Solution
= Solution
The definitions through the Stokes stream function give
$$
{1\over r}\partial_r(ru)+\partial_zw
=-{1\over r}\partial_r\partial_zΨ+{1\over r}\partial_z\partial_rΨ=0,
$$
so <incompressible flow>[incompressibility] is automatic.
Balancing inertia $A^2/r^5$ with viscosity $νU/δ^2$ gives
$$
δ(r)\sim(νr^3/A)^{1/2}.
$$
The streamfunction scale is $rUδ\sim(Aνr)^{1/2}$, motivating
$$
\boxed{Ψ=(Aνr)^{1/2}F(η),\qquad η=z(A/(νr^3))^{1/2}.}
$$