Consider an incompressible, axisymmetric jet with surface . Once the axial velocity is specified, continuity determines the radial velocity. Requiring regularity at the axis removes the integration constant, and the surface kinematics then give conservation of cross-sectional area.
We use dimensional coordinates, as in the leading-order derivation. Primes denote . If , the dimensional powers of carry the slender ordering; no additional powers of multiply them.
1. Integrate continuity from the axis
Axisymmetric incompressibility requires
Expand the axial velocity as
where the coefficients depend on and . Substitution into continuity gives
Integrating with respect to ,
A nonzero would give a singular velocity at the axis. Regularity therefore requires , giving
The radial velocity is odd in , as required by axis regularity. In particular, a positive axial strain rate produces an inward radial velocity.
2. Apply the surface kinematics
The surface moves with the liquid, so
At leading order this becomes
Multiplying by and collecting the axial derivative gives
We can check the coefficient by integrating the axial velocity over a section. The exact volume flux is
so its expansion is
The exact area balance is . Truncating this flux after recovers the second-order equation above. Under the regular slender ordering, the contribution is a relative correction, while the contribution is relative . Consequently, the centreline velocity and the mean velocity agree only at leading order.
