Modelling viscoplastic surface flows

Molten alloys tend to oxidise very quickly when first exposed to the atmosphere at ambient conditions. Thin oxide films develop at the liquid metal-air interface which, in turn, affects casting processes. As a consequence of the oxide film, the selection of the right boundary condition to be applied along the surface of the liquid metal flow is not straightforward. Irrespective of the behaviour of the oxide layer that covers the surface, one of two boundary conditions are usually considered by default: either a slip condition, or a no-slip condition when the oxide film is considered to behave as a deformable wall. The reality is somewhat in between these two conditions: the continuous formation of an oxide film at the metal-air interface leads to a local change in the boundary condition that can strongly affect both the bulk and surface flow. Recent experimental investigations have shed new light on the complex dynamics associated with these liquid-metal-type flow problems. Both the non-Newtonian behaviour of the oxide layer over the liquid metal surface and the curvature of the interface due to wetting effects were observed via scanning electron microscopy techniques. In this study we develop a mathematical model to describe these flows. Working in the low Reynolds number regime, we make use of the Bingham fluid model to capture viscoplastic surface effects. We develop both asymptotic and numerical analyses and report on the surface flow characteristics as well as the important effects of surface curvature.