 Solution

For the simple shear flow, the vertical vorticity isFor $\beta \ne 0$, this is not an irrotational flow. A complex potential of a two-dimensional potential flow $w=\varphi +i\psi$, analytic in the flow region, has complex velocity $w^{\prime }=v_x-i{v}_y$. The Cauchy-Riemann equations imply both incompressibility and zero vorticity, so this shear has no such complex potential. It does have a Cartesian streamfunction, $\psi =\beta y^2/2$. In the degenerate case $\beta =0$, the resting fluid has a constant complex potential.