The first radial correction to axial velocity is small, but its viscous contribution need not be. Two radial derivatives of leave the finite term . We derive this term and the next coefficient using the same dimensional coordinates as the leading-order slender-jet equation.
For an incompressible liquid with constant solvent viscosity , the axial component of the solvent-stress divergence is . In axisymmetric cylindrical coordinates,
The axial component has the scalar Laplacian shown here. The radial component of the vector Laplacian has an additional term.
1. Differentiate the radial expansion
Write
with coefficients depending on and . Primes denote . Then
Adding gives
The apparent singularity is removable because vanishes linearly at the axis. More generally, for ,
Polymer stress adds to this coefficient equation; it is separate from the solvent contribution derived here. At the next radial order, the solvent term is . The full equations are given in the second-order hierarchy.
3. Check the length scales
Let the axial velocity scale be , with radial scale . In the regular slender expansion, , so
Thus, the small correction to the axial velocity profile and the leading axial velocity contribute at the same order after viscous differentiation. The surface traction must determine before we eliminate it from the leading momentum balance.
If instead we introduce scaled coordinates and , the dimensional operator becomes
This is the origin of the factor in the scaled equation. It belongs to derivatives with respect to , not to the dimensional operator used above.
