4. Parametric study of the GPTT model

We will now present a detailed parametric study on the influence of the new parameters α, β (arising from the Mittag-Leffler function) on the rheological behavior of the generalized exponential PTT model.

### 4.1. Steady-state shear flows

\_ \_ ðγ\_Þ ¼ σxyð Þ γ γ with As shown in [18], the steady shear viscosity is given by η =

$$\sigma\_{\text{xy}}(\dot{\gamma}) = \frac{\eta \dot{\gamma} - \sigma\_{\text{xx}} \dot{\text{Wi}} \xi}{\Gamma \{ \beta \} E\_{\alpha, \beta} \left( \frac{\varepsilon \lambda}{\eta} \left( \frac{2 - 2\xi}{2 - \xi} \right) \sigma\_{\text{xx}} \right)} \,\tag{22}$$

and σxx is given by the solution of

$$
\Gamma^2(\beta) E\_{a,\beta} \left( \frac{\varepsilon \lambda}{\eta} \left( \frac{2 - 2\xi}{2 - \xi} \right) \sigma\_{\text{xx}} \right)^2 \sigma\_{\text{xx}} = (2 - \xi) (\lambda \dot{\gamma})^2 \left[ \frac{\eta}{\lambda} - \sigma\_{\text{xx}} \xi \right]. \tag{23}
$$

Here Wi ¼ λ γ is the dimensionless strength of the shear flow and η, λ, ε, ξ, α, β are the constitutive parameters of the generalized PTT (or GPTT) model. \_

\_ Since we consider a simple plane shear flow aligned with the x-axis, we have that ðγÞ ¼ σxxξ= ðγÞ ¼ 0 (see [18] for more details). σ \_ yy ð2 � ξÞ and σzz

Eqs. (22) and (23) can readily be solved using the Newton-Raphson method (solving first Eq. (23) and then substituting the numerical values obtained for σxx into Eq. (22)).

Figure 6 shows the dimensionless steady shear viscosity obtained for the different parameters of the Mittag-Leffler function, α, β. On the left, we show the influence of α by keeping constant all other parameters. On the right, we show the influence of β (it should be remarked that when α, β ¼ 1 the exponential PTT model is recovered). We observe that when compared to the

Figure 6. Dimensionless shear viscosity obtained for the different parameters of the Mittag-Leffler function; (a) varying α holding the other five parameters constant and (b) varying β.

Figure 7. Dimensionless shear viscosity obtained for the different parameters of the Mittag-Leffler function varying: (a) Constant ε; (b) Varying ε.

classical exponential PTT model, when α, β < 1, shear-thinning occurs for lower dimensionless shear rates and when α, β > 1 there is a delay in the shear-thinning effect. For α, β > 1 the shear viscosity increases, especially for high shear rates. Also, when we increase α, the slope of the shear viscosity curve for high dimensionless shear rates decreases (observed in Figure 6(a)), while varying β, the slope seems to be the same, but a higher viscosity is obtained (observed in Figure 6(b)).

Figure 7(a) shows the dimensionless steady shear viscosity, now obtained for different values of α, β and ε. These plots allow one to see that the ε parameter may not be compared directly to the value used in the classical models (featuring linear and exponential functions of the trace of the stress tensor). For comparison purposes, we plot again the curve obtained for the exponential PTT model with ε ¼ 0:25 (α ¼ β ¼ 1) by the dash-dot lines.

Note that (see Figure 7(b)) small variations of the parameter ε allows one to control the rate of transition to the shear-thinning at high Wi while maintaining a similar shear thinning set point.

Figure 7 shows that by setting different combinations of α, β we may obtain different slopes at higher dimensional shear rates. For low β and high α, we obtain a lower slope but a premature shear viscosity thinning, while for high β and low α, we obtain a higher slope but a delayed shear-thinning.

## 4.2. Steady-state elongational flows

\_ \_ ˜ ° The steady unidirectional extensional viscosity is defined as <sup>η</sup> <sup>ς</sup>, where <sup>ς</sup> is the <sup>E</sup> <sup>¼</sup> <sup>σ</sup>xx � <sup>σ</sup>yy <sup>=</sup> imposed elongation rate [18], and can be obtained by solving the system of equations (for a simpler technique that does not involve an iterative procedure, please consult [18])

$$
\sigma\_{\rm xx} \left( \Gamma \{ \beta \} E\_{a,\beta} \left( \frac{\varepsilon \lambda}{\eta} \sigma\_{kk} \right) - 2 \lambda \dot{\varepsilon} (1 - \xi) \right) = 2 \eta \dot{\varepsilon},
\tag{24}
$$

Recent Advances in Complex Fluids Modeling 19 http://dx.doi.org/10.5772/intechopen.82689

$$
\sigma\_{\rm xx} \left( \Gamma(\beta) E\_{a,\beta} \left( \frac{\varepsilon \lambda}{\eta} \sigma\_{\rm k\bar{\ell}} \right) + \lambda \dot{\varepsilon} (1 - \xi) \right) = -\eta \dot{\varepsilon}, \tag{25}
$$

with σkk ¼ σxx þ 2σyy.

Figure 8 shows the dimensionless steady elongational viscosity obtained for different parameters of the Mittag-Leffler function. In Figure 8(a), we show the influence of α by keeping constant all other parameters. In Figure 8(b), we show the influence of β. Note that we have used the same parameters as in the shear viscosity case.

Note that when we increase α, β, we observe an increase of the elongational viscosity, with the maximum value being reached for higher dimensionless extensional rates. Again, we observe different asymptotic slopes for high extension rates (when varying α). Note that there is no overshoot for low values of β.

We may conclude that by varying α, β, we change both the shear and elongational viscosities, and therefore the fit to experimental data should be performed with care, taking into account this dependence.

Figure 9 shows the effect of the parameters used in Figure 7, for the case of elongational viscosity. The results are qualitatively similar to the ones obtained in Figure 7, that is, in terms of changes to the asymptotic slopes at high deformation rates and premature/delayed thinning. It can be observed that the elongational viscosity is more sensitive to changes in the parameters α, β and ε. This result is to be expected since this is a strong flow, and, the exponential PTT model was originally proposed to be able to describe the response of complex fluids in strong flows. Figure 9(a) shows that the overshoot can be suppressed using a low β and high α values. Note that the maximum extensional viscosity is obtained for the exponential PTT model, and that the values of α, β have a strong influence on the asymptotic slope of the curve for high extensional rates. Figure 9(b) shows three different curves for different combinations of α, β and ε. Note that for α ¼ 0:1, β ¼ 0:1 and ε ¼ 10 we can also suppress the

Figure 8. Dimensionless elongational viscosity obtained for different parameters of the Mittag-Leffler function: (a) Varying α; (b) Varying β.

Figure 9. Dimensionless elongational viscosity obtained for the different parameters of the Mittag-Leffler function: (a) Constant ε; (b) Varying ε.

overshoot in the extensional viscosity, and for α ¼ 2, β ¼ 0:1 and ε ¼ 0:05 we can decrease the curvature of the overshoot, and at the same time decrease the slope of curve.

### 4.3. Steady-state shear and elongational flows

Until now, we have explored generally the influence of the different model parameters on the behavior of the GPTT model for steady flows, but, a more quantitative side-by-side comparison

Figure 10. Comparison of the dimensionless elongational and shear viscosity obtained for different parameters of the Mittag-Leffler function, varying ε, and the classical exponential PTT model (α = 1, β = 0).

between the shear and elongational flow curves was not performed, and the limited flexibility of the classical exponential PTT model for fitting experimental data (when compared to the GPTT) was not explored. In Figure 10, we try to illustrate the advantages of using the Mittag-Leffler function instead of the classical exponential one. To this end, we present the viscometric predictions obtained for both shear and elongational flows for both models (GPTT and exponential PTT).

Figure 10 illustrates the additional flexibility of using the Mittag-Leffler function, by showing that we can manipulate the magnitude of the increase in the elongational viscosity and at the same time only slightly change the shear viscosity. This allows better fits to rheological data when using the Mittag-Leffler function [18]. Note that in the exponential PTT model, when we increase the ε parameter, both the shear and elongational viscosities increase concomitantly.
