Mixed convection in an air-filled annular enclosure between vertical concentric cylinders, which is driven by combined heating and axial translation of the inner cylinder, is simulated numerically. Both cylinders are assumed to be isothermal with an imposed temperature difference between them. The flow is assumed to be axisymmetric and the Boussinesq approximation is employed. The conservation equations for momentum and energy are solved using a spectral Chebyshev collocation technique utilizing a pressure Poisson influence matrix approach to maintain a solenoidal velocity field. The buoyancy driven flow observed at low Re is displaced by a forced convection cell that grows with increasing translational velocity until it fills the entire cavity at high Re. High heat transfer rates are observed for cases in which the air flow is dominated by either buoyancy or shear forces while low heat transfer rates result from flow driven by competing forces.