Pure Rotation.

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 ( $\nabla\cdot\vec{v} = 0$). We can specify the value of the angular velocity as $\Omega$. Then the velocity field has to satisfy the equations


\begin{displaymath}\frac{\partial V_{y}}{\partial x} = \Omega
\end{displaymath}

and


\begin{displaymath}\frac{\partial V_{x}}{\partial y} = -\Omega.
\end{displaymath}

Therefore, the general form of the velocity field under such restrictions are given by the equations


\begin{displaymath}V_{x} = -\Omega y+Cx
\end{displaymath}

and


\begin{displaymath}V_{y} = \Omega x -Cy,
\end{displaymath}

where $C$ is a constant. The part of the vector field multiplied by the constant $C$ 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


\begin{displaymath}V_{x} = -\Omega y
\end{displaymath}

and


\begin{displaymath}V_{y} = \Omega x.
\end{displaymath}

As an example of a vector field that only causes rotation to a material element of the flow, we can set $\Omega =1$ 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 $-y$ in the text box Input $V_{x}$ and type $x$ in the text box Input $V_{y}$. 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.



Karl P Burr
2003-07-07