Time-dependent viscous flows driven by capillarity act to minimize the surface area of a fluid bounded in a plane geometry with initial gradients in surface curvature. These free-surface flows are solved by a finite-element model applied to describe the viscous sintering of two-dimensional ceramic particles. The numerical model is validated by comparison to the analytical solution obtained by Hopper (1990) for the coalescence of two infinite cylinders of equal cross section and is applied to several other geometries pertinent to the study of particle sintering for which analytical results are not available. Details of the flowfields and morphological evolution lend insight to the physical behavior of these systems and provide a basis for the more complete understanding of viscous sintering phenomena.