Аннотация:We here deal with the Stokes-Leibenson problem for a punctual Hele-Shaw flow.
By using a geometrical transformation inspired by Helmholtz-Kirchhoff method, we
introduce an integro-differential problem which leads to the construction of a discrete
model. We first give a short recall about the source-case: global in time existence
and uniqueness result for an initial contour close to a circular one, investigation of the
evolutionary structure of the solution. Our main subject concerns the development of a
numerical model in order to get some qualitative properties of the motion. This model
provides numerical experiments which confirm the existence of a critical manifold of
codimension 1 in some space of contours. This manifold contains one attractive point
in the source-case corresponding to a circular contour centered at the source-point. In
the sink-case, every point of this manifold seems to be attractive. In particular, we
present some numerical experiments linked to fingering effects.