On the gravity-driven draining of a rivulet of fluid with temperature-dependent viscosity down a uniformly heated or cooled substrate

We use the lubrication approximation to investigate the unsteady gravity-driven draining of a thin rivulet of Newtonian fluid with temperature-dependent viscosity down a substrate that is either uniformly hotter or uniformly colder than the surrounding atmosphere. First we derive the general nonlinear evolution equation for a thin film of fluid with an arbitrary dependence of viscosity on temperature. Then we show that at leading order in the limit of small Biot number the rivulet is isothermal, as expected, but that at leading order in the limit of large Biot number (in which the rivulet is not isothermal) the governing equation can, rather unexpectedly, always be reduced to that in the isothermal case with a suitable rescaling. These results are then used to give a complete description of steady flow of a slender rivulet in the limit of large Biot number in two situations in which the corresponding isothermal problem has previously been solved analytically, namely non-uniform flow down an inclined plane, and locally unidirectional flow down a slowly varying substrate. In particular, we find that if a suitably defined integral measure of the fluidity of the film is a decreasing function of the temperature of the atmosphere (as it is for all three specific viscosity models we consider) then decreasing the temperature of the atmosphere always has the effect of making the rivulet wider and deeper.