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– The standard drainage equation applies only to foam columns of constant crosssectional 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. Introduction. – Free drainage is the unavoidable fate of aqueous foams under Earth’s gravity [1, 2]. Because of the density mismatch between gas and liquid, the bubbles rise and collect at the top, while the liquid falls through the random network of plateau borders and accumulates at the bottom. This behavior can be understood in general terms via a nonlinear partial differential “drainage equation” that expresses liquid conservation as flow proceeds in response to gravity, capillary, and viscous forces [3–5]. However, in spite of its obvious importance for applications, and its apparent simplicity, the problem of free drainage in foams is not well understood yet [6]. One reason is that analytic predictions of the full time dependence cannot be achieved unless capillary terms, the most egregious nonlinearity, are dropped from the drainage equation [7]. This cannot be a valid approximation for short columns, at late times, or even at early times for dry foams [6]. Another reason is that the mechanism of viscous dissipation is not clearly established; shear flow may occur within the plateau borders at which three films meet [3], or only within the vertices at which four plateau borders meet [8]. Only recently have experiments been carried out systematically as a function of uniform initial liquid fraction and sample height [6]. Even then it was not possible to isolate and identify the individual effects of capillarity, dissipation, and potentially coarsening, because of their strong nonlinear couplings. In this paper we attack the free-drainage problem with an experiment designed to minimize capillary effects and, thereby, to isolate and identify dissipation mechanisms. This is accomplished by altering the sample geometry in such a way that drainage may proceed without vertical gradients in liquid fraction. Since the standard drainage equation applies only to straight columns of constant cross-section, we begin by generalizing to account for arbitrary container shape. We then develop an exact solution for special containers that flare out exponentially towards the bottom, much like the Eiffel tower. At each height, we predict, more liquid can be passed down than received from above, so that the foam becomes uniformly drier with time. Finally, we compare with experiments using specially constructed “Eiffel

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