= Solution
{c}
For a planar <incompressible flow>, the <divergence> of the <velocity field> must vanish. Here
$$
\nabla\cdot\mathbf u=\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=-1+1=0.
$$
Equivalently, the <Jacobian determinant> of the <deformation map> is
$$
\det\frac{\partial(x,y)}{\partial(X,Y)}
=e^{-t}e^t(\cos^2\Omega t+\sin^2\Omega t)=1,
$$
so every material area is preserved. However, the out-of-plane <vorticity> is
$$
\boxed{\omega_z=\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}
=-\Omega(e^{2t}+e^{-2t})=-2\Omega\cosh(2t)}.
$$
Since $\Omega>0$, this <vorticity> never vanishes. The motion is an <incompressible flow> but is not an <irrotational flow>; its local spin is clockwise.
Back to article page
