On the similarities between the simplified Phan-Thien
Tanner (sPTT) and FENE-P models for polymeric fluids
M Davoodi1, K Zografos2 and R
J Poole3
1Schlumberger Cambridge Research, UK
2Department of Mechanical and Aerospace Engineering, University of Strathclyde, Glasgow, UK
3School of Engineering, University of Liverpool, UK
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.
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