Two-dimensional thermocapillary convection in an open cylindrical annulus heated from the inside wall is computed. The deformable free surface is obtained as a solution of the coupled transport equations, assuming pinned contact points, at Prandtl number of 30 and prescribed geometry. Only steady convection is possible at any Reynolds number (Re) in the axisymmetric computations with either nondeformable or deformable surfaces. Dynamic free surface deformations do not induce transitions to oscillatory convection even at large Re and capillary numbers (Ca). Free surfaces are convex near the cold wall stagnation point and concave near the hot wall. Two peaks appear at the free surface at low Re while four peaks are possible at larger Re. Free surface shapes and convection in the interior are insensitive to variations in Ca while the magnitudes of surface ripples increase with Ca. At Ca = 0 convection is calculated assuming nondeformable concave surfaces as function of the liquid volume (V) and the contact angle (theta) at the inner boundary. At constant V, peaks of surface velocity increase while central surface temperatures decrease with increasing theta. Curvature significantly influences convection which is more vigorous with increasing V/theta at constant theta/V. (C) 2004 Elsevier Ltd. All rights reserved.