= 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> $\gamma/a$, $3\pi\mu a^2w_z$ 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 gives
$$
2\eta_t=-a_0w_z,
\qquad
C'_t=-C_0\left(w_z+\frac{\eta_t}{a_0}\right)
=\frac{C_0}{a_0}\eta_t.
$$
Thus
$$
\frac\partial{\partial t}(a_0C'-C_0\eta)=0,
$$
and hence
$$
\boxed{a_0C'(z,t)=C_0\eta(z,t)+\psi(z)},
$$
where $\psi$ is fixed by the initial data.
Linearizing the displayed net axial force and using $\gamma'=-AC'$ gives
$$
3\pi\mu a_0^2w_z+\pi\gamma_0\eta
-\pi Aa_0C'=0.
$$
Eliminating $w_z$ and $C'$ therefore gives directly
$$
6\mu a_0\eta_t
=(\gamma_0-AC_0)\eta-A\psi(z).
$$
Thus the three displayed evolution equations imply
$$
\boxed{
\eta_t=s\eta-\frac{A\psi(z)}{6\mu a_0},
\qquad
s=\frac{\gamma_0-AC_0}{6\mu a_0}}.
$$
The 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 $s=(\gamma_0-AC_0)/(6\pi\mu a_0)$.
Initially $\eta=0$ and $C'>0$, 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 at
$$
\eta_\infty=\frac{A\psi}{\gamma_0-AC_0}<0.
$$
The concentration perturbation becomes negative, raising the local surface tension until its axial force balances that of the wider regions. This is stabilization by a <Marangoni effect>[surfactant-induced Marangoni stress].
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