Asymptotic basis of the L closure for finitely extensible dumbbells in suddenly started uniaxial extension

This paper applies the recent L closure [G. Lielens, P. Halin, I. Jaumain, R. Keunings, V. Legat, J. Non-Newtonian Fluid Mech., 76 (1998), 249-279], which was originally developed for FENE dumbbells with fixed friction, to two alternative dumbbell models of polymers in a dilute solution undergoing uniaxial extension: (A) a linear-locked dumbbell with fixed friction and (B) a FENE dumbbell with variable friction. The simplified box-spike representation of the probability density function (PDF) - used for Model B to close conformational averages of nonlinear quantities in terms of a reduced set of state variables (moments of the PDF) - is justified through detailed asymptotic analysis (singular perturbations combined with multiple scales) of the Smoluchowski equation, in the limit of large extensibility parameter at fixed elongation rate. Both dumbbell models A and B are actually more amenable to the L closure than the FENE to which it had previously been applied. The resulting closure relations compare favorably with corresponding integrals of asymptotic or numerical PDF's (the latter obtained via atomistic SPH simulations of the Smoluchowski equation). Example calculations show the L closure to yield reasonably accurate stress-extension curves, even for a moderate (dimensionless) limit of extension (L=4).