Now we consider velocity fields which cause pure rotation only. There is no volume change of material elements, which implies that the velocity field has zero divergence (
). We can specify the value of the angular velocity as
.
Then the velocity field has to satisfy the equations
and
Therefore, the general form of the velocity field under such restrictions are given by the equations
and
where
is a constant. The part of the vector field multiplied by the constant
is responsible only for strain of a material element of the flow, given no contribution to rotation. Then, velocity field which has rotation as its main effect is of the form
and
As an example of a vector field that only causes rotation to a material element of the flow, we can set
in the equations above. To illustrate the effect of this velocity field to a material contour, the user can use the vector field manipulation application. Type
in the text box Input
and type
in the text box Input
. Then input in the display area, by mouse clicking, a square contour. Then change the value of time in the text box Input t, and then click the Draw button. As a result the contour suffers rotation by the vector field. Repeat it as many times you want.