A free surface, placed in a strong viscous flow (such that viscous
forces overwhelm surface tension), often develops ends with very sharp
tips. We show that the axisymmetric shape of the ends, non-dimensionalized
by the tip curvature, is governed by a universal similarity solution.
The shape of the similarity solution is close to a cone, but whose slope
varies with the square root of the logarithmic distance from the tip.
Pursuing the calculation of the tip similarity solution to next order,
we demonstrate matching to previous slender-body analyses, which fail
near the tip. This allows us to resolve the long-standing problem,
first raised by G.I. Taylor, of finding the global solution of a bubble
in a strong hyperbolic flow. Our results are shown to agree in detail with
experiment as well as full numerical simulations of the Stokes equation.