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. For the bubble problem, the deformation becomes more severe as the vortex moves further into the corner until eventually the free surface makes contact with the walls. For the magnetic shaping problem, columns approach a limiting configuration featuring either a finite number of cusps or a fixed number of trapped bubbles along the perimeter. The division between these two different behaviors is explained by means of an exact solution for zero surface tension.