  This paper is concerned with the mathematical derivation of the
inhomoge-neous incompressible Navier-Stokes equations (INS) from the
compressible Navier-Stokes equations (CNS) in the large volume viscosity limit.
We first prove a result of large time existence of regular solutions for (CNS).
Next, as a consequence, we establish that the solutions of (CNS) converge to
those of (INS) when the volume viscosity tends to infinity. Analysis is
performed in the two dimensional torus, for general initial data. In
particular, we are able to handle large variations of density.
