this term therefore re ects the fact that the curl of velocity

denotes the (local) intensity of rotation in the ow.

5.4. THE SECOND SOURCE OF VORTICITY SHEAR 45

y

( z-upward)

*

â„¦

x

v

Figure 5.3: The vorticity for a rigid rotation.

5.4 The second source of vorticity shear

In addition to rigid rotation, shear is another cause of vorticity. In order to see this

we consider a uid in which the ow is only in the x-direction and where the ow

depends on the y-coordinate only: vy = vz = 0, vx = f(y).

Problem a: Show that this ow does not describe a rigid rotation. Hint: how long

does it take before a uid particle returns to its original position?

Problem b: Show that for this ow

v = @f ^ :

@y z (5.9)

r ;

As a special example consider the velocity given by:

2=L2

vx = f(y) = v0 exp : (5.10)

;y

This ow eld is sketched in gure (5.4).

Problem c: Verify for yourself that paddle-wheels placed in the ow rotate in the

sense indicated in gure (5.4)

Problem d: Compute v for this ow eld and verify that both the curl and

r

the rotation vector of the paddle wheels are aligned with the z-axis. Show that

the vorticity is positive where the paddle-wheels rotate in the counterclockwise

direction and that the vorticity is negative where the paddle-wheels rotate in

the clockwise direction.

It follows from the example of this section and the example of section (5.3) that both

rotation and shear cause a nonzero vorticity. Both phenomena lead to the rotation

of imaginary paddle-wheels embedded in the vector eld. Therefore, the curl of a

CHAPTER 5. THE CURL OF A VECTOR FIELD

46

y

( z -upward)

* x

*

Figure 5.4: Sketch of the ow eld for a shear ow.

vector eld measures the local rotation of the vector eld (in a literal sense). This

explains why in some languages (i.e. Dutch) the notation rot v is used rather than

curl v. Note that this interpretation of the curl as a measure of (local) rotation

ñòð. 14 |