# Analytical Solutions of Upper Convected Maxwell Fluid with Exponential Dependence of Viscosity under the Influence of Pressure

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## Abstract

**:**

## 1. Introduction

**T**is the Cauchy stress tensor,

**I**the unit tensor, $\mathit{A}=\mathit{L}+{\mathit{L}}^{T}$ is the first Rivlin-Ericksen tensor, in which

**L**is the gradient of the velocity vector, and $\eta (p)$ is the fluid viscosity. From the previous constitutive relation, it results that the frictional forces exerted by adjacent layers of the fluid depend on the normal force that acts between the layers Fusi [9].

^{−1}for the polymer melts (Carreras et al. [25] and Sorrentino and Pantani [26]), 10–70 GPa

^{−1}for lubricants (Kottke et al. [27]) and 10–20 GPa

^{−1}for mineral oils (Venner and Lubrecht [28]). These values are also valid for small or medium pressure differences $p-{p}_{0}$.

## 2. Presentation of the Problem

_{2}. If $\alpha \to 0$ in this relation, then $\eta (p)\to \mu $, and Equation (3) reduces to the constitutive equations of the ordinary incompressible UCM fluids. In addition, $\eta (p)\to \infty $ if $p\to \infty $, which is a property that has been experimentally proved.

_{1}in (3)

_{2}and bearing in mind the above assumption, we find that

_{2,}it clearly results from Equations (9)

_{2}, (9)

_{3}, and (10)

_{2}that ${S}_{yy}={S}_{yz}={S}_{zz}=0$. Using ${S}_{yz}=0$ in Equation (10)

_{3,}it also results that ${S}_{xz}=0$.

_{2}between 0 and d, it results that

_{2}and (13)

_{1}and using the expressions of $\eta (p)$ and $p(y)$ from Equations (2) and (14), one obtains the governing equation

## 3. Exact Expressions for the Dimensionless Velocity and Shear Stress Fields

_{1}. To avoid confusion, the starting solutions of the two distinct motion problems will be denoted by ${u}_{c}(y,t)$, ${\tau}_{c}(y,t)$ and ${u}_{s}(y,t)$, ${\tau}_{s}(y,t)$. In order to determine them, we shall use suitable changes of the spatial variable and the unknown function and the Laplace transform technique.

#### 3.1. Calculation of the Velocity Field ${u}_{c}(y,t)$

#### 3.2. Calculation of the Velocity Field ${u}_{s}(y,t)$

#### 3.3. Calculation of the Shear Stress ${\tau}_{c}(y,t)$

_{1}, we find that

_{2}and its correspondent for the function ${Y}_{1}(\cdot )$, the result is

#### 3.4. Calculation of the Shear Stress ${\tau}_{s}(y,t)$

## 4. Limiting Cases

#### 4.1. Case $\omega \to 0$ (Lower Plate Applies an Exponential Shear Stress to the Fluid)

_{1}). This is not a surprising case, because the ordinary differential equations governing the steady motions of the incompressible UCM or Newtonian fluids with pressure-dependent viscosity are identical. What is very important is the fact that the non-dimensional steady shear stress corresponding to this motion is constant on the entire flow domain, although the adequate velocity field depends of the spatial variable. This constant is just the dimensionless shear stress applied to the fluid by the lower plate.

#### 4.2. Case $\beta \to 0$ (Flows of Ordinary Incompressible UCM Fluids)

#### 4.3. Case $\mathrm{We}\to 0$ (Flows of Incompressible Newtonian Fluids with Exponential Dependence of Viscosity on the Pressure)

## 5. Some Numerical Results and Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$\mathit{T}$ | Cauchy stress tensor | $\mathit{S}$ | Extra stress tensor |

$\mathit{A}$ | First Rivlin–Ericksen tensor | $\mathit{v}$ | Velocity vector |

L | The velocity gradient | u | Fluid velocity |

p | Pressure | $x,y,z$ | Cartesian coordinates |

$\tau $ | Non-trivial shear stress | ${\sigma}_{x}$ | Non-trivial normal stress |

t | Time | g | Gravitational acceleration |

$\lambda $ | Relaxation time | $\rho $ | Fluid density |

$\mu $ | Fluid viscosity at the reference pressure | $\nu $ | Kinematic viscosity |

$\alpha $ | Pressure–viscosity coefficient | S | Constant shear-stress |

d | Distance between plates | We | Weissenberg number |

## Appendix A

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**Figure 1.**The profiles of velocity ${u}_{c}(y,t)$ for We = 0.5 and different values of the pressure–viscosity parameter $\beta $ at (

**a**) t = 0.05, (

**b**) t = 0.10, (

**c**) t = 0.15.

**Figure 2.**The profiles of velocity ${u}_{c}(y,t)$ for We = 0.75 and different values of the pressure–viscosity parameter $\beta $ at (

**a**) t = 0.05, (

**b**) t = 0.10, (

**c**) t = 0.15.

**Figure 3.**Spatial variation of the velocity ${u}_{c}(y,t)$ in the channel (

**a**), and its profiles from four transversal sections (

**b**), when the Weissenberg parameter belongs to the interval [0.1, 3].

**Figure 4.**Profiles of the transient velocity ${u}_{ct}(y,t)$ versus t, for three different values of pressure–viscosity parameter $\beta $ at (

**a**) y = 0.4, We = 0.5, and (

**b**) y = 0.4, We = 0.75.

**Figure 5.**Spatial profile of the frictional force ${\tau}_{c}(1,t)$ on the upper plate for $t\in [0,4]$ and $\mathrm{We}\in [0.1,3]\hspace{0.17em}.$

**Figure 6.**The profiles of the frictional force ${\tau}_{c}(1,t)$ on the upper plate (

**a**) versus time t, and (

**b**) versus Weissenberg number We.

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Fetecau, C.; Vieru, D.; Abbas, T.; Ellahi, R.
Analytical Solutions of Upper Convected Maxwell Fluid with Exponential Dependence of Viscosity under the Influence of Pressure. *Mathematics* **2021**, *9*, 334.
https://doi.org/10.3390/math9040334

**AMA Style**

Fetecau C, Vieru D, Abbas T, Ellahi R.
Analytical Solutions of Upper Convected Maxwell Fluid with Exponential Dependence of Viscosity under the Influence of Pressure. *Mathematics*. 2021; 9(4):334.
https://doi.org/10.3390/math9040334

**Chicago/Turabian Style**

Fetecau, Constantin, Dumitru Vieru, Tehseen Abbas, and Rahmat Ellahi.
2021. "Analytical Solutions of Upper Convected Maxwell Fluid with Exponential Dependence of Viscosity under the Influence of Pressure" *Mathematics* 9, no. 4: 334.
https://doi.org/10.3390/math9040334