We study a thin liquid film on a vertical fibre. Without gravity, there
is a Rayleigh-Plateau instability in which surface tension reduces the
surface area of the initially cylindrical film. Spherical drops cannot
form because of the fibre, and instead, the film forms bulges of
roughly twice the initial thickness. Large bulges then grow very slowly
through a ripening mechanism. A small non-dimensional gravity moves the
bulges. They leave behind a thinner film than that in front of them, and
so grow. As they grow into large drops, they move faster and grow
faster. When gravity is stronger, the bulges grow only to finite
amplitude solitary waves, with equal film thickness behind and in front.
We study these solitary waves, and the effect of shear-thinning and
shear-thickening of the fluid. In particular, we will be interested in
solitary waves of large amplitudes, which occur near the boundary
between large and small gravity. Frustratingly, the speed is only
determined at the third term in an asymptotic expansion. The case of
Newtonian fluids requires four terms.