A simple time-dependent model is proposed for the process of painting a metal surface by electrodeposition. The problem is then to solve the Laplace equation in a given region with a free boundary on a part of its surface. A related steady problem is considered. Numerical solutions to examples of both problems are presented. We also discuss the existence and regularity of the solutions to the steady problem and find the exact solution to a one-dimensional unsteady problem. In certain cases we are able to derive approximate solutions for other unsteady problems. © 1984 Academic Press Inc. (London) Limited.