Theory of bubble tips in strong viscous flows

A free surface, placed in a strong viscous flow (such that viscous forces overwhelm surface tension), often develops ends with very sharp tips. In [Courrech du Pont and Eggers, PNAS {\bf 117}, 32238 (2020)] we have shown 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. Here we develop the calculation of the tip similarity solution to next order, using which 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. We also calculate the tip curvature quantitatively, beyond the scaling behavior of the leading order solution. Our results are shown to agree in detail with full numerical simulations of the Stokes equation.