On the similarities of the sPTT and FENE-P models for polymeric fluids

For many commonly used single mode viscoelastic constitutive equations of differential type, it is well known that they share many features. For example, in certain parameter limits the models due to Giesekus, Phan-Thien Tanner and FENE-type models approach the Oldroyd-B model. In this talk, I'll compare the response of the linear form of the simplified Phan-Thien Tanner model [due to Phan-Thien and Tanner, Journal of Non-Newtonian Fluid Mechanics, 2 (1977) 353–365] (the "sPTT") and the Finitely Extensible Nonlinear Elastic model that follows the Peterlin approximation [due to Bird et al., Journal of Non-Newtonian Fluid Mechanics, 7 (1980) 213–235] (the "FENE-P"). I'll show that for steady homogeneous flows such as steady simple shear flow or pure extension the response of both models is identical under certain conditions. For more general flows we show analytically that the results from the two models only formally approach each other when both the polymer concentration and Weissenberg number is small. We then use a numerical approach to investigate the response of the two models when the flow is "complex" under two different definitions: firstly, when the applied deformation field is homogenous in space but transient in time (so called "start-up" shear and planar extensional flow) and then for "complex" flows (through a range of geometries) which, although Eulerian steady, are unsteady in a Lagrangian sense. Under the limit that the flows remain Eulerian steady (so the Weissenberg number is typically small), we see once again a very close agreement between the FENE-P and sPTT models.