= Solution
The condition $a\gamma=a_0\gamma_0$ makes the leading extensional axial force uniform, but it does not make the <capillary pressure> uniform. Since
$$
p_c=\frac\gamma a=\frac{a_0\gamma_0}{a^2},
$$
a nonuniform radius gives a nonuniform pressure. Its axial gradient must drive flow, so the prediction $w=0$ is inconsistent.
For $a=a_0+\eta$ varying over the axial scale $L$,
$$
p_c'\sim\frac{\gamma_0\eta}{a_0^2},
\qquad
\frac{\partial p_c}{\partial z}
\sim\frac{\gamma_0\eta}{a_0^2L}.
$$
An axial <Hagen-Poiseuille flow> in a cylinder has speed scale
$$
w\sim\frac{a_0^2}{\mu}\frac{\partial p_c}{\partial z}
\sim\frac{\gamma_0\eta}{\mu L}.
$$
Cross-sectional <mass conservation> gives $\eta_t\sim a_0w/L$, and consequently
$$
\boxed{
\frac{\partial\eta}{\partial t}
\sim\frac{\gamma_0a_0}{\mu L^2}\eta}.
$$
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