  We consider a model of congestion dynamics with chemotaxis, where the density
of cells follows the chemical signal it generates, while observing an
incompressibility constraint. We show that when the chemical diffuses slowly
and attracts the cells strongly, then the dynamics of the congested cells is
well approximated by a surface-tension driven free boundary problem. More
precisely, we show that in this limit the density of cell converges to the
characteristic function of a set whose evolution is described by a Hele-Shaw
free boundary problem with surface tension. Our problem is set in a bounded
domain, which leads to an interesting analysis on the limiting boundary
conditions for the density function. Namely, we prove that the assumption of
Robin boundary conditions for the chemical potential leads to a contact angle
condition for the free interface.
