Towards a 'moving-point' formulation for the modelling of oscillation-mark formation in the continuous casting of steel

In the continuous casting of steel, solidification begins at a triple point where solid steel, molten steel and molten flux meet; the motion of this point determines how surface defects known as oscillation marks (OSMs) are formed. Here, under a number of simplifying assumptions, we derive an asymptotic model in 15 dimensionless parameters that describes the relevant momentum and heat transfer for the process, and involves both surface tension at the meniscus between molten flux and molten steel and solidification; further development couples an earlier lubrication-theory model for the region below the triple point to a reduced model for the region above that is based around a regular perturbation treatment of the Navier Stokes equations in terms of the capillary number. The resulting problem is then broken up into a hierarchy of four sub-problems; the first one – for the velocity field in the molten flux – is considered in depth. A numerical algorithm is developed for an isothermal situation in which the triple point moves only perpendicular to the casting direction; this involves the solution of a novel “moving point” problem to determine motion of the triple point. Comparison of model and experimental results indicates that this reduced model could produce OSMs having the observed depth if the triple point were not too far below the top of the meniscus; otherwise, the computed marks would be too deep. The analysis also tentatively relates the location of the triple point to the fold- and overflow-type OSMs that generally form in practice.