Accurate asymptotic formulas for the transient PDF of a FENE dumbbell in suddenly started uniaxial extension followed by relaxation

Singular perturbation theory is combined with the method of multiple scales to derive an asymptotic solution for the transient, one-dimensional probability density function (PDF) of a FENE dumbbell in a suddenly started uniaxial extensional flow. We consider the dual asymptotic limit of large dimensionless spring length, L=ɛ−1L=ɛ−1, and large dimensionless elongation rate (or Weissenberg number), Γ=γ/ɛ, with these two quantities remaining in a fixed (arbitrary) proportion, γγ. The analytical formula for the transient PDF agrees closely with numerics in both (i) the central, approximately Gaussian core, and (ii) a thin boundary layer near the limit of extension [see, e.g., R. Keunings, J. Non-Newtonian Fluid Mech. 68 (1997) 85–100]. Stress buildup and stress–extension curves are well predicted. We also explain the collapse of different dumbbell lengths onto a single stress–extension line, τ=2Γ〈x2〉, which has been observed numerically in connection with transient stress-birefringence [P.S. Doyle, E.S.G. Shaqfeh, G.H. McKinley, S.H. Spiegelberg, J. Non-Newtonian Fluid Mech. 76 (1998) 79–110]. This result agrees with the large-strain plateau in the transient stress-optic coefficient for large Weissenberg number [R. Sizaire, G. Lielens, I. Jaumain, R. Keunings, V. Legat, J. Non-Newtonian Fluid Mech. 82 (1999) 233–253]. For relaxation of the FENE dumbbell from its fully stretched configuration, a Gaussian approximation of the PDF—whose time-dependent position and width are given by closed analytical formulas—matches the numerical results extremely well. The slope of the advective velocity is interpreted as a negative contribution to the effective diffusion coefficient. The asymptotic theory supports the recently proposed L closure [G. Lielens, P. Halin, I. Jaumain, R. Keunings, V. Legat, J. Non-Newtonian Fluid Mech. 76 (1998) 249–279].