A two-dimensional arbitrary Lagrangian-Eulerian (ALE) model is established to investigate the migration and melting of a single solid particle in a pressure-driven flow between two vertical parallel plates. The developed sharp interface tracking method is advantageous as it has direct access to the transient transport fluxes at the solid-liquid interface without using averaging techniques. At each time step, the particle displacement and rotation, and the interfacial heat transfer on the particle surface are calculated to capture the interaction between the cylindrical particle and primary liquid phase accurately. The influence of Gr number on the flow and thermal fields is investigated. The model was validated by comparing the predicted melting rate and settling velocity of a particle against available computational results in the literature. It was found that, by increasing Gr, the melting time is reduced as the natural convection is strengthened. It was also observed that as the Gr is increased, the particle migration exhibits an increasingly oscillatory behavior that eventually stabilizes at an equilibrium position.