the divergence condition) $$\Psivec = - \Psi(x, y,t) \underline{e}_z$$ Next we note that under the assumption that $\Psi$ satisfies $\del.\Psivec = 0$ $$\omvec = -\delsq \Psivec =  \underline{e}_z \delsq \Psi$$ and the vorticity equation becomes

$$\nu \nabla^4 \Psi - \frac{\partial\ }{\partial t}\nabla^2 \Psi=  \frac{\partial(\Psi, \delsq \Psi)}{\partial(X, Y)}$$

For problems where it is possible to identify a natural length scale $L$ and a natural speed $U$, it is normal practice to perform a non-dimensionalization of the variables and introduce the Reynolds number $R_e = UL/\nu$. Then the NavierStokes equation becomes (after rescaling the variables suitably to $\psi, x, y, \tau$): $$\nabla^4 \psi - R_e  \frac{\partial\ }{\partial \tau}\nabla^2 \psi= R_e \frac{\partial(\psi, \delsq \psi)}{\partial(x, y)}$$ Throughout this paper we shall work in units in which the [*radius*]{} $a$ of the cylinder is taken to be unity. In such units the Reynolds number is here based on the radius[^4] and is given by $R_e = U/\nu$. The two stream functions are related by $\Psi = U \psi$ and furthermore $X=x, Y=y$ etc.

The old historical approach to the limiting case when $R_e \rightarrow 0$ and the flow is time-independent is to take the view that the non-linearities may then be ignored (provided the non-linear 