Competition between conductive cooling and advective heating occurs whenever hot fluid invades a cold environment. Here the solidification of hot viscous flow driven by a fixed pressure drop through an initially planar or cylindrical channel embedded in a cold rigid solid is analysed. At early times, or far from the channel entrance, the flow starts to solidify and block the channel, thus reducing the flow rate. Close to the channel entrance, and at later times, the supply of new hot fluid starts to melt back the initial chill. Eventually, either solidification or meltback becomes dominant throughout the channel, and flow either ceases or continues until the source is exhausted. The evolution of the dimensionless system, which is characterized by the initial Péclet number Pe, the Stefan number S and the dimensionless solidification temperature Θ, is calculated numerically and by a variety of asymptotic schemes. The results show the importance of variations along the channel and caution against models based on a single ‘representative’ width. The critical Péclet number Pec, which marks the boundary between eventual solidification and eventual meltback, is determined for a wide range of parameters and found to be much larger for cylindrical channels than for planar channels, owing to the slower rate of decay of the heat flux into the solid in a cylindrical geometry. For a planar channel Pec is given by the simple algebraic result Pec ∼ 0.46[Θ2/(1 - Θ)(S + 2/π)]3 when (1 - Θ)−1 [Gt ] S [Gt ] 1, but in general it requires numerical solution. Similar analyses, in which there is a spatially varying and time-dependent interaction between the rates of solidification and flow, have a range of applications to geological and industrial processes.