A new method is described for the iterative solution of two-dimensional free-surface problems, with arbitrary initial geometries, in which the interior of the domain is represented by an unstructured, triangular Eulerian mesh and the free surface is represented directly by the piecewise-quadratic edges of the isoparametric quadratic-velocity, linear-pressure Taylor-Hood elements. At each time step, the motion of the free surface is computed explicitly using the current velocity field and, once the new free-surface location has been found, the interior nodes of the mesh are repositioned using a continuous deformation model that preserves the original connectivity.\ud \ud In the event that the interior of the domain must be completely remeshed, a standard Delaunay triangulation algorithm is used, which leaves the initial boundary discretisation unchanged. The algorithm is validated via the benchmark viscous flow problem of the coalescence of two infinite cylinders of equal radius, in which the motion is due entirely to the action of capillary forces on the free surface. This problem has been selected for a variety of reasons: the initial and final (steady state) geometries differ considerably; in the passage from the former to the latter, large free-surface curvatures - requiring accurate modelling - are encountered; an analytical solution is known for the location of the free surface; there exists a large body of literature on alternative numerical simulations. A novel feature of the present work is its geometric generality and robustness; it does not require a priori knowledge of either the evolving domain geometry or the solution contained therein
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