Predicting the free-surface shape of a liquid bridge has been the focus of several recent models of containerless semiconductor crystal-growth methods, such as float-zone processing. We wish to sort out the discrepancies in the predictions of the numerical models by investigating the physics of a simplified system, the half-zone in microgravity. In the absence of gravity, the deformation of the free surface is small. Therefore, we first calculate the flow for a cylindrical melt region, corresponding to a large reference surface tension. This problem is well benchmarked, and the results are in nearly universal agreement. We then investigate a small perturbation of the free-surface shape using the flow calculated for the undeformed liquid bridge. Applying asymptotic expansions, we can predict the leading order of the free-surface velocities and deformation. In this formulation, it is easy to understand the relevance of each term, including the dynamic pressure variation. This solution is also more efficient than the numerical schemes that iterate between the shape of the free surface and the associated flow field. Furthermore, it provides physical insight that is difficult to extract from a purely numerical solution. Conversely, it is an approximation and therefore neglects terms of importance to a highly deformed free surface. Where possible, we will compare the leading-order free-surface shape to that predicted by numerical models, and discuss the advantages and disadvantages of this technique.
Calculation of the free surface shape in the electromagnetic processing of liquid metals
