 Solution

The expression in braces in the first equation is the net axial force. The term $-\pi a^2\gamma /a$ is the compressive force from the capillary pressure $\gamma /a$, $3\pi \mu a^2{w}_z$ is the Newtonian extensional tension with Trouton ratio three, and $2\pi a\gamma$ is the axial pull of surface tension around the circumference. Its $z$-derivative vanishes because axial force is conserved.

The final equation is conservation of an insoluble surfactant. The surface divergence $w_z+u/a$ is the sum of axial and circumferential extension rates. Positive surface divergence increases interfacial area and dilutes $C$; negative divergence concentrates it. The absence of a diffusion term expresses the assumption of negligible surface diffusion.

Linearizing the area and surfactant equations givesThusand hencewhere $\psi$ is fixed by the initial data.

Linearizing the displayed net axial force and using ${\gamma }^{\prime }=-A{C}^{\prime }$ givesEliminating $w_z$ and $C^{\prime }$ therefore gives directlyThus the three displayed evolution equations implyThe target formula printed later in the paper contains an additional factor of $\pi$ in both denominators. That factor does not follow from the displayed equations because every term in the axial-force balance contains the same factor $\pi$. If the target formula is adopted as the intended normalization, its corresponding value is $s=({\gamma }_0-A{C}_0)/(6\pi \mu a_0)$.

Initially $\eta =0$ and $C^{\prime }>0$, so $\psi >0$ and ${\eta }_t<0$: a surfactant-rich, low-tension region begins to neck as neighboring higher tension pulls fluid away. The accompanying axial extension dilutes the surfactant.

If $A<{\gamma }_0/C_0$, then $s>0$. The Rayleigh–Plateau instability overwhelms the weak surface-elastic response: the necking perturbation grows in the linear model while the original concentration excess is diluted and eventually changes sign.

If $A>{\gamma }_0/C_0$, then $s<0$. Strong surface elasticity arrests the disturbance atThe concentration perturbation becomes negative, raising the local surface tension until its axial force balances that of the wider regions. This is stabilization by a surfactant-induced Marangoni stress.