# Two-Phase Couette Flow of Couple Stress Fluid with Temperature Dependent Viscosity Thermally Affected by Magnetized Moving Surface

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Analysis

#### Boundary Conditions

## 3. Results and Discussion

#### 3.1. Variable Viscosity

#### 3.2. Numerical Procedure

#### 3.3. Graphical Illustration

#### 3.4. Validation

## 4. Conclusions

- ⮚
- The flow of couple stress fluid resists for increasing values of Hartmann number.
- ⮚
- The temperature effectively variates the viscosity of the fluid to cause the shear thinning effects.
- ⮚
- The temperature of the flow mounts in response of higher values of Brinkman number.
- ⮚
- Attenuation of the viscosity results to expedite the flows.
- ⮚
- Viscosity parameter brings celerity in the velocity of bi-phase fluid due to high temperature difference.
- ⮚
- Molecules additives of base fluid reduce the force of friction and hence in both phases the velocity is galvanized.
- ⮚
- Due to the immense applications of multiphase flows in industrial and pharmaceutical, the proposed theoretical model is now available to vet relevant experimental investigations.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Farooq, M.; Rahim, M.T.; Islam, S.; Siddiqui, A.M. Steady Poiseuille flow and heat transfer of couple stress fluids between two parallel inclined plates with variable viscosity. J. Assoc. Arab Univ. Basic Appl. Sci.
**2013**, 14, 9–18. [Google Scholar] [CrossRef] [Green Version] - Mahabaleshwar, U.S.; Sarris, I.E.; Hill, A.; Lorenzini, G.; Pop, I. An MHD couple stress fluid due to a perforated sheet undergoing linear stretching with heat transfer. Int. J. Heat Mass. Transf.
**2017**, 105, 157–167. [Google Scholar] [CrossRef] [Green Version] - Saad, H.; Ashmawy, E.A. Unsteady plane Couette flow of an incompressible couple stress fluid with slip boundary conditions. Int. J. Med. Health Sci.
**2016**, 3, 85–92. [Google Scholar] [CrossRef] - Akhtar, S.; Shah, N.A. Exact solutions for some unsteady flows of a couple stress fluid between parallel plates. Ain Shams Eng. J.
**2018**, 9, 985–992. [Google Scholar] [CrossRef] - Khan, N.A.; Khan, H.; Ali, S.A. Exact solutions for MHD flow of couple stress fluid with heat transfer. J. Egypt. Math. Soc.
**2016**, 24, 125–129. [Google Scholar] [CrossRef] [Green Version] - Asghar, S.; Ahmad, A. Unsteady Couette flow of viscous fluid under a non-uniform magnetic field. Appl. Math. Lett.
**2012**, 25, 1953–1958. [Google Scholar] [CrossRef] [Green Version] - Shaowei, W.; Mingyu, X. Exact solution on unsteady Couette flow of generalized Maxwell fluid with fractional derivative. Acta Mech.
**2006**, 187, 103–112. [Google Scholar] [CrossRef] - Eegunjobi, A.S.; Makinde, O.D.; Tshehla, M.S.; Franks, O. Irreversibility analysis of unsteady Couette flow with variable viscosity. J. Hydrodyn. B
**2015**, 27, 304–310. [Google Scholar] [CrossRef] - Ellahi, R.; Wang, X.; Hameed, M. Effects of heat transfer and nonlinear slip on the steady flow of Couette fluid by means of Chebyshev Spectral Method. Z. Naturforsch A.
**2014**, 69, 1–8. [Google Scholar] [CrossRef] - Ellahi, R.; Shivanian, E.; Abbasbandy, S.; Hayat, T. Numerical study of magnetohydrodynamics generalized Couette flow of Eyring-Powell fluid with heat transfer and slip condition. Int. J. Numer. Method Heat Fluid Flow
**2016**, 26, 1433–1445. [Google Scholar] [CrossRef] - Zeeshan, A.; Shehzad, N.; Ellahi, R. Analysis of activation energy in Couette-Poiseuille flow of nanofluid in the presence of chemical reaction and convective boundary conditions. Results Phys.
**2018**, 8, 502–512. [Google Scholar] [CrossRef] - Shehzad, N.; Zeeshan, A.; Ellahi, R. Electroosmotic flow of MHD Power law Al2O3-PVC nanofluid in a horizontal channel: Couette-Poiseuille flow model. Commun. Theor. Phys.
**2018**, 69, 655–666. [Google Scholar] [CrossRef] - Hussain, F.; Ellahi, R.; Zeeshan, A.; Vafai, K. Modelling study on heated couple stress fluid peristaltically conveying gold nanoparticles through coaxial tubes: A remedy for gland tumors and arthritis. J. Mol. Liq.
**2018**, 268, 149–155. [Google Scholar] [CrossRef] - 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. [Google Scholar] [CrossRef] - Ellahi, R.; Bhatti, M.M.; Fetecau, C.; Vafai, K. Peristaltic flow of couple stress fluid in a non-uniform rectangular duct having compliant walls. Commun. Theor. Phys.
**2016**, 65, 66–72. [Google Scholar] [CrossRef] - Poply, V.; Singh, P.; Yadav, A.K. A study of Temperature-dependent fluid properties on MHD free stream flow and heat transfer over a non-linearly stretching sheet. Procedia Eng.
**2015**, 127, 391–397. [Google Scholar] [CrossRef] - Ellahi, R.; Raza, M.; Vafai, K. Series solutions of non-Newtonian nanofluids with Reynolds model and Vogel’s model by means of the homotopy analysis method. Math. Comput. Model.
**2012**, 55, 1876–1891. [Google Scholar] [CrossRef] - Disu, A.B.; Dada, M.S. Reynolds model viscosity on radiative MHD flow in a porous medium between two vertical wavy walls. J. Taibah Univ. Sci.
**2017**, 11, 548–565. [Google Scholar] [CrossRef] - Nadeem, S.; Ali, M. Analytical solutions for pipe flow of a fourth-grade fluid with Reynolds and Vogel’s models of viscosities. Commun. Nonlin. Sci. Numer. Simulat.
**2009**, 14, 2073–2090. [Google Scholar] [CrossRef] - Ellahi, R.; Zeeshan, A.; Hussain, F.; Abbas, T. Thermally charged MHD bi-phase flow coatings with non-Newtonian nanofluid and Hafnium particles through slippery walls. Coatings
**2019**, 9, 300. [Google Scholar] [CrossRef] - Mahmoud, M.A. Chemical reaction and variable viscosity effects on flow and mass transfer of a non-Newtonian visco-elastic fluid past a stretching surface embedded in a porous medium. Meccanica
**2010**, 45, 835–846. [Google Scholar] [CrossRef] - Makinde, O.D. Laminar falling liquid film with variable viscosity along an inclined heated plate. Appl. Math. Comput.
**2006**, 175, 80–88. [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] - Ellahi, R. A study on the convergence of series solution of non-Newtonian third grade fluid with variable viscosity: By means of homotopy analysis method. Adv. Math. Phys.
**2012**, 2012, 634925. [Google Scholar] [CrossRef] - Karimipour, A.; Orazio, A.D.; Shadloo, M.S. The effects of different nano particles of Al
_{2}O_{3}and Ag on the MHD nano fluid flow and heat transfer in a microchannel including slip velocity and temperature jump. Phys. E**2017**, 86, 146–153. [Google Scholar] [CrossRef] - Hosseini, S.M.; Safaei, M.R.; Goodarzi, M.; Alrashed, A.A.A.A.; Nguyen, T.K. New temperature, interfacial shell dependent dimensionless model for thermal conductivity of nanofluids. Int. J. Heat Mass Transf.
**2017**, 114, 207–210. [Google Scholar] [CrossRef] - Nasiri, H.; Jamalabadi, M.Y.A.; Sadeghi, R.; Safaei, M.R.; Nguyen, T.K.; Shadloo, M.S. A smoothed particle hydrodynamics approach for numerical simulation of nano-fluid flows. J. Therm. Anal. Calorim.
**2018**, 1–9. [Google Scholar] [CrossRef] - Safaei, M.R.; Ahmadi, G.; Goodarzi, M.S.; Shadloo, M.S.; Goshayeshi, H.R.; Dahari, M. Heat transfer and pressure drop in fully developed turbulent flows of graphene nanoplatelets–silver/water nanofluids. Fluids
**2016**, 1, 20. [Google Scholar] [CrossRef] - Sadiq, M.A. MHD stagnation point flow of nanofluid on a plate with anisotropic slip. Symmetry
**2019**, 11, 132. [Google Scholar] [CrossRef] - Rashidi, S.; Esfahani, J.A.; Ellahi, R. Convective heat transfer and particle motion in an obstructed duct with two side by side obstacles by means of DPM model. Appl. Sci.
**2017**, 7, 431. [Google Scholar] [CrossRef] - Shehzad, N.; Zeeshan, A.; Ellahi, R.; Rashidid, S. Modelling study on internal energy loss due to entropy generation for non-Darcy Poiseuille flow of silver-water nanofluid: An application of purification. Entropy
**2018**, 20, 851. [Google Scholar] [CrossRef] - Hassan, M.; Ellahi, R.; Bhatti, M.M.; Zeeshan, A. A comparative study of magnetic and non-magnetic particles in nanofluid propagating over a wedge. Can. J. Phys.
**2019**, 97, 277–285. [Google Scholar] [CrossRef] - Zeeshan, A.; Shehzad, N.; Abbas, A.; Ellahi, R. Effects of radiative electro-magnetohydrodynamics diminishing internal energy of pressure-driven flow of titanium dioxide-water nanofluid due to entropy generation. Entropy
**2019**, 21, 236. [Google Scholar] [CrossRef] - Ashrafi, N.; Khayat, R.E. A low-dimensional approach to nonlinear plane-Couette flow of viscoelastic fluids. Phys. Fluids.
**2000**, 12, 345–365. [Google Scholar] [CrossRef] - Srivastava, L.M.; Srivastava, V.P. Peristaltic transport of a particle-fluid suspension. J. Biomech. Eng.
**1989**, 111, 157–165. [Google Scholar] [CrossRef] - Tam, C.K.W. The drag on a cloud of spherical particles in a low Reynolds number flow. J. Fluid Mech.
**1969**, 38, 537–546. [Google Scholar] [CrossRef] - Ellahi, R. The effects of MHD and temperature dependent viscosity on the flow of non-Newtonian nanofluid in a pipe: Analytical solutions. Appl. Math. Model.
**2013**, 37, 1451–1457. [Google Scholar] [CrossRef] - Hossain, M.A.; Subba, R.; Gorla, R. Natural convection flow of non-Newtonian power-law fluid from a slotted vertical isothermal surface. Int. J Numer. Methods Heat Fluid Flow
**2009**, 19, 835–846. [Google Scholar] [CrossRef] - Makinde, O.D.; Onyejekwe, O.O. A numerical study of MHD generalized Couette flow and heat transfer with variable viscosity and electrical conductivity. J. Magn. Magn. Mater.
**2011**, 323, 2757–2763. [Google Scholar] [CrossRef] - Coelho, M.P.; Faria, J.S. On the generalized Brinkman number definition and its importance for Bingham fluids. J. Heat Transf.
**2011**, 133, 545051–545055. [Google Scholar] [CrossRef] - Swarnalathamma, B.V.; Krishna, M.V. Peristaltic hemodynamic flow of couple stress fluid through a porous medium under the influence of magnetic field with slip effect. AIP Conf. Proc.
**2016**, 1728, 0206031–0206039. [Google Scholar] [CrossRef] - Charm, S.E.; Kurland, G.S. Blood Flow and Microcirculation; Wiley: New York, NY, USA, 1974. [Google Scholar]

$\mathit{y}$ | ${\mathit{u}}_{\mathit{p}}$ Newtonian Fluid $\left(\mathit{\gamma}=0.0\right)$ | ${\mathit{u}}_{\mathit{p}}$ Couple Stress Fluid $\left(\mathit{\gamma}=2.0\right)$ | ${\mathit{u}}_{\mathit{f}}$ Newtonian Fluid $\left(\mathit{\gamma}=0.0\right)$ | ${\mathit{u}}_{\mathit{f}}$ Couple Stress Fluid $(\mathit{\gamma}=2.0)$ |
---|---|---|---|---|

−1.0 | 1.0000 | 1.0000 | 0.0000 | 0.0000 |

−0.6 | 1.2000 | 1.3221 | 0.2000 | 0.3221 |

−0.2 | 1.4000 | 1.5826 | 0.4000 | 0.5826 |

0.2 | 1.6000 | 1.7698 | 0.6000 | 0.7698 |

0.6 | 1.8000 | 1.8998 | 0.8000 | 0.8998 |

1.0 | 2.0000 | 2.0000 | 1.0000 | 1.0000 |

$\mathit{y}$ | ${\mathit{u}}_{\mathit{f}}$ Single Phase $\left(\mathit{C}=0.0\right)$ | ${\mathit{u}}_{\mathit{p}}$ Solid–Liquid Phase $\left(\mathit{C}=0.4\right)$ | ${\mathit{u}}_{\mathit{f}}$ Solid–Liquid Phase $\left(\mathit{C}=0.4\right)$ |
---|---|---|---|

−1.0 | 0.0000 | 1.0000 | 0.0000 |

−0.6 | 0.2741 | 1.3221 | 0.3221 |

−0.2 | 0.5117 | 1.5826 | 0.5826 |

0.2 | 0.7047 | 1.7698 | 0.7698 |

0.6 | 0.8618 | 1.8998 | 0.8998 |

1.0 | 1.0000 | 2.0000 | 1.0000 |

$\mathit{y}$ | $\mathbf{\Theta}$ ${\mathit{B}}_{\mathit{r}}=0.0$ | $\mathbf{\Theta}$ ${\mathit{B}}_{\mathit{r}}=2.0$ | $\mathbf{\Theta}$ $\mathbf{\gamma}=0.0$ | $\mathbf{\Theta}$ $\mathit{C}=0.0$ |
---|---|---|---|---|

−1.0 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

−0.6 | 0.2000 | 0.3916 | 0.3512 | 0.3578 |

−0.2 | 0.4000 | 0.6066 | 0.5629 | 0.5870 |

0.2 | 0.6000 | 0.7528 | 0.7095 | 0.7504 |

0.6 | 0.8000 | 0.8785 | 0.8446 | 0.8830 |

1.0 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

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.
https://doi.org/10.3390/sym11050647

**AMA Style**

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(5):647.
https://doi.org/10.3390/sym11050647

**Chicago/Turabian Style**

Ellahi, Rahmat, Ahmed Zeeshan, Farooq Hussain, and Tehseen Abbas.
2019. "Two-Phase Couette Flow of Couple Stress Fluid with Temperature Dependent Viscosity Thermally Affected by Magnetized Moving Surface" *Symmetry* 11, no. 5: 647.
https://doi.org/10.3390/sym11050647