= Zero-flux thermocapillary film profiles
{title2=$H_X^2/2+H(\ln H-1)=E$}
Positive steady solutions of the balanced <thermocapillary thin-film equation> with zero <volume flux> satisfy $H_{XX}+\ln H=0$ after choosing the thickness scale. Their first integral has potential $V(H)=H(\ln H-1)$, with minimum $V(1)=-1$ and limiting value $V(0^+)=0$. At $E=-1$ the film is uniform; for $-1<E<0$ it is positive and periodic. At $E=0$ a limiting drop has peak $e$, half-width $\sqrt{\pi e}$, zero limiting <contact angle> and divergent curvature at zero thickness. Small periodic profiles have <period> $2\pi$ and approach the neutral, rather than fastest-growing, <normal mode> of the <thermocapillary film instability>.
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