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.
Short-Time Structural Stability of Compressible Vortex Sheets with Surface Tension