# Peristaltic Blood Flow of Couple Stress Fluid Suspended with Nanoparticles under the Influence of Chemical Reaction and Activation Energy

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## Abstract

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## 1. Introduction

## 2. Formulism

## 3. Results

## 4. Discussion

#### 4.1. Axial Velocity

#### 4.2. Thermal Distribution

#### 4.3. Nanoparticle Concentration Profile

#### 4.4. Trapping Phenomenon/ Streamline Configuration

## 5. Conclusions

- Strong buoyant force results in retarded axial velocity for the thermophoresis parameter.
- Peristaltic movement of the outer tube enhances the Brownian motion and raises the temperature of the nanofluid.
- Activation energy entering the process maximizes the concentration boundary layer thickness.
- The reaction rate constant increases concentration at the catheter, which decreases the concentration of nanoparticles.
- The thermophoresis parameter shrinks the size of the bolus by strengthening isotherms and closed paths of concentration lines.
- The couple stress parameter and reaction rate constant give freedom to the bolus to swell by binding the stream lines closer to each another.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$V$ | Nanofluid velocity |

G | Gravitational acceleration |

$u$ | Radial velocity Component (fixed frame) |

$w$ | Axial velocity component (Fixed frame) |

${u}^{*}$ | Radial velocity component (wave frame) |

${w}^{*}$ | Axial velocity component (Wave frame) |

$\overline{u}$ | Dimensionless radial velocity component |

$\overline{w}$ | Dimensionless lateral velocity component |

$d$ | Amplitude of peristaltic wave |

$t$ | Time |

${k}_{r}$ | Rate of reaction |

$c$ | Propagating velocity of wave |

${N}_{t}$ | Thermophoresis parameter |

$k$ | Thermal conductivity |

${N}_{b}$ | Brownian motion parameter |

${G}_{r}$ | Grashof number |

${D}_{t}$ | Thermophoretic diffusion coefficient |

${D}_{b}$ | Brownian motion coefficient |

$d$ | Amplitude of peristaltic wave |

$t$ | Time |

${R}_{2}$ | Dimensionless radius of outer tube |

${R}_{1}$ | Dimensionless radius of inner tube |

${P}^{*}$ | Dimensional pressure |

${B}_{r}$ | Brownian diffusion constant |

${A}^{*}$ | Reaction rate constant |

${E}^{*}$ | Activation energy (Dimensionless) |

${E}_{a}$ | Activation energy (Dimensional) |

n | Fitted rate constant |

Greek Symbols | |

$\xi $ | Radial direction of the flow (Fixed frame) |

$\eta $ | Axial direction of the flow (Fixed frame) |

${\xi}^{*}$ | Radial direction of the flow (Wave frame) |

${\eta}^{*}$ | Axial direction of the flow (Wave frame) |

$\overline{\xi}$ | Radial direction of the flow (Dimensionless) |

$\overline{\eta}$ | Axial direction of the flow (Dimensionless) |

${\xi}_{1}$ | Radius of inner tube (Dimensional) |

${\xi}_{2}$ | Radius of outer tube (Dimensional) |

$\overrightarrow{\phi}$ | Nanoparticle concentration (Fixed frame) |

$\overrightarrow{\nu}$ | Nanofluid temperature (Fixed frame) |

${\phi}^{*}$ | Nanoparticle concentration (Wave frame) |

${\nu}^{*}$ | Nanofluid temperature (Wave frame) |

$\overline{\phi}$ | Nanoparticle concentration (Dimensionless) |

$\overline{\nu}$ | Nanofluid temperature (Dimensionless) |

${\gamma}_{1}$ | Couple stress fluid’s constant |

$\gamma $ | Couple stress parameter |

$\tau $ | A ratio defined as $\frac{{\left(\tilde{\rho}c\right)}_{f}}{{\left(\tilde{\rho}c\right)}_{p}}$ |

${\beta}^{*}$ | Temperature ratio |

${\rho}_{p}$ | Density of nanoparticle at reference temperature |

${\rho}_{f}$ | Density of nanofluid at reference temperature |

${\left(\rho c\right)}_{f}$ | Heat capacity of base fluid |

${\left(\rho c\right)}_{p}$ | Heat capacity of particle |

$\mu $ | Dynamic Viscosity |

$\nu $ | Kinematic viscosity |

$\lambda $ | Wavelength |

$\alpha $ | Ratio defined as $\frac{k}{{\left(\rho c\right)}_{f}}$ |

$\overline{\u03f5}$ | A constant ratio |

${\beta}_{T}$ | Volumetric coefficient of expansion |

${\phi}_{w}$ | Reference concentration |

${\nu}_{w}$ | Reference temperature |

${\phi}_{m}$ | Mass concentration |

${\nu}_{m}$ | Fluid temperature |

Subscripts | |

$f$ | Base fluid |

$p$ | Particle |

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**Figure 13.**Stream lines for Brownian diffusion constant. (

**a**): For ${\mathrm{B}}_{\mathrm{r}}=0.2$; (

**b**): For ${\mathrm{B}}_{\mathrm{r}}=0.2$.

**Figure 14.**Stream lines for Grashof number. (

**a**): For ${G}_{\mathrm{r}}=0.1$; (

**b**): For ${G}_{\mathrm{r}}=0.3$.

**Figure 15.**Stream lines for couple stress parameter. (

**a**): For $\mathsf{\gamma}=1.0$; (

**b**): For $\mathsf{\gamma}=2.0$.

**Figure 18.**Contour plot for reaction rate constant. (

**a**): For ${\mathrm{A}}^{*}=0.5$; (

**b**): For ${\mathrm{A}}^{*}=1.0$.

**Figure 19.**Contour plot for Thermophoresis parameter. (

**a**): For ${N}_{t}=0.2$; (

**b**): For ${N}_{t}=0.5$.

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**MDPI and ACS Style**

Ellahi, R.; Zeeshan, A.; Hussain, F.; Asadollahi, A.
Peristaltic Blood Flow of Couple Stress Fluid Suspended with Nanoparticles under the Influence of Chemical Reaction and Activation Energy. *Symmetry* **2019**, *11*, 276.
https://doi.org/10.3390/sym11020276

**AMA Style**

Ellahi R, Zeeshan A, Hussain F, Asadollahi A.
Peristaltic Blood Flow of Couple Stress Fluid Suspended with Nanoparticles under the Influence of Chemical Reaction and Activation Energy. *Symmetry*. 2019; 11(2):276.
https://doi.org/10.3390/sym11020276

**Chicago/Turabian Style**

Ellahi, Rahmat, Ahmed Zeeshan, Farooq Hussain, and A. Asadollahi.
2019. "Peristaltic Blood Flow of Couple Stress Fluid Suspended with Nanoparticles under the Influence of Chemical Reaction and Activation Energy" *Symmetry* 11, no. 2: 276.
https://doi.org/10.3390/sym11020276