code for the evaluation of the stream function and velocity field is given in the Appendix.

Choice of $\lambda$ and $\beta$
-------------------------------

At this stage we have no knowledge about how to fix the parameter $\epsilon =  \sqrt{\lambda}R_e^{\beta}$, and we now turn our attention to this issue and a more detailed analysis of Eq. (\[expandedns\]). We note first that our working assumption of a power dependence of $\epsilon$ on $R_e$ has not actually been necessary. All we need is that $\epsilon \rightarrow 0_+$ as $R_e \rightarrow 0$. We also note that the choice $\beta=1$ and $\lambda = 1/(4 e)$ recovers the result (Van Dyke, 1964) that the limiting stream function for fixed $r$ and small $R_e$ is, from Eq. (\[nearcyl\]) $$\psi \sim \frac{\sin\theta}{\log(4/R_e) + \frac{1}{2} - \gamma}  \left(r \log r - \frac{1}{2}r + \frac{1}{2r}\right)$$ so that the low $R_e$ limit matches exactly the first term of the MAE result.[^5] Clearly other choices of $\lambda$ can be considered, as can other powers of $R_e$ or even a more general $\epsilon$ still. However, in Section 8 we will establish a theoretical basis for estimating these parameters, and argue that $\beta=1$.

Analysis of the remainder
=========================

We must now analyze the right side 