3.2. Generalized Phan-Thien and Tanner model

The previous integral model given by Eq. (18) allows a good fit to experimental rheological data, in flows with defined kinematics where C-<sup>1</sup> can be computed explicitly; but, it would be desirable to obtain also an improved frame-invariant differential model, that is easier to handle both mathematically and numerically, when compared to integral models, for solving complex flow problems with spatially varying kinematics. The model to be presented was recently proposed by our research group [18], and basically takes advantage of the flexible functional form of the Mittag-Leffler function by inserting this function into the already well-known Phan-Thien and Tanner (PTT) model, replacing the classical linear and exponential functions of the trace of the stress tensor.

The original exponential PTT model [19, 20] is given by.

$$\exp\left(\frac{\varepsilon\lambda}{\eta}\mathfrak{o}\_{kk}\right)\mathfrak{o} + \lambda\overleftarrow{\mathfrak{G}} = \eta\dot{\gamma},\tag{20}$$

□ <sup>T</sup> with <sup>σ</sup> <sup>¼</sup> <sup>∂</sup>σ=∂<sup>t</sup> <sup>þ</sup> <sup>u</sup> • <sup>∇</sup><sup>σ</sup> - ð∇uÞ • <sup>σ</sup> - <sup>σ</sup> • <sup>∇</sup><sup>u</sup> <sup>þ</sup> <sup>ξ</sup>ð<sup>D</sup> • <sup>σ</sup> - <sup>σ</sup> • <sup>D</sup><sup>Þ</sup> being the Gordon-Schowalter derivative and σkk the trace of the stress tensor. Here, the parameter ξ accounts for slip between the molecular network and points in the continuous medium. The model was derived from a Lodge-Yamamoto type of network theory for polymeric fluids, in which the network junctions are not assumed to move strictly as points of the continuum but instead they are allowed a certain effective slip as well as a rate of destruction that depends on the state of stress in the network. Phan-Thien proposed that an exponential function form would be quite adequate to represent the rate of destruction of junctions and in [17] it was shown that the Mittag-Leffler function could improve the quality of model fits to real data by allowing different forms for the rates of destruction.

The model is then given by

$$
\Gamma\left(\beta\right) E\_{\alpha,\beta} \left(\frac{\varepsilon\lambda}{\eta}\mathfrak{o}\_{kk}\right) \mathfrak{o} + \lambda \stackrel{\bigtriangleup}{\mathfrak{o}} = \eta \dot{\mathfrak{y}} \tag{21}
$$

˛ ˝ where the factor <sup>Γ</sup>(β) is used to ensure that <sup>Γ</sup> <sup>β</sup> <sup>E</sup>α,βð0Þ ¼ 1.

This new model can further improve the accuracy of the description of real data obtained with the original exponential function of the trace of the stress tensor, as shown in [18].
