The characteristics of the forced convection melting process of a circular vertical coolant channel wall induced by the flowing fluid through a channel have been analyzed by solving an unsteady, two dimensional, non-linear momentum and energy equations along with the continuity and the interface energy balance equations. The solution is facilitated by the coordinate transformation that immobilizes the moving phase boundary. The transformed equations and the corresponding boundary and initial conditions are recast as finite difference analougues. These set of finite difference equations are then solved by the simplified Marker and Cell method with modification to the pressure iteration scheme as suggested by Viecelli. The analysis is confined to situations where the superheat of the flowing fluid is sufficiently high so that there is no potential for solidification at the liquid-solid interface. Sample numerical solutions are obtained using R-11 ($CCl_3F$) and ice tube as the flowing fluid and the melting wall, respectively:Dimensionless melt-layer thickness and liquid-solid interface shapes at various dimensionless times for various combinations of two most important paramenters, Stefan and Reynolds numbers, are presented in graphical form along with dimensionless velocity and temperature profiles. In the laminar forced convection melting, the thickness of the melt layer was found to vary along the channel, with the smallest layer thickness at the bottom and the greatest thickness at the top, as was the case for the melting in the presence of natural convection. this contrasts with a uniform melt layer thickness predicted by the conduction solution.