1.11 Vorticity and Circulation

Any motion of a small region of a fluid can be thought of as a combination of translation, rotation, and deformation. Translation is described by velocity of a point. Deformation is described by rates of deformation, as in Section 1.7. Here, we consider the rotation of a fluid element.

In rigid-body mechanics, the concept of angular rotation is an extremely important one and rather intuitive. Along with translational velocity, it is one of the basic descriptors of the motion. In fluid mechanics we can introduce a similar concept in the following manner.

Consider again the two-dimensional picture shown in Figure 1.7.1. In Section 1.7, we saw that the instantaneous rates of rotation of lines AB and BC were ₁ and ₂, given by equations (1.7.5) and (1.7.6).

Since equations (1.7.5) and (1.7.6) generally will differ, it is seen that the angular velocity of a line depends on the initial orientation of that line. To develop our analogy of angular velocity, we want a definition that is independent of orientation and direction at a point, and depends only on local conditions at the point itself. In considering the transformation of ABC into A ?B ?C ?, it is seen that two things have happened: The angle has changed, or deformed, by an amount d ? ₁+ d ? ₂, and the bisector of the angle ABC has rotated an amount 0 .5( d ? ₁ ? d ?