# Peristaltic Pumping of Nanofluids through a Tapered Channel in a Porous Environment: Applications in Blood Flow

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## Abstract

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## 1. Introduction

## 2. Mathematical Formulation

## 3. Convective Boundary Conditions

## 4. Non-Dimensional Analysis

## 5. Analytical Solution

## 6. Computational Results and Discussion

#### 6.1. Thermal and Concentration Profiles

#### 6.2. Nanoparticle Heat Transfer Coefficient

#### 6.3. Trapping

#### 6.4. Validation

## 7. Conclusions

- Nanoparticle heat transfer between the tapered walls strongly depends on Brinkman number because the tissue presents the chief resistance to heat flow.
- Thermal radiation contains the potential to contribute a significant change in the nanoparticle temperature distribution.
- With increasing the radiation parameter, the nanoparticle temperature and heat transfer coefficient enhance.
- The nanoparticle temperature reduces with enhancing the Prandtl number, however, reverse behavior is noticed for nanoparticle concentration.
- Heat transfer coefficient depends on the flow, thermal and geometrical nature of flow regime.
- The trapping phenomenon also alters with changing the magnitude of slip and permeability parameters.
- The findings of the present models can be utilized to engineer smart peristaltic pumps which can be applicable for transporting drugs and delivery of nanoparticles.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Symbol | description | Unit |

$\left({a}_{1},\text{}{a}_{2}\right)$ | Dimensional amplitude of the lower and upper walls | m |

$c$ | Wave speed | m/s |

$\overline{C}$ | Nanoparticle volumetric volume fraction | Kg/m^{3} |

${C}_{0},\text{}{C}_{1}$ | Nanoparticle concentration at the lower and upper walls | Kg/m^{3} |

${D}_{B}$ | Brownian diffusion coefficient | m/s |

${D}_{T}$ | Themophoretic diffusion coefficient | m^{2}/s |

$\overline{d}$ | Dimensionless half width of the channel | m |

${\overline{h}}_{h}$ | Heat transfer coefficient | W/m^{2}K (or) kg/s^{3}K |

${\overline{h}}_{m}$ | Mass transfer coefficient | m/s |

$k$ | Permeable of porous medium | H/m |

$\overline{k}$ | Permeability of the porous wavy wall | Darcy (or) m^{2} |

${\overline{k}}_{h}$ | Thermal conductivity of wavy wall | W/mK |

${\overline{k}}_{m}$ | Mass conductivity of wavy wall | W/mK |

$\overline{m}$ | Dimensional non-uniform parameter | m |

$\overline{P}$ | Dimensional pressures | Pa (or) N/m^{2} (or) kg/ms^{2} |

${\overline{q}}_{r}$ | Uni-directional thermal radiative flux | kg/s^{3} (or) W/m^{2} |

${t}^{\prime}$ | Dimensional time | s |

$\overline{T}$ | Nanoparticle temperature | K |

${T}_{m}$ | Mean temperature | K |

$\left({T}_{0},{T}_{1}\right)$ | Temperature at the lower and upper walls | K |

$\overline{U},\overline{V}$ | Velocity components in the wave frame | m/s |

$\overline{\xi},\overline{\eta}$ | Rectangular coordinates | m |

${\rho}_{f}$ | Density of the fluid | Kg/m^{3} |

${\rho}_{p}$ | Density of the particle | Kg/m^{3} |

$\mu $ | Dynamic Viscosity | kg/m.s |

$\kappa $ | Thermal conductivity of the fluid | m^{2}/s |

$\lambda $ | Wave length | m |

## Dimensionless parameters:

$\overline{\alpha}$ | Slip coefficient at the surface of the porous walls |

$A$ | Blood flow constant |

$\left(a,\text{}b\right)$ | Dimensionless amplitude of the lower and upper walls |

${B}_{h}$ | Heat transfer Biot number |

${B}_{m}$ | Mass transfer Biot number |

$Br$ | Brinkman number |

$Ec$ | Eckert number |

$F$ | Dimensionless flow rate |

$\left({\overline{H}}_{1},\text{}{\overline{H}}_{2}\right)$ | Lower and upper wall boundaries of the micro- asymmetric channel |

$\left({h}_{1},\text{}{h}_{2}\right)$ | Dimensionless lower and upper wall shapes in wave frame |

$L$ | Slip parameter |

$m$ | Dimensionless non-uniform parameter |

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

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

$p$ | Dimensionless pressure |

$\mathrm{Pr}$ | Prandtl number |

R | Reynolds number |

${R}_{n}$ | Thermal radiation |

$Sc$ | Schmidt number |

$t$ | Dimensionless time |

$\left(u,\text{}v\right)$ | Velocity components in the wave frame $\left(x,\text{}y\right)$ |

$\Theta $ | Constant flow rate |

$\sigma $ | Dimensionless rescaled nanoparticle volume fraction |

$\theta $ | Dimensionless nanoparticle temperature |

$\psi $ | Stream function |

$\varphi $ | Phase difference |

$K$ | Permeability parameter |

$\delta $ | Wave number |

## Appendix A

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**Figure 19.**Streamlines when $a=0.4$, $b=0.3$, $\varphi =\pi /2$, $t=0.3$, $A=0.2$, (

**a**) $m=0$, $L=0.1$, $K=0.1$, $\Theta =1.25$, (

**b**) $m=0.1$, $L=0.1$, $K=0.1$, $\Theta =1.25$, (

**c**) $m=0.1$, $L=0.15$, $K=0.1$, $\Theta =1.25$, (

**d**) $m=0.1$, $L=0.15$, $K\to \infty $, $\Theta =1.25$, (

**e**) $m=0.1$, $L=0.15$, $K\to \infty $, $\Theta =1.5$.

**Figure 20.**Comparison between numerical and present solutions for temperature and nanoparticle volume fraction profiles.

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**MDPI and ACS Style**

Prakash, J.; Tripathi, D.; Tiwari, A.K.; Sait, S.M.; Ellahi, R.
Peristaltic Pumping of Nanofluids through a Tapered Channel in a Porous Environment: Applications in Blood Flow. *Symmetry* **2019**, *11*, 868.
https://doi.org/10.3390/sym11070868

**AMA Style**

Prakash J, Tripathi D, Tiwari AK, Sait SM, Ellahi R.
Peristaltic Pumping of Nanofluids through a Tapered Channel in a Porous Environment: Applications in Blood Flow. *Symmetry*. 2019; 11(7):868.
https://doi.org/10.3390/sym11070868

**Chicago/Turabian Style**

Prakash, J., Dharmendra Tripathi, Abhishek Kumar Tiwari, Sadiq M. Sait, and Rahmat Ellahi.
2019. "Peristaltic Pumping of Nanofluids through a Tapered Channel in a Porous Environment: Applications in Blood Flow" *Symmetry* 11, no. 7: 868.
https://doi.org/10.3390/sym11070868