SUMMARYIn the present paper the thermo-solutal-capillary migration of a dissolving liquid drop, composed by a binary mixture having a miscibility gap, injected in a closed cavity with di erentially heated end walls, is studied. The main goal of the analysis is to clarify if and how the drop migration is a ected by the dissolution process. The numerical code is based on a ÿnite volume formulation. A level-set technique is used for describing the dynamics of the interface separating the di erent phases. A thermodynamic constraint ÿxes the concentration jump between the interface sides. This jump, together with that of the concentration normal derivatives, in turn deÿnes the entity of the dissolution cross-ow through the interface and the interface velocity relative to the uid. Since the jump singularity of normal derivatives cannot be easily molliÿed, while retaining the necessary accuracy, a scheme for the species equation is elaborated that allows sharp jumps and has subcell resolution. Steady migration speeds are determined after the start-up phase for di erent radii and temperature di erences. The results will be used for the preparation of a sounding rocket space experiment.