= Near-axis radial velocity in incompressible axisymmetric flow
{title2=$u_r=-\tfrac12r\,u_z'(0,z)+O(r^3)$}
For a smooth <axisymmetric flow>, write $u_z(r,z)=w(z)+O(r^2)$ near the axis. The <incompressible flow> condition is $r^{-1}\partial_r(ru_r)+\partial_zu_z=0$. Integrating radially and excluding a singular line source at the axis gives $u_r=-rw'(z)/2+O(r^3)$. The signed axial velocity is essential: specifying only its magnitude does not fix the sign of the radial velocity.
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