This paper presents a mathematical model for heat or mass transfer from an open cavity. It is assumed that the Péclet number, based on conditions at the cavity, and the Prandtl number are both large. The model assumes heat- or mass-transfer boundary layers at the rim of the cavity vortex flow. Heat or mass exchange with the surrounding fluid occurs in a free boundary layer which spans the mouth of the cavity. It is shown that the solution depends upon a single parameter only. This parameter is determined by the flow field. For small and large values of matched asymptotic expansions are presented. The model is illustrated for a few simple flows in closed cavities. Etching, clot formation in flowing blood, lubrication and cooling of rough surfaces are mentioned as possible fields of application.