We present a model for ice formation in a thin, viscous liquid film driven by a
Blasius boundary layer after heating is switched off along part of the flat plate. The
flow is assumed to initially be in the Nelson et al. (J. Fluid Mech., vol. 284, 1995,
pp. 159–169) steady-state configuration with a constant flux of liquid supplied at
the tip of the plate, so that the film thickness grows like x
1/4
in distance along
the plate. Plate cooling is applied downstream of a point, Lx0, an O(L)-distance
from the tip of the plate, where L is much larger than the film thickness. The
cooling is assumed to be slow enough that the flow is quasi-steady. We present a
thorough asymptotic derivation of the governing equations from the incompressible
Navier–Stokes equations in each fluid and the corresponding Stefan problem for ice
growth. The problem breaks down into two temporal regimes corresponding to the
relative size of the temperature difference across the ice, which are analysed in detail
asymptotically and numerically. In each regime, two distinct spatial regions arise, an
outer region of the length scale of the plate, and an inner region close to x0 in which
the film and air are driven over the growing ice layer. Moreover, in the early time
regime, there is an additional intermediate region in which the air–water interface
propagates a slope discontinuity downstream due to the sudden onset of the ice at
the switch-off point. For each regime, we present ice profiles and growth rates, and
show that for large times, the film is predicted to rupture in the outer region when
the slope discontinuity becomes sufficiently enhanced.