Axisymmetric flow

An axisymmetric flow is invariant under rotations about one fixed axis, so scalar fields and cylindrical-coordinate velocity components do not depend on the azimuthal angle.

For a smooth axisymmetric flow, write $u_z(r,z)=w(z)+O(r^2)$ near the axis. The incompressible flow condition is $r^{-1}{\partial }_r(r{u}_r)+{\partial }_z{u}_z=0$. Integrating radially and excluding a singular line source at the axis gives $u_r=-r{w}^{\prime }(z)/2+O(r^3)$. The signed axial velocity is essential: specifying only its magnitude does not fix the sign of the radial velocity.