# Mathematical Analysis on an Asymmetrical Wavy Motion of Blood under the Influence Entropy Generation with Convective Boundary Conditions

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## Abstract

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## 1. Introduction

## 2. Governing Equations

## 3. Entropy Generation Analysis

## 4. Series Solution

## 5. Discussion

## 6. Conclusions

- (i)
- It was noticed that the temperature profile revealed an increasing behavior by increasing the amplitude in the upper and lower region;
- (ii)
- The Biot number and Brinkman number significantly enhanced the temperature profile, whereas the behavior is converse for the phase difference parameter and Weissenberg number;
- (iii)
- Entropy profile represented an increment profile for higher values of Brinkmann number and Biot number, and a decrement behavior for the Weissenberg number;
- (iv)
- The Weissenberg number boosedt the Bejan number profile, whereas it decreased due to the Biot number and Brinkman number;
- (v)
- Trapping mechanism showed that the phase difference parameter affected the magnitude of the trapped bolus, while the Weissenberg number not only affected the magnitude of the trapped bolus and the number of trapped boluses reduced in the lower region;
- (vi)
- The non-Newtonian results in the present study could be reduced to Newtonian fluid flow by taking $We=0.$

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

$\overline{{d}_{1}},\overline{{d}_{2}}$ | channel width |

T | temperature |

c | wave speed |

$\overline{t}$ | time |

$\overline{X},\overline{Y}$ | coordinate system |

$\overline{U},\overline{V}$ | velocity components |

$\overline{\mathbf{S}}$ | stress tensor |

${S}_{h}$ | specific heat |

K | thermal conductivity |

P | pressure |

$a,b$ | wave amplitude |

$\mathrm{Re}$ | Reynold’s number |

${E}_{c}$ | Eckert number |

${P}_{r}$ | Prandtl number |

${B}_{r}$ | Brinkmann number |

${B}_{i}$ | Biot number |

$We$ | Weissenberg number |

$Be$ | Bejan number |

${S}_{gen}^{{}^{\prime}}$ | entropy |

## Greek Symbol

$\varphi $ | phase difference |

$\rho $ | density |

$\lambda $ | wavelength |

$\mu $ | viscosity |

$\overline{\Gamma}$ | time constant |

$\delta $ | wave number |

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**Figure 2.**Temperature distribution for different values of a and b. Solid line: $a=0.1$, dashed line: $a=0.15$ and dot-dashed line: $a=0.2$.

**Figure 3.**Temperature distribution for different values of ${B}_{i}$ and ${B}_{r}$. Solid line: ${B}_{i}=0.1$, dashed line: ${B}_{i}=0.25$ and dot-dashed line: ${B}_{i}=0.3$.

**Figure 4.**Temperature distribution for different values of Q and d. Solid line: $Q=1.0$, dashed line: $Q=1.2$ and dot-dashed line: $Q=1.4$.

**Figure 5.**Temperature distribution for different values of $\varphi $ and $We$. Solid line: $\varphi =0.1$, dashed line: $\varphi =0.5$ and dot-dashed line: $\varphi =0.9$.

**Figure 6.**Entropy profile for different values of a and b. Solid line: $a=0.1$, dashed line: $a=0.15$, and dot-dashed line: $a=0.2$.

**Figure 7.**Entropy profile for different values of $We$ and ${B}_{r}$. Solid line: ${B}_{r}=1.0$, dashed line: ${B}_{r}=1.2$, and dot-dashed line: ${B}_{r}=1.4$.

**Figure 8.**Entropy profile for different values of ${B}_{i}$ and ${B}_{r}$. Solid line: ${B}_{i}=0.1$, dashed line: ${B}_{i}=0.25$, and dot-dashed line: ${B}_{i}=0.3$.

**Figure 9.**Entropy profile for different values of $\Delta $. Solid line: $\Delta =0.1$, dashed line: $\Delta =0.2$, dot-dashed line: $\Delta =0.3$, and dot-dashed line: $\Delta =0.4$.

**Figure 10.**Bejan number for different values of $We$ and ${B}_{r}$. Solid line: ${B}_{r}=1.0$, dashed line: ${B}_{r}=1.2$, and dot-dashed line: ${B}_{r}=1.4$.

**Figure 11.**Bejan number for different values of $\varphi $ and ${B}_{i}$. Solid line: $\varphi =0.1$, dashed line: $\varphi =0.5$, and dot-dashed line: $\varphi =0.9$.

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**MDPI and ACS Style**

Riaz, A.; Bhatti, M.M.; Ellahi, R.; Zeeshan, A.; M. Sait, S.
Mathematical Analysis on an Asymmetrical Wavy Motion of Blood under the Influence Entropy Generation with Convective Boundary Conditions. *Symmetry* **2020**, *12*, 102.
https://doi.org/10.3390/sym12010102

**AMA Style**

Riaz A, Bhatti MM, Ellahi R, Zeeshan A, M. Sait S.
Mathematical Analysis on an Asymmetrical Wavy Motion of Blood under the Influence Entropy Generation with Convective Boundary Conditions. *Symmetry*. 2020; 12(1):102.
https://doi.org/10.3390/sym12010102

**Chicago/Turabian Style**

Riaz, Arshad, Muhammad Mubashir Bhatti, Rahmat Ellahi, Ahmed Zeeshan, and Sadiq M. Sait.
2020. "Mathematical Analysis on an Asymmetrical Wavy Motion of Blood under the Influence Entropy Generation with Convective Boundary Conditions" *Symmetry* 12, no. 1: 102.
https://doi.org/10.3390/sym12010102