Angular Deformation.

Now we consider the case of angular deformation. There is no volume change, so the velocity field has zero divergence ( $\nabla\cdot\vec{v} = 0$). There is also no rotation of any material element of the flow, so the vorticity is zero, which implies that the vector field has to satisfy


\begin{displaymath}\frac{\partial V_{y}}{\partial x}-\frac{\partial V_{x}}{\partial y} = 0.
\end{displaymath}

As a result, the vector field has the form


\begin{displaymath}V_{x} = Cy+C_{1}x
\end{displaymath}

and


\begin{displaymath}V_{y} = Cx-C_{1}y,
\end{displaymath}

where $C$ and $C_{1}$ are constants. The part of the vector field multiplied by the constant $C_{1}$ gives only pure strain. Therefore, a vector field which has angular deformation as its main effect has the form


\begin{displaymath}V_{x} = Cy
\end{displaymath}

and


\begin{displaymath}V_{y} = Cx.
\end{displaymath}

As an example of a vector field that only causes angular deformation to a material element of the flow, consider the case where the constants $C$ assumes the values $C = 1$. 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 angular deformation by the vector field. Repeat it as many times you want. Notice that strain is also observed.



Karl P Burr
2003-07-07