# Semi-Analytical Solutions for the Poiseuille–Couette Flow of a Generalised Phan-Thien–Tanner Fluid

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## Abstract

**:**

## 1. Introduction

## 2. Analytical Solution for the gPTT Model in Couette flow

**Remark**

**1.**

## 3. Analytical Solution for the gPTT Model in Pure Couette flow.

## 4. Discussion of Results

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**2019**, 269, 88–99. [Google Scholar] [CrossRef] - Phan-Thien, N.; Tanner, R. A new constitutive equation derived from network theory. J. Non-Newton. Fluid Mech.
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**Figure 1.**Dimensionless material properties in steady-state Couette flow using the three versions of the sPTT and for the gPTT model: (

**a**) $\beta =1$; and (

**b**) $\alpha =1$.

**Figure 2.**Fitting of the shear viscosity and the first normal stress difference coefficient to rheological data from Laun [7]. The generalised PTT model only considers the one-parameter Mittag–Leffler function, ${E}_{\alpha}$. By adding only one parameter, we obtain a fitting error (Equation (17)) of $29.7$ and 6 for the exponential and gPTT models, respectively. The symbols represent the experimental data from Laun [7] for a low density polyethylene melt.

**Figure 3.**Geometry of: (

**a**) the pure Couette flow; and (

**b**) the Couette flow with an imposed pressure gradient (Poiseuille–Couette flow).

**Figure 4.**Comparison between the gPTT model and exponential PTT considering a Poiseuille flow with different values of $\epsilon W{i}^{2}$ and different values of imposed $\overline{{P}_{x}}$.

**Figure 5.**Velocity profiles obtained for the Poiseuille–Couette flow considering different values of $\epsilon {Wi}^{2}$ and different values of $\alpha $ ($\beta =1$): (

**a**) $\overline{{P}_{x}}=-1$; and (

**b**) $\overline{{P}_{x}}=-2$.

**Figure 6.**Velocity profiles obtained for the Poiseuille–Couette flow considering different values of $\epsilon {Wi}^{2}$ and different values of $\beta $ ($\alpha =1$): (

**a**) $\overline{{P}_{x}}=-1$; and (

**b**) $\overline{{P}_{x}}=-2$.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ribau, Â.M.; Ferrás, L.L.; Morgado, M.L.; Rebelo, M.; Afonso, A.M. Semi-Analytical Solutions for the Poiseuille–Couette Flow of a Generalised Phan-Thien–Tanner Fluid. *Fluids* **2019**, *4*, 129.
https://doi.org/10.3390/fluids4030129

**AMA Style**

Ribau ÂM, Ferrás LL, Morgado ML, Rebelo M, Afonso AM. Semi-Analytical Solutions for the Poiseuille–Couette Flow of a Generalised Phan-Thien–Tanner Fluid. *Fluids*. 2019; 4(3):129.
https://doi.org/10.3390/fluids4030129

**Chicago/Turabian Style**

Ribau, Ângela M., Luís L. Ferrás, Maria L. Morgado, Magda Rebelo, and Alexandre M. Afonso. 2019. "Semi-Analytical Solutions for the Poiseuille–Couette Flow of a Generalised Phan-Thien–Tanner Fluid" *Fluids* 4, no. 3: 129.
https://doi.org/10.3390/fluids4030129