Multiple bubbles and fingers in a Hele-Shaw channel: complete set of steady solutions

Giovani L. Vasconcelos
  • Journal of Fluid Mechanics, September 2015, Cambridge University Press
  • DOI: 10.1017/jfm.2015.469

Multiple bubbles in a Hele-Shaw cell

What is it about?

A complete set of exact solutions is presented for multiple bubbles steadily propagating in a Hele-Shaw cell. The solutions are written in explicit form in terms of a conformal mapping involving a novel class of special transcendetal functions, the so-called secondary Schottky-Klein prime functions, recently introduced by the author and collaborators. To motivate the introduction of these special functions, a generalized method of images is first employed, which (we hope) makes the paper accessible to a larger audience. It is also hoped that the mathematical formalism described in the paper may find application in other areas of potential theory and fluid mechanics.

Why is it important?

Because it solves the long-standing problem of the steady motion of multiple bubbles (including the special case of multiple fingers and bubbles) in a Hele-Shaw channel. All previous solutions found for this problem are special cases of the general solutions presented in our paper. Our formalism also paves the way for constructing exact time-dependent solutions for the evolution of multiple bubbles, starting from an arbitrary initial configuration.

Perspectives

Professor Giovani L. Vasconcelos (Author)
Universidade Federal de Pernambuco

The next step is to construct time-dependent exact solution for multiple bubbles and to study the selection problem. Preliminary results (in collaboration with M. Mineev-Weinstein) show that steady solutions with velocity U=2 is generally selected in this case as well.

The following have contributed to this page: Professor Giovani L. Vasconcelos