= Capillary instability of a thin film on a rigid cylinder
{title2=$h_t+[h^3(h_z+h_{zzz})]_z=0$}
A thin axisymmetric coating of physical thickness $d=\epsilon a h$ on a rigid cylinder of radius $a$ has leading curvature $a^{-1}-d/a^2-d_{\bar z\bar z}$. With constant <surface tension> $\gamma$, negligible <Newtonian gravity>, <no-slip boundary condition> at the wall and negligible exterior stress, <lubrication theory> gives $q=-d^3p_{\bar z}/(3\mu)$. Set $z=\bar z/a$ and $t=\gamma\epsilon^3\bar t/(3\mu a)$ to obtain the displayed <thin-film equation>. A perturbation of a uniform film $h_0$ proportional to $e^{st+ikz}$ has
$$
s=h_0^3(k^2-k^4).
$$
Thus the fastest <Rayleigh–Plateau instability> has $|k|=1/\sqrt2$, or physical wavelength $2\pi\sqrt2a$.
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