A boundary-element method is applied to solve the equations describing the deformation of a two-dimensional liquid region under the influence of gradients of the curvature of its outer boundary. This research is motivated by a desire to obtain a better understanding of viscous sintering processes in which a granular compact is heated to a temperature at which the viscosity of the constituent material becomes low enough for surface tension to cause adjacent particles to deform and coalesce. The boundary-element method is capable of showing how a moderately curved initial shape transforms itself into a circle. Initial shapes showing more extreme curvature gradients, which are relevant in the initial stages of a sintering process, cannot be dealt with by the boundary-element method in its present form. The numerical solution of the continuous model shows a tendency to create oscillations in the outer boundary of the liquid region. On the other hand, an analytical small-amplitude analysis shows that rapid oscillations vanish exponentially fast.