Anisotropy parameter restrictions for the eXtended Pom-Pom model

作者：Baltussen Michiel G H M; Verbeeten Wilco M H; Bogaerds Arjen C B; Hulsen Martien A; Peters Gerrit W M^*

来源：Journal of Non-Newtonian Fluid Mechanics, 2010, 165(19-20): 1047-1054.

DOI：10.1016/j.jnnfm.2010.05.002

• eXtended Pom-Pom model
• Giesekus model
• Shear viscosity
• Extensional viscosity
• Second normal stress coefficient
• Restricted anisotropy parameter

A significant step forward in modelling polymer melt rheology has been the introduction of the Pom-Pom constitutive model of McLeish and Larson [T.C.B. McLeish, R.G. Larson, Molecular constitutive equations for a class of branched polymers: the Pom-Pom polymer, J. Rheol. 42 (1) (1998) 81-110]. Various modifications of the Pom-Pom model have been published over the years in order to overcome several inconveniences of the original model. Amongst those modified models, the eXtended Pom-Pom (XPP) model of Verbeeten et. al. [W.M.H. Verbeeten, G.W.M. Peters, F.P.T. Baaijens, Differential constitutive equations for polymer melts: the extended Pom-Pom model, J. Rheol. 45 (4) (2001) 823-843] has received quite some attention. However, the XPP model has been criticized for the generation of multiple and unphysical solutions. This paper deals with two issues. First, in the XPP model, anisotropy is implemented in a Giesekus-like manner which is known to result in unphysical solutions for non-linear parameter values alpha >= 0.5. Hence, we put forward the conjecture that a similar limitation holds for the XPP model. In the present paper, the limits for the anisotropy parameter are elaborated on and result to be most restraining at high deformation rates where the backbone tube is oriented and the backbone tube stretch approaches the number of arms q. By restricting the anisotropy parameter to a maximum critical value the XPP model produces only one solution, which is the correct physical rheology. In the second part we show that, contrary to the results published by Inkson and Phillips [N.J. Inkson, T.N. Phillips, Unphysical phenomena associated with the extended Pom-Pom model in steady flow, J. Non-Newton. Fluid 145 (2-3)(2007)92-101], for the special case where the anisotropy parameter equals zero, only one physically relevant solution exists in unaxial extensional. In addition to this physically relevant solution, also solutions exist in the physically unattainable part of the conformation space. However, the existence of these physically unattainable solutions is not a unique feature of the XPP model but rather general for non-linear differential type rheological equations.

• 出版日期2010-10