# Re-Entrant Corner for a White-Metzner Fluid

^{1}

^{2}

^{*}

^{†}

^{‡}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### Governing Equations

## 3. Results

#### 3.1. Boundary Layer Analysis

#### Outer Solution

- (I)
- $m>1$,
- (II)
- $\alpha \left(\right)open="("\; close=")">2+m(q-n)$,
- (III)
- $\alpha <2{m}^{-1}$,
- (IV)
- $\frac{1}{2}<\alpha <1$.

#### 3.2. Upstream Boundary Layer

#### 3.3. Similarity Solution

#### Near Wall Analysis

#### 3.4. Upstream Boundary Layer

## 4. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

PTT | Phan-Thien-Tanner |

UCM | upper-convective Maxwell |

WM | White-Metzner |

## Appendix A. Series Expansion Terms

## Appendix B. Parameters

## Appendix C. Constraint

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**Figure 2.**Solution of the upstream boundary layer with $a=-1$ and ${p}_{0}=1$ for $q=n$. The results are shown for $n=0.2,1$ and 2 which are denoted by the solid, dashed, and dotted lines respectively.

**Figure 3.**Solution of the scaled upstream boundary layer with $a=-1$ and ${p}_{0}=1$ for $q=n$. The results, which are scaled by the outer solution, are shown for $n=0.2,1$ and 2 which are denoted by the solid, dashed and dot-dashed lines, respectively.

**Table 1.**Tabulated values of parameters for the WM model. Fluid A is polyisobutylene solution in decalin solvent [32]. The parameters for Fluid B are those fitted from a pseudo-Carreau Cross WM model to a low-density polyethylene melt [33]. Fluids C and D are polyethylene and polystyrene melts [34].

Fluid | n | q |
---|---|---|

A | 0.33 | 0.56 |

B | 0.4 | 0.0 |

C | 0.29 | 0.11 |

D | 0.17 | 0.064 |

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**MDPI and ACS Style**

Chaffin, S.; Monk, N.; Rees, J.; Zimmerman, W.
Re-Entrant Corner for a White-Metzner Fluid. *Fluids* **2021**, *6*, 241.
https://doi.org/10.3390/fluids6070241

**AMA Style**

Chaffin S, Monk N, Rees J, Zimmerman W.
Re-Entrant Corner for a White-Metzner Fluid. *Fluids*. 2021; 6(7):241.
https://doi.org/10.3390/fluids6070241

**Chicago/Turabian Style**

Chaffin, Stephen, Nicholas Monk, Julia Rees, and William Zimmerman.
2021. "Re-Entrant Corner for a White-Metzner Fluid" *Fluids* 6, no. 7: 241.
https://doi.org/10.3390/fluids6070241