Irrotational Flow

A fluid element can be subjected to three types of flow, namely: translation, rotation and deformation. These types of elemental motions are depicted in figure 2.11. The concept of translational motion is self-evident.. The deformational motion will occur when relative orientation of the axis changes as shown in figure 2.11. Deformational flow also exists if one or more axis are stretched or compressed. Here we focus our attention on rotational flow, which is depicted by the fluid element of figure 2.11 turning around.

Figure 2.11: Three types of fluid motion

Let us examine the flow in a fluid with circular streamlines as shown in figure 2.12. The fluid rotates like a rigid body. each element turns around at a certain angular velocity. The arrows shown rotate at the same rate. This is unquestionably a case of rotational flow.

Now let us consider the flow between the two horizontal flat plates, where the bottom plate is stationary while the top one moves at velocity v0 as shown in figure 2.13. we note that the horizontal arrow is simply translated, while

Figure 2.12: Fluid rotating like a rigid body
Figure 2.13: Shearing flow between two flat plates

the vertical one turns. It is not clear wether this is a case of rotational flow or not. To determine that we propose to use the average rate at rotation of the two arrows as a measure. If the average rate of rotation is zero the flow is said to be irrotational, if not the flow is rotational. For generality we refer to figure 2.14.

Figure 2.14: Change of relative positions in an arbitrary flow field

We define the angular velocity uz about the axis z as the average rate of counterclockwise rotation of the two lines:

In the limit of small angles we would have:

d_a = = di) = dvy dt dt dx dx dß dfä) dvx dt dt dy dy

Thus,

Similarly:

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