On the mechanisms of icicle evolution

We present a study of a cylinder of ice melting in warm air in order to quantify the heat-transfer mechanisms controlling the evolution of its shape, which are inherent in a range of phenomena involving phase change and fluid flow. Motivated by the initial melting at the top of a flat-topped cylinder of ice, we analyse laminar, natural convection above a cooled, finite, horizontal plate (or below a heated, finite, horizontal plate) and show that, to a very good approximation, the partial-differential, boundary-layer equations can be separated with self-similar vertical profiles scaled by the boundary-layer thickness. We find that the horizontal evolution of the boundary-layer thickness is governed by equations describing a steady, viscous gravity current fed by diffusive entrainment, and therefore describe such flows as diffusive gravity currents. We first use the predictions of our model to examine previous experimental results in two dimensions. Our experimental results relating to the melting of ice in air are then compared with predictions based on our analysis of the axisymmetric thermal boundary layer. This comparison confirms the vertical thermal structure and shows that melting is governed in roughly equal measure by heat transfer from the air, the latent heat of condensation of water vapour, and the net radiative heat transfer from the surroundings to the ice