  We explore the behaviour of global-in-time weak solutions to a class of
bead-spring chain models, with finitely extensible nonlinear elastic (FENE)
spring potentials, for dilute polymeric fluids. In the models under
consideration the solvent is assumed to be a compressible, isentropic, viscous,
isothermal Newtonian fluid, confined to a bounded open domain in
$\mathbb{R}^3$, and the velocity field is assumed to satisfy a complete slip
boundary condition. We show that as the Mach number tends to zero the system is
driven to its incompressible counterpart.
