= Solution
Axisymmetric incompressibility is $\partial_r(ru_r)+r\partial_zu_z=0$. Multiply the radial momentum equation by $r$ and use this to put advection in conservative form:
$$
\partial_r(ru_r^2)+r\partial_z(u_ru_z)=\nu r\partial_z^2u_r.
$$
Integrating through the layer gives
$$
\frac{dF}{dr}=r[\nu\partial_zu_r-u_ru_z]_{-\infty}^{\infty}=0.
$$
The boundary terms vanish because radial velocity and its shear decay into the surrounding fluid; an entrainment velocity need not itself vanish as long as its product with $u_r$ does. Hence
$$
\boxed{F=\int_{-\infty}^{\infty}ru_r^2\,dz\text{ is independent of }r.}
$$
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