  The freezing of a water rivulet begins with a water thread flowing over a
very cold surface, is naturally followed by the growth of an ice layer and ends
up with a water rivulet flowing on a static thin ice wall. The structure of
this final ice layer presents a surprising linear shape that thickens with the
distance. This paper presents a theoretical model and experimental
characterisation of the ice growth dynamics, the final ice shape and the
temperature fields. In a first part, we establish a 2D model, based on the
advection-diffusion heat equations, that allows us to predict the shape of the
ice structure and the temperature fields in both the water and the ice. Then,
we study experimentally the formation of the ice layer and we show that both
the transient dynamics and the final shape are well captured by the model. In a
last part, we characterise experimentally the temperature fields in the ice and
in the water, using an infrared camera. The model shows an excellent agreement
with the experimental fields. In particular, it predicts well the linear
decrease of the water surface temperature observed along the plane, confirming
that the final ice shape is a consequence of the interaction between the
thermal boundary layer and the free surface.
