# Heat Transfer Analysis of a Magneto-Bio-Fluid Transport with Variable Thermal Viscosity Through a Vertical Ciliated Channel

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## Abstract

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## 1. Introduction

## 2. Statement of the Problem

## 3. Solution Methodology

## 4. Results and Discussions

#### 4.1. Velocity Distribution

#### 4.2. Pressure Distribution

#### 4.3. Volume Flow Rate

#### 4.4. Trapping Phenomenon

## 5. Application: Physiological Transport of Semen

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

## References

- Blake, J.R.; Sleigh, M.A. Mechanics of Ciliary Locomotion. Biol. Rev.
**1974**, 49, 85–125. [Google Scholar] [CrossRef] [PubMed] - Liron, N.; Mochon, S. Stokes flow for a stokeslet between two parallel flat plates. J. Eng. Math.
**1976**, 10, 287. [Google Scholar] [CrossRef] - Sleigh, M.A. The Biology of Cilia and Flagella; MacMillian: New York, NY, USA, 1962. [Google Scholar]
- Ibanez-Tallon, I.; Heintz, N.; Omran, H. To beat or not to beat: Role of cilia in development and disease. Hum. Mol. Genet.
**2003**, 12, 27–35. [Google Scholar] [CrossRef] [PubMed] - Dauptain, A.; Favier, J.; Battaro, A. Hydrodynamics of ciliary propulsion. J. Fluids Struct.
**2008**, 24, 1156–1165. [Google Scholar] [CrossRef][Green Version] - Velez-Cordero, J.R.; Lauga, E. Waving transport and propulsion in a generalized Newtonian fluid. J. Non-Newton. Fluid Mech.
**2013**, 199, 37–50. [Google Scholar] [CrossRef][Green Version] - Brown, J.M.; Witman, G.B. Cilia and Diseases. Biosciences
**2014**, 64, 1126–1137. [Google Scholar] [CrossRef] - Hanasoge, S.; Hesketh, P.J.; Alexeev, A. Metachronal motion of artificial cilia. Soft Matter.
**2018**, 14, 3689–3693. [Google Scholar] [CrossRef] - Lardner, T.J.; Shack, W.J. Cilia Transport. Bull. Math. Biophys.
**1972**, 34, 25–35. [Google Scholar] [CrossRef] - Agrawal, L.; Anawaruddin. Cilia transport of bio-fluid with variable viscosity. Indian J. Pure Appl. Math.
**1984**, 15, 1128–1139. [Google Scholar] - Mills, Z.G.; Aziz, A.; Alexeev, A. Beating synthetic cilia enhance heat transport in micro fluidic channels. Soft Matter.
**2012**, 8, 11508–11513. [Google Scholar] [CrossRef] - Siddiqui, A.M.; Farooq, A.A.; Rana, M.A. Hydromagnetic flow of Newtonian fluid due to ciliary motion in a channel. Magnetohydrodynamics
**2014**, 50, 109–122. [Google Scholar] - Nadeem, S.; Sadaf, H. Trapping study of nanofluids in an annulus with cilia. AIP Adv.
**2015**, 5, 127204. [Google Scholar] [CrossRef][Green Version] - Sadaf, H.; Nadeem, S. Influences of slip and Cu-blood nanofluid in a physiological study of cilia. Comput. Methods Programs Biomed.
**2016**, 131, 169–180. [Google Scholar] [CrossRef] [PubMed] - Farooq, A.A.; Siddiqui, A.M. Mathematical model for the ciliary induced transport of seminal liquids through the ductuli efferentes. Int. J. Biomath.
**2017**, 10, 1750031. [Google Scholar] [CrossRef] - Farooq, A.A. On the transport of epididymal fluid induced by metachronal wave of cilia. J. Eng. Math.
**2018**, 110, 167–180. [Google Scholar] [CrossRef] - Ponalagusamy, R. Mathematical analysis of flow of non-Newtonian fluid due to metachronal beating of cilia in a tube and its physiological applications. Appl. Math. Comput.
**2018**, 337, 545–561. [Google Scholar] [CrossRef] - Nadeem, S.; Sadaf, S. Metachronal wave of cilia transport in a curved channel. Z. Naturforsch.
**2015**, 70, 33–38. [Google Scholar] [CrossRef] - Farooq, A.A.; Tripathi, D.; Elnaqeeb, T. On the propulsion of micropolar fluid inside a channel due to ciliary induced metachronal wave. Appl. Math. Comput.
**2019**, 347, 225–235. [Google Scholar] [CrossRef] - Akbar, N.S.; Tripathi, D.; Beg, O.A.; Khan, Z.H. MHD dissipative flow and heat transfer of Casson fluid due to metachronal wave propulsion of beating cilia with thermal and velocity slip effects under oblique magnetic field. Acta Astronaut.
**2016**, 128, 1–12. [Google Scholar] [CrossRef] - Shaheen, A.; Nadeem, S. Metachronal wave analysis for non-Newtonian fluid under Thermophoresis and Brownian motion effects. Results Phys.
**2017**, 7, 2950–2957. [Google Scholar] [CrossRef] - Akbar, N.S.; Khan, Z.H. Influence of magnetic field for metachoronical beating of cilia for nanofluid with Newtonian heating. J. Magn. Magn. Mater.
**2015**, 381, 235–242. [Google Scholar] [CrossRef] - Akbar, N.S.; Khan, Z.H.; Nadeem, S. Metachronal beating of cilia under influence of Hartmann layer and heat transfer. Eur. Phys. J. Plus
**2014**, 129, 176. [Google Scholar] [CrossRef] - Tripathi, D.; Bég, O.A. A study of unsteady physiological magneto-fluid flow and heat transfer through a finite length channel by peristaltic pumping. Proc. Inst. Mech. Eng. Part H J. Eng. Med.
**2012**, 226, 631–644. [Google Scholar] [CrossRef] - Nadeem, S.; Hayat, T.; Akbar, N.S.; Malik, M.Y. On the influence of heat transfer in peristalsis with variable viscosity. Int. J. Heat Mass Transf.
**2009**, 52, 4722–4730. [Google Scholar] [CrossRef] - Nadeem, S.; Akbar, N.S. Influence of temperature dependent viscosity on peristaltic transport of a Newtonian fluid: Application of an endoscope. Appl. Math. Comput.
**2010**, 216, 3606–3619. [Google Scholar] [CrossRef] - Nadeem, S.; Akbar, N.S. Effects of heat transfer on the peristaltic transport of MHD Newtonian fluid with variable viscosity: Application of Adomian decomposition method. Commun. Nonlinear Sci. Numer. Simul.
**2009**, 14, 3844–3855. [Google Scholar] [CrossRef] - Shit, G.C.; Ranjit, N.K.; Shina, A. Adomian Decomposition Method for Magnetohydrodynamic flow of blood induced by peristaltic waves. J. Mech. Med. Biol.
**2017**, 17, 1750007. [Google Scholar] [CrossRef] - Rathod, V.P.; Laxni, D. Effects of heat transfer on the peristaltic MHD flow of a Binghamfluid through a porous medium in a channel. Int. J. Biomath.
**2014**, 7, 1450060. [Google Scholar] [CrossRef] - Manzoor, N.; Maqbool, K.; Beg, O.A.; Shaheen, S. Adomian decomposition solution for propulsion of dissipative magnetic Jeffrey biofluid in a ciliated channel containing a porous medium with forced convection. Heat Transf. Asian Res
**2018**, 48, 556–581. [Google Scholar] [CrossRef] - Kefayati, G. Lattice Boltzmann simulation of double-diffusive natural convection of viscoplastic fluids in a porous cavity. Phys. Fluids
**2019**, 31, 1. [Google Scholar] [CrossRef] - Kefayati, G.H.R.; Tang, H. Three-dimensional Lattice Boltzmann simulation on thermosolutal convection and entropy generation of Carreau-Yasuda fluids. Int. J. Heat Mass Transf.
**2019**, 131, 346–364. [Google Scholar] [CrossRef] - Kefayati, G.H.; Hosseinizadeh, S.F.; Gorji, M.; Sajjadi, H. Lattice Boltzmann simulation of natural convection in tall enclosures using water/SiO2 nanofluid. Int. Commun. Heat Mass Transf.
**2011**, 38, 798–805. [Google Scholar] [CrossRef] - Ellahi, R.; Sait, S.M.; Shehzad, N.; Mobin, N. Numerical Simulation and Mathematical Modeling of Electro-Osmotic Couette–Poiseuille Flow of MHD Power-Law Nanofluid with Entropy Generation. Symmetry
**2019**, 11, 1038. [Google Scholar] [CrossRef] - Ellahi, R.; Zeeshan, A.; Hussain, F.; Abbas, T. Two-Phase Couette Flow of Couple Stress Fluid with Temperature Dependent Viscosity Thermally Affected by Magnetized Moving Surface. Symmetry
**2019**, 11, 647. [Google Scholar] [CrossRef] - Jawad, M.; Shah, Z.; Islam, S.; Majdoubi, J.; Tlili, I.; Khan, W.; Khan, I. Impact of Nonlinear Thermal Radiation and the Viscous Dissipation Effect on the Unsteady Three-Dimensional Rotating Flow of Single-Wall Carbon Nanotubes with Aqueous Suspensions. Symmetry
**2019**, 11, 207. [Google Scholar] [CrossRef] - Saeed, A.; Islam, S.; Dawar, A.; Shah, Z.; Kumam, P.; Khan, W. Influence of Cattaneo–Christov Heat Flux on MHD Jeffrey, Maxwell, and Oldroyd-B Nanofluids with Homogeneous-Heterogeneous Reaction. Symmetry
**2019**, 11, 439. [Google Scholar] [CrossRef] - Alzahrani, E.; Shah, Z.; Alghamdi, W.; Zaka Ullah, M. Darcy–Forchheimer Radiative Flow of Micropoler CNT Nanofluid in Rotating Frame with Convective Heat Generation/Consumption. Processes
**2019**, 7, 666. [Google Scholar] [CrossRef] - Shah, Z.; Bonyah, E.; Islam, S.; Khan, W.; Ishaq, M. Radiative MHD thin film flow of Williamson fluid over an unsteady permeable stretching sheet. Heliyon
**2018**, 4, e00825. [Google Scholar] [CrossRef][Green Version] - Ullah, A.; Shah, Z.; Kumam, P.; Ayaz, M.; Islam, S.; Jameel, M. Viscoelastic MHD Nanofluid Thin Film Flow over an Unsteady Vertical Stretching Sheet with Entropy Generation. Processes
**2019**, 7, 262. [Google Scholar] [CrossRef] - Sheikholeslami, M.; Shah, Z.; Shafee, A.; Khan, I.; Tlili, I. Uniform magnetic force impact on water based nanofluid thermal behavior in a porous enclosure with ellipse shaped obstacle. Sci. Rep.
**2019**, 9, 1196. [Google Scholar] [CrossRef] - Marin, M.; Vlase, S.; Ellahi, R.; Bhatti, M. On the Partition of Energies for the Backward in Time Problem of Thermoelastic Materials with a Dipolar Structure. Symmetry
**2019**, 11, 863. [Google Scholar] [CrossRef] - Ullah, A.; Alzahrani, E.; Shah, Z.; Ayaz, M.; Islam, S. Nanofluids thin film flow of Reiner-Philippoff fluid over an unstable stretching surface with Brownian motion and thermophoresis effects. Coatings
**2019**, 9, 21. [Google Scholar] [CrossRef] - Shah, Z.; Islam, S.; Ayaz, H.; Khan, S. Radiative heat and mass transfer analysis of micropolar nanofluid flow of Casson fluid between two rotating parallel plates with effects of Hall current. J. Heat Transf.
**2019**, 141, 022401. [Google Scholar] [CrossRef] - Ameen, I.; Shah, Z.; Islam, S.; Nasir, S.; Khan, W.; Kumam, P.; Thounthong, P. Hall and Ion-Slip Effect on CNTS Nanofluid over a Porous Extending Surface through Heat Generation and Absorption. Entropy
**2019**, 21, 801. [Google Scholar] [CrossRef] - Mebarek-Oudina, F. Convective heat transfer of Titania nanofluids of di_erent base fluids in cylindrical annulus with discrete heat source. Heat Tran. Asian Res.
**2019**, 48, 135–147. [Google Scholar] [CrossRef] - Mebarek-Oudina, F.; Bessaih, R. Oscillatory Magnetohydrodynamic Natural Convection of Liquid Metal between Vertical Coaxial Cylinders. J. Appl. Fluid Mech.
**2016**, 9, 1655–1665. [Google Scholar] [CrossRef] - Saeed, A.; Shah, Z.; Islam, S.; Jawad, M.; Ullah, A.; Gul, T.; Kumam, P. Three-Dimensional Casson Nanofluid Thin Film Flow over an Inclined Rotating Disk with the Impact of Heat Generation/Consumption and Thermal Radiation. Coatings
**2019**, 9, 248. [Google Scholar] [CrossRef] - Ahmad, M.W.; Kumam, P.; Shah, Z.; Farooq, A.A.; Nawaz, R.; Dawar, A.; Islam, S.; Thounthong, P. Darcy–Forchheimer MHD Couple Stress 3D Nanofluid over an Exponentially Stretching Sheet through Cattaneo–Christov Convective Heat Flux with Zero Nanoparticles Mass Flux Conditions. Entropy
**2019**, 21, 867. [Google Scholar] [CrossRef] - Shah, Z.; Ullah, A.; Bonyah, E.; Ayaz, M.; Islam, S.; Khan, I. Hall effect on Titania nanofluids thin film flow and radiative thermal behavior with different base fluids on an inclined rotating surface. AIP Adv.
**2019**, 9, 055113. [Google Scholar] [CrossRef][Green Version]

**Figure 2.**$u(y)$ for (

**a**) $M$ (

**b**) $\alpha $ (

**c**) ${g}_{t}$ (

**d**) ${S}_{t}$. The other parameters are fixed as $\delta =0.4$, $\beta =0.4$, $\epsilon =0.3$, $Q=0.9$, $M=2$, $\alpha =0.05$, ${g}_{t}=5$, ${S}_{t}=5$.

**Figure 3.**$dp/dx$ for (

**a**) $M$ (

**b**) $\alpha $ (

**c**) ${g}_{t}$ (

**d**) ${S}_{t}$. The other parameters are fixed as $\delta =0.4$, $\beta =0.4$, $\epsilon =0.3$, $Q=0.9$, $M=2$, $\alpha =0.05$, ${g}_{t}=5$, ${S}_{t}=5$.

**Figure 4.**Pressure difference ($\Delta p$) versus the flow rate ($Q$) for (

**a**) $M$ (

**b**) $\alpha $ (

**c**) ${g}_{t}$ (

**d**) ${S}_{t}$. The other parameters are fixed as $\delta =0.4$, $\beta =0.4$, $\epsilon =0.3$, $M=2$, $\alpha =0.05$, ${g}_{t}=5$, ${S}_{t}=5$.

**Figure 5.**Volume flow rate ($Q$) versus $\epsilon $ for (

**a**) $M$, (

**b**) $\alpha $, (

**c**) ${g}_{t}$, (

**d**) ${S}_{t}$. The other parameters are fixed as $\delta =0.4$, $\beta =0.4$, $\Delta p=-3$, $M=0.5$, $\alpha =0.05$, ${g}_{t}=3$, ${S}_{t}=3$.

**Figure 6.**Volume flow rate ($Q$) versus $\delta $ for (

**a**) $M$ (

**b**) $\epsilon $. The other parameters are fixed as $\epsilon =0.3$, $\beta =0.4$, $\Delta p=-3$, $M=0.5$, $\alpha =0.05$, ${g}_{t}=3$, ${S}_{t}=3$.

**Figure 7.**Streamlines for (

**a**) $M=1.5$, (

**b**) $M=2.5$, (

**c**) $M=3.5$. The other parameters are fixed as $\delta =0.2$, $\beta =0.2$, $\epsilon =0.3$, $Q=0.9$, $M=1.5$, $\alpha =0.03$, ${g}_{t}=5$, ${S}_{t}=5$.

**Figure 8.**Streamlines for (

**a**) $\alpha =0.0$, (

**b**) $\alpha =0.03$, (

**c**) $\alpha =0.05$. The other parameters are fixed as $\delta =0.2$, $\beta =0.2$, $\epsilon =0.3$, $Q=0.9$, $M=1.5$, ${g}_{t}=5$, ${S}_{t}=5$.

**Figure 9.**Streamlines for (

**a**) ${g}_{t}=0$, (

**b**) ${g}_{t}=3$, (

**c**) ${g}_{t}=5$. The other parameters are fixed as $\delta =0.2$, $\beta =0.2$, $\epsilon =0.3$, $Q=0.9$, $M=1.5$, $\alpha =0.03$, ${g}_{t}=3$, ${S}_{t}=3$.

**Figure 10.**Streamlines for (

**a**) ${S}_{t}=0$, (

**b**) ${S}_{t}=3$, (

**c**) ${S}_{t}=5$. The other parameters are fixed as $\delta =0.2$, $\beta =0.2$, $\epsilon =0.3$, $Q=0.9$, $M=1.5$, $\alpha =0.03$, ${g}_{t}=3$, ${S}_{t}=3$.

**Table 1.**Numerical results of $Q$ and ${Q}^{*}=(Q\times 2\pi {d}^{2}c)$, when $\delta =0.1$, $\epsilon =0.1$, $\beta =1.0$, and $\Delta p=0$.

$M$ | $\alpha $ | ${g}_{t}$ | ${S}_{t}$ | $Q$ | ${Q}^{*}$$\left(\mathrm{mL}/\mathrm{h}\right)$ |
---|---|---|---|---|---|

0 | 0 | 0 | 0 | 0.02204 | 0.00012 |

0.01 | 0.1 | 0.1 | 0.05957 | 0.00032 | |

0.02 | 0.1 | 0.1 | 0.06443 | 0.00035 | |

0.03 | 0.1 | 0.1 | 0.07071 | 0.00038 | |

0.1 | 0.01 | 0.1 | 0.1 | 0.05932 | 0.00032 |

0.02 | 0.1 | 0.1 | 0.06411 | 0.00034 | |

0.03 | 0.1 | 0.1 | 0.07029 | 0.00031 | |

0.01 | 0.3 | 0.1 | 0.13398 | 0.00073 | |

0.01 | 0.5 | 0.1 | 0.20864 | 0.00113 | |

0.01 | 0.1 | 0.3 | 0.06249 | 0.00034 | |

0.01 | 0.1 | 0.5 | 0.06569 | 0.00035 | |

0.3 | 0.01 | 0.1 | 0.1 | 0.05716 | 0.00031 |

0.5 | 0.01 | 0.1 | 0.1 | 0.05208 | 0.00028 |

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**MDPI and ACS Style**

Ahmad Farooq, A.; Shah, Z.; Alzahrani, E.O. Heat Transfer Analysis of a Magneto-Bio-Fluid Transport with Variable Thermal Viscosity Through a Vertical Ciliated Channel. *Symmetry* **2019**, *11*, 1240.
https://doi.org/10.3390/sym11101240

**AMA Style**

Ahmad Farooq A, Shah Z, Alzahrani EO. Heat Transfer Analysis of a Magneto-Bio-Fluid Transport with Variable Thermal Viscosity Through a Vertical Ciliated Channel. *Symmetry*. 2019; 11(10):1240.
https://doi.org/10.3390/sym11101240

**Chicago/Turabian Style**

Ahmad Farooq, Ali, Zahir Shah, and Ebraheem O. Alzahrani. 2019. "Heat Transfer Analysis of a Magneto-Bio-Fluid Transport with Variable Thermal Viscosity Through a Vertical Ciliated Channel" *Symmetry* 11, no. 10: 1240.
https://doi.org/10.3390/sym11101240