Understanding the rates of cooling and solidification in laminar flows down sloping channels is central to predicting the advance of lava flows. The mechanisms involved include thermal convection and a competition between shear strain rate and the rate of formation of solid at the chilled surface of the flow. We report experiments in which polyethylene glycol wax flows in a laminar fashion down an inclined, open channel of rectangular cross-section under cold water. Two distinctly different flow regimes are recognized: ‘tube’ flow in which solidification of the flow surface creates a stationary roof while melt continues to flow through a relatively well-insulated ‘tube’ beneath, and a ‘mobile crust’ regime in which a solid surface crust develops only in the centre of the channel. In the latter regime the crust is carried down the channel, separated from the walls by crust-free shear regions in which cooling produces only dispersed fragments of solid owing to the effects of shearing. This flow structure is quasi-invariant over a large distance downstream. We show that thermal convection takes place in organized rolls that have axes aligned with the shear flow, and conclude that transition between the two flow regimes occurs at a critical value of the combined parameter $\vartheta \,{=}\, \psi (\hbox{\it Ra}{/}R_0)^{1/3}$, where $\psi \,{=}\, U_0 t_s{/}H_0$ is the ratio of a surface solidification timescale $t_{s}$ to a shearing timescale $H_0 {/} U_0 $, $H_{0}$ and $U_{0}$ are the flow depth and centreline surface velocity in the absence of solidification, {\it Ra} is a Rayleigh number and $R_{0}$ is a constant.