Definition of rotation at a point

Rotation

           Figure 8.3 represent the situation of rotation

          Observations from the figure:

• The transverse displacement of B with respect to A and the lateral displacement of D with respect to A (Fig. 8.3) can be considered as the rotations of the linear segments AB and AD about A.
  
• This brings the concept of rotation in a flow field.

         Definition of rotation at a point:        

The rotation at a point is defined as the arithmetic mean of the angular velocities of two perpendicular linear segments meeting at that point.

         Example: The angular velocities of AB and AD about A are
 

           and     respectively.
 

Considering the anticlockwise direction as positive, the rotation at A can be written as,

(8.5a)

or

(8.5b)

The suffix z in ω represents the rotation about z-axis.

When u = u (x, y) and v = v (x, y) the rotation and angular deformation of a fluid element exist simultaneously. 

          Special case : Situation of pure Rotation 

 ,        and    

          Observation:

• The linear segments AB and AD move with the same angular velocity (both in magnitude and direction).
  
• The included angle between them remains the same and no angular deformation takes place. This situation is known as pure rotation.