NEW CLOSURE APPROXIMATIONS FOR THE KINETIC-THEORY OF FINITELY EXTENSIBLE DUMBBELLS

We address the closure problem for the most elementary non-linear kine tic model of a dilute polymeric solution, known as the Warner finitely extensible non-linear elastic (FENE) dumbbell model. In view of the c losure problem, the FENE theory cannot be translated into an equivalen t macroscopic constitutive equation for the polymer contribution to th e stress tensor. We present a general framework for developing closure approximations, based on the concept of canonical distribution subspa ce first introduced by Verleye and Dupret (in: Developments in Non-New tonian Flows, AMD-Vol. 175, ASME, New York, 1993, 139-163) in the cont ext of fiber suspension modeling. The classical consistent pre-averagi ng approximation due to Peterlin (that yields the FENE-P constitutive equation) is obtained from the canonical approach as the simplest firs t-order closure model involving only the second moment of the configur ation distribution function. A second-order closure model (referred to as FENE-P-2) is derived, which involves the second and fourth moments of the distribution function. We show that the FENE-P-2 model behaves like the FENE-P equation with a reduced extensibility parameter. In t his respect, it is a close relative of the FENE-P equation proposed b y van Heel et al. (J. Non-Newton. Fluid Mech., 1998, in press). Inspir ed by stochastic simulation results for the FENE theory, we propose a more sophisticated second-order closure model (referred to as FENE-L). The rheological response of the FENE-P, FENE-P-2 and FENE-L closure m odels are compared to that of the FENE theory in various time-dependen t, one-dimensional elongational flows. Overall, the FENE-L model is fo und to provide the best agreement with the FENE results. In particular , it is capable of reproducing the hysteretic behaviour of the FENE mo del, also observed in recent experiments involving polystyrene-based B oger fluids (Doyle et al., J. Non-Newton. Fluid Mech., submitted), in stress versus birefringence curves during startup of flow and subseque nt relaxation. (C) 1998 Elsevier Science B.V. All rights reserved.