How a Long Bubble Shrinks: a Numerical Method for an Unforced Hele-Shaw   Flow
  We develop a numerical method for solving a free boundary problem which describes shape relaxation, by surface tension, of a long and thin bubble of an inviscid fluid trapped inside a viscous fluid in a Hele-Shaw cell. The method of solution of the exterior Dirichlet problem employs a classical boundary integral formulation. Our version of the numerical method is especially advantageous for following the dynamics of a very long and thin bubble, for which an asymptotic scaling theory has been recently developed. Because of the very large aspect ratio of the bubble, a direct implementation of the boundary integral algorithm would be impractical. We modify the algorithm by introducing a new approximation of the integrals which appear in the Fredholm integral equation and in the integral expression for the normal derivative of the pressure at the bubble interface. The new approximation allows one to considerably reduce the number of nodes at the almost flat part of the bubble interface, while keeping a good accuracy. An additional benefit from the new approximation is in that it eliminates numerical divergence of the integral for the tangential derivative of the harmonic conjugate. The interface's position is advanced in time by using explicit node tracking, whereas the larger node spacing enables one to use larger time steps. The algorithm is tested on two model problems, for which approximate analytical solutions are available. 
Arkady Vilenkin and Baruch Meerson