The free surface determines both the pressure and the first radial correction to axial velocity. Normal traction gives the pressure at leading order. Tangential traction determines , including the effect of polymer shear and the projection of polymer normal stress onto a tilted interface.
Consider an axisymmetric, swirl-free liquid with surface . We use dimensional coordinates, with primes denoting . The solvent viscosity and surface tension are constant. The surrounding gas exerts a uniform pressure and negligible viscous stress. We use the meridional polymer-stress symmetry specified in the regularity note.
1. Fix the stress and curvature signs
Write the absolute liquid pressure as . The liquid and gas stresses are
where . With the unit normal pointing out of the liquid,
The normal and tangential projections are therefore
For , the exact geometry is
A cylindrical surface has , and a sphere of radius has . These limits fix the curvature sign in the normal balance.
Let . With and , the expansion is
The last two terms are relative corrections to . We retain only the leading curvature in the calculation below.
Derivatives are taken at fixed coordinates before evaluating at . In the normal gradients, the displayed terms are second-order corrections. In the shear gradient, both displayed terms contribute at the first nonzero order. The polymer stresses at this accuracy are
3. Use normal traction to determine the pressure
The exact normal projection gives
At leading order, , and the pressure is uniform across the section. Thus,
Defining the pressure relative to the gas as , we obtain the convention used in the main notes:
At second order, the surface and centreline pressures differ by , so this leading expression cannot be used unchanged at that accuracy.
4. Use tangential traction to determine
The exact tangential projection is
Pressure cancels from . At the first nonzero order, the solvent contribution is therefore
The first bracketed term supplies . The second gives
The polymer projection contributes
Combining these terms gives the leading tangential condition,
For , we can rearrange it as
The tilted interface projects normal stress into the tangential direction. Consequently, and must both be retained; a shear-free surface does not require the polymer shear stress itself to vanish.
5. Keep track of the accuracy
The leading normal condition neglects relative corrections. The first tangential balance is itself on the characteristic stress scale, and its next contribution is on that scale.
In the notation of the second-order derivation, the full retained tangential condition is . Using only is sufficient for leading-order momentum, but loses the correction at the next order. Together with the normal-traction correction , the second-order axial equation therefore contains
The full expressions for these corrections, including the radial polymer-stress coefficients, are given in the main second-order note.
