= Solution
Measure $z$ from the rigid tube wall towards the air core, so $0<z<h$. To leading order in $h_0/a$ and small interface slope, the <curvature> entering the <Young–Laplace equation> is
$$
\kappa=\frac1{a-h}+h_{xx}+\cdots,\qquad
p=p_{\rm air}-\sigma_0\kappa.
$$
The air <pressure> is spatially uniform at the order retained. Thus the linear liquid <pressure> perturbation associated with thickness amplitude $H$ is $p_1=-\sigma_0(a^{-2}-k^2)H$. The circumferential <curvature> favors growth at long <wavelength>; the <curvature> opposes it at short <wavelength>.
The <lubrication theory> momentum equation is $\mu u_{zz}=p_x$. With the wall <no-slip boundary condition> $u(0)=0$ and zero surface <shear stress>, $u_z(h)=0$, its solution and flux are
$$
u=\frac{p_x}{2\mu}(z^2-2hz),\qquad
q=\int_0^h u\,dz=-\frac{h^3}{3\mu}p_x.
$$
Leading thin-film <mass conservation> is $h_t+q_x=0$. For a mode $e^{\beta t+ikx}$, this gives the <capillary instability of an annular liquid lining>
$$
\boxed{\beta=\frac{\sigma_0h_0^3}{3\mu}k^2(a^{-2}-k^2).}
$$
Therefore \b[the unstable band is $\boxed{0<k^2<a^{-2}}$]. Maximization with respect to $k^2$ gives $k_*^2=1/(2a^2)$ and
$$
\boxed{\beta_{\max}=\frac{\sigma_0h_0^3}{12\mu a^4}.}
$$
These <curvature> coefficients use $a$ instead of $a-h_0$ consistently at leading order in the thin-lining approximation.
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