  We investigate experimentally the formation of the particular ice structure
obtained when a capillary trickle of water flows on a cold substrate. We show
that after a few minutes the water ends up flowing on a tiny ice wall whose
shape is permanent. We characterize and understand quantitatively the formation
dynamics and the final thickness of this ice structure. In particular, we
identify two growth regimes. First, a 1D solidification diffusive regime, where
ice is building independently of the flowing water. And second, once the ice is
thick enough, the heat flux in the water comes into play, breaking the 1D
symmetry of the problem, and the ice ends up thickening linearly downward. This
linear pattern is explained by considering the confinement of the thermal
boundary layer in the water by the free surface.
