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Two different physical problems are considered: the magnetic shaping of a liquid metal column and the distortion of a bubble in a corner vortex flow. It is shown that the two problems can be modeled with a virtually identical set of equations. These equations are solved numerically using a conformal mapping and a series truncation method, which permits fast and efficient computation of the bubble or column shapes. It is found that the two problems exhibit different limiting configurations. Fordoi:10.1137/050622353 fatcat:5xawgz53ibfibaqbvngvn57uju