A simple phenomenological non-Newtonian fluid model
Introduction
As early as in 1948 Rivlin observed that for an isotropic generalized Newtonian fluid the viscosity is a function of only three principal invariants of the deformation rate tensor, [1], [2]. However, for incompressible non-Newtonian liquids (for which the first invariant of the deformation rate tensor is zero) viscosity is usually considered to be a function of only the second and third principal invariants of the deformation rate tensor [3], [4], [5], [6], [7], [8], [9], [10]. The main weakness of generalized Newtonian law model, in which the stress tensor is directly proportional to the deformation rate tensor through , is its incapability to correctly represent the flow behavior in extensional flows, especially in plane flows, where the third invariant of deformation rate tensor IIID is 0. In order to overcome this problem, new definition of the non-Newtonian fluid viscosity is proposed here and tested for three different polymer melts (highly branched LDPE, slightly branched mLLDPE and linear HDPE) whose rheological characteristics were taken from the open literature [11], [12].
Section snippets
Model development and testing
The simplest non-Newtonian fluid model is generalized Newtonian law in the following form:where τ means the stress tensor, D represents the deformation rate tensor and η stands for the viscosity which is not constant (unlike in the standard Newtonian law) and is allowed to vary with the first, second and third invariant of the deformation rate tensor, , and , respectively. Let us consider viscosity as a variable which depends on three principal
Conclusion
The simple phenomenological generalized Newtonian law model proposed and tested for different polymer melts in this work has proved a good capability to describe the strain rate dependent steady shear and uniaxial extensional viscosities for linear and branched polyolefines. Moreover, it has shown correct predictive behavior for steady planar and equibiaxial extensional viscosities and ability to control their strain hardening level independently of uniaxial extensional viscosity. This supports
Acknowledgments
The author wishes to acknowledge Czech Science Foundation and the Ministry of Education, Youth and Sports of CR for the financial support (Grant No. 103/09/2066 and MSM 7088352101, respectively).
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