Abstract The boundary-layer flow over a heated horizontal plane boundary is analysed. Temperature variations along the boundary induce a pressure gradient that drives the flow. From consideration of an exact solution it is shown that no steady boundary-layer solution exists at a point where the temperature is a maximum. This is confirmed from the unsteady flow development that, at such a point, reveals a singular behaviour at a finite time. Steady, spatially periodic flows are considered for which it is shown that the boundary-layer solution terminates in a collision at points where the temperature is a maximum.