= Solution
Let the signed axial <velocity> on the axis be $w(z)$. Smooth <axisymmetric flow> has $u_z(r,z)=w(z)+O(r^2)$ and regular radial velocity at $r=0$. The <incompressible flow> condition in <cylindrical coordinates> is
$$
\frac1r\frac{\partial(ru_r)}{\partial r}+\frac{\partial u_z}{\partial z}=0.
$$
To leading order near the axis, integrate $\partial_r(ru_r)=-rw'(z)$ from zero to $r$. The integration constant vanishes by regularity, giving the <near-axis radial velocity in incompressible axisymmetric flow>,
$$
u_r=-\frac r2w'(z)+O(r^3).
$$
With the usual signed interpretation $w(z)=A/z^3$,
$$
\boxed{u_r(r,z)\simeq\frac{3Ar}{2z^4},\qquad r\ll|z|.}
$$
For $A>0$ this is outward: the outward axial flow slows as its distance from the origin grows, and incompressibility requires radial spreading. Here $A$ has units $\mathrm{m^4\,s^{-1}}$, so the answer is a speed.
The source specifies a magnitude rather than an unambiguous signed component. On $z>0$, reversing the axial flow while keeping its magnitude would reverse the radial answer. The general relation $u_r=-rw'/2$ makes that dependence explicit; the boxed expression adopts $u_z=A/z^3$, outward on both halves of the axis.
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