The standard drainage equation applies only to foam columns of constant cross-sectional area. Here, we generalize to include the effects of arbitrary container shape and develop an exact solution for an exponential, "Eiffel Tower", sample. This geometry largely eliminates vertical wetness gradients, and hence capillary effects, and should permit a clean test of dissipation mechanisms. Agreement with experiment is not achieved at late times, however, highlighting the importance of both boundary conditions and coarsening.