We consider the evolution under the action of surface tension of wedges and cones of viscous fluid whose initial semi-angles are close to π/2. A short time after the fluid is released from rest, there is an inner region, where surface tension and viscosity dominate, and an outer region, where inertia and viscosity dominate. We also find that the velocity of the tip of the wedge or cone is singular, of O(log(1/t)), as time, t, tends to zero. After a long time, the free surface asymptotes to a similarity form where deformations are of O(t2/3), and capillary waves propagate away from the tip. However, a distance of O(t3/4) away from the tip, viscosity acts to damp out the capillary waves.

We solve the linearized governing equations using double integral transforms, which we calculate numerically, and use asymptotic techniques to approximate the solutions for small and large times. We also compare the asymptotic solution for the inviscid fat wedge with a numerical solution of the nonlinear inviscid problem for wedges of arbitrary semi-angle.