Singularities in Hele--Shaw Flows Driven by a Multipole

We study, analytically and numerically, singularity formation in an interface flow driven by a multipole for a two-dimensional Hele–Shaw cell with surface tension. Our analysis proves that singularity formation is inevitable in the case of a dipole. For a multipole of a higher order, we show that the solution does not tend to any stationary solution as time goes to infinity if its initial center of mass is not at the multipole; it is therefore very likely that this solution will develop finite time singularities. Our extensive numerical studies suggest that a solution develops finite time singularities via the interface reaching the multipole while forming a corner at the tip of the finger that touches the multipole. In addition, it is observed that the interface approaches the multipole from directions which can be predicted beforehand. We also estimate as a function of time the distance between the finger tip and multipole, and the results are in excellent agreement with numerical computations.