  We investigate the global existence of weak solutions to a free boundary
problem governing the evolution of finitely extensible bead-spring chains in
dilute polymers. We construct weak solutions of the two-phase model by
performing the asymptotic limit as the adiabatic exponent $\gamma$ goes to
$\infty$ for a macroscopic model which arises from the kinetic theory of dilute
solutions of nonhomogeneous polymeric liquids. In this context the polymeric
molecules are idealized as bead-spring chains with finitely extensible
nonlinear elastic (FENE) type spring potentials. This class of models involves
the unsteady, compressible, isentropic, isothermal Navier-Stokes system in a
bounded domain $\Omega$ in $\mathbb{R}^d,$ $d=2, 3$ coupled with a
Fokker-Planck-Smoluchowski-type diffusion equation (cf. Barrett and S\"{u}li
[4], [5], [9]). The convergence of these solutions, up to a subsequence, to the
free-boundary problem is established using weak convergence methods,
compactness arguments which rely on the monotonicity properties of certain
quantities in the spirit of [19].
