# Statistical Analysis of the Mathematical Model of Entropy Generation of Magnetized Nanofluid

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation

## 3. Solution of Problem

## 4. Results and Discussion

## 5. Conclusions

- Temperature profile increases with higher values of ${N}_{b}$ and ${N}_{t}$.
- Pressure distribution and friction have an inverse conduct for bigger estimations of the magnetic parameter, Brownian movement parameter and thermophoresis parameter.
- The variability of entropy generation is 81% for the values of M, while 99% variability for the parameter ${N}_{b}$.
- The variability of entropy generation is 40% for the values of ${N}_{t},$ while 100% variability for the parameter ${B}_{r}$.

## Author Contributions

## Conflicts of Interest

## Nomenclature

$\tilde{u},\tilde{v}$ | velocity components $\left(\mathrm{m}/\mathrm{s}\mathrm{\right)}$ |

$\tilde{x},\tilde{y}$ | Cartesian coordinate $\left(\mathrm{m}\right)$ |

$\tilde{p}$ | pressure in fixed frame $\left(\mathrm{N}/{\mathrm{m}}^{2}\right)$ |

$\tilde{a}$ | wave amplitude $\left(\mathrm{m}\right)$ |

$b\left(\tilde{x}\right)$ | width of the channel $\left(\mathrm{m}\right)$ |

$\tilde{c}$ | wave velocity $\left(\mathrm{m}/\mathrm{s}\right)$ |

$\mathrm{Pr}$ | Prandtl number |

$\mathrm{Re}$ | Reynolds number |

$\mathrm{Rn}$ | Radiation parameter |

$\tilde{t}$ | time $\left(\mathrm{s}\right)$ |

$G{r}_{F}$ | basic density Grashof number |

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

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

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

$\overline{{\rm K}}\left(\ll 1\right)$ | constant |

${B}_{0}$ | magnetic field $\left(\mathrm{T}\right)$ |

$We$ | Weissenberg number |

$Q$ | volume flow rate $\left({\mathrm{m}}^{3}/\mathrm{s}\right)$ |

$T,F$ | temperature $\left(\mathrm{K}\right)$ and concentration |

$g$ | acceleration due to gravity $\left(\mathrm{m}/{\mathrm{s}}^{2}\right)$ |

${\mathrm{D}}_{\mathrm{B}}$ | Brownian diffusion coefficient $\left({\mathrm{m}}^{2}/\mathrm{s}\right)$ |

${D}_{T}$ | thermophoretic diffusion coefficient $\left({\mathrm{m}}^{2}/\mathrm{s}\right)$ |

$K$ | mean absorption constant |

$M$ | Hartman number |

$S$ | stress tensor |

$\tilde{k}$ | porosity parameter |

## Greek Symbols

$\kappa $ | nanofluid thermal conductivity $\left(\mathrm{W}/\mathrm{m}\mathrm{K}\right)$ |

$\mu $ | viscosity of the fluid $\left(\mathrm{N}\mathrm{s}/{\mathrm{m}}^{2}\right)$ |

$\Phi $ | nanoparticle volume fraction |

$\sigma $ | electrical conductivity $\left(\mathrm{S}/\mathrm{m}\right)$ |

$\delta $ | wave number $\left({\mathrm{m}}^{-1}\right)$ |

${c}_{p}$ | effective heat capacity of nanoparticle $\left(\mathrm{J}/\mathrm{K}\right)$ |

$\nu $ | nanofluid kinematic viscosity $\left({\mathrm{m}}^{2}/\mathrm{s}\right)$ |

${\left(\rho \right)}_{p}$ | nanoparticle mass density $\left(\mathrm{kg}/{\mathrm{m}}^{3}\right)$ |

${\rho}_{f}$ | fluid density $\left(\mathrm{kg}/{\mathrm{m}}^{3}\right)$ |

${\rho}_{{f}_{0}}$ | fluid density at the reference temperature $\left({T}_{0}\right)\left(\mathrm{kg}/{\mathrm{m}}^{3}\right)$ |

$\zeta $ | volumetric expansion coefficient of the fluid |

${\left(\rho c\right)}_{f}$ | heat capacity of fluid $\left(\mathrm{J}/\mathrm{K}\right)$ |

$\lambda $ | wavelength $\left(\mathrm{m}\right)$ |

$\varphi $ | amplitude ratio |

${\mu}_{nf}$ | viscosity of nanofluid |

$\theta $ | Temperature |

## References

- Villone, M.M.; Greco, F.; Hulsen, M.A.; Maffettone, P.L. Simulation of an elastic particle in Newtonian and Viscoelastic fluids subjected to confined shear flow. J. Non-Newton. Fluid Mech.
**2014**, 210, 47–55. [Google Scholar] [CrossRef] - Hatami, M.; Domairry, G. Transient vertically motion of a soluble particle in a Newtonian fluid media. J. Powder Technol.
**2014**, 253, 481–485. [Google Scholar] [CrossRef] - Liu, J.; Zhu, C.; Fu, T.; Ma, Y.; Li, H. Numerical simulation of the interaction between three equal interval parallel bubbles rising in non-Newtonian fluid. Chem. Eng. Sci.
**2013**, 93, 55–66. [Google Scholar] [CrossRef] - Hatami, M.; Ganji, D.D. Natural convection of sodium alginate Non-Newtonian nanofluid flow between two vertical flat plates by analytical and numerical methods. Case Stud. Therm. Eng.
**2014**, 2, 14–22. [Google Scholar] [CrossRef] - Chamkha, A.J. On laminar hydromagnetic mixed convection flow in a vertical channel with symmetric and asymmetric wall heating conditions. Int. J. Heat Mass Transf.
**2002**, 45, 2509–2525. [Google Scholar] [CrossRef] - Umavathi, J.C.; Kumar, J.P.; Chamkha, A.J.; Pop, I. Mixed convection in a vertical porous channel. Transp. Porous Media
**2005**, 61, 315–335. [Google Scholar] [CrossRef] - Chamkha, A.J. Unsteady laminar hydromagnetic fluid–particle flow and heat transfer in channels and circular pipes. Int. J. Heat Fluid Flow
**2000**, 21, 740–746. [Google Scholar] [CrossRef] - Ghalambaz, M.; Behseresht, A.; Behseresht, J.; Chamkha, A. Effects of nanoparticles diameter and concentration on natural convection of the Al
_{2}O_{3}–water nanofluids considering variable thermal conductivity around a vertical cone in porous media. Adv. Powder Technol.**2015**, 26, 224–235. [Google Scholar] [CrossRef] - Reddy, P.S.; Chamkha, A.J. Soret and Dufour effects on MHD convective flow of Al
_{2}O_{3}–water and TiO_{2}–water nanofluids past a stretching sheet in porous media with heat generation/absorption. Adv. Powder Technol.**2016**, 27, 1207–1218. [Google Scholar] [CrossRef] - Chamkha, A. Fully developed free convection of a micropolar fluid in a vertical channel. Int. Commun. Heat Mass Transf.
**2002**, 29, 1119–1127. [Google Scholar] [CrossRef] - Kumar, J.P.; Umavathi, J.C.; Chamkha, A.J.; Pop, I. Fully-developed free-convective flow of micropolar and viscous fluids in a vertical channel. Appl. Math. Model.
**2010**, 34, 1175–1186. [Google Scholar] [CrossRef] - Chamkha, A.J.; Grosan, T.; Pop, I. Fully developed mixed convection of a micropolar fluid in a vertical channel. Int. J. Fluid Mech. Res.
**2003**, 30, 251–263. [Google Scholar] [CrossRef] - Choi, S.U.S.; Eastman, J.A. Enhancing thermal conductivity of fluids with nanoparticles. Mater. Sci.
**1995**, 231, 99–105. [Google Scholar] - Eastman, J.A.; Choi, U.S.; Li, S.; Soyez, G.; Thompson, L.J.; DiMelfi, R.J. Novel thermal properties of nanostructured materials. Mater. Sci. Forum
**1999**, 312, 629–634. [Google Scholar] [CrossRef] - Latham, T.W. Fluid Motions in a Peristaltic Pump. Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA, USA, 1966. [Google Scholar]
- Abbas, M.A.; Bai, Y.Q.; Rashidi, M.M.; Bhatti, M.M. Application of drug delivery in magnetohydrodynamics peristaltic blood flow of nanofluid in a non-uniform channel. J. Mech. Med. Biol.
**2016**, 16, 1650052. [Google Scholar] [CrossRef] - Bhatti, M.M.; Abbas, M.A.; Rashidi, M.M. Combine effects of magnetohydrodynamics (MHD) and partial slip on peristaltic blood flow of Ree–Eyring fluid with wall properties. Eng. Sci. Technol. Int. J.
**2016**, 19, 1497–1502. [Google Scholar] [CrossRef] - Salleh, S.; Bachok, N.; Arifin, N.; Ali, F.; Pop, I. Magnetohydrodynamics flow past a moving vertical thin needle in a nanofluid with stability analysis. Energies
**2018**, 11, 3297. [Google Scholar] [CrossRef] - Cong, R.; Ozaki, Y.; Machado, B.; Das, P. Constructal Design of a Rectangular Fin in a Mixed Convective Confined Environment. Inventions
**2018**, 3, 27. [Google Scholar] [CrossRef] - Sadiq, M.; Alsabery, A.; Hashim, I. MHD Mixed Convection in a Lid-Driven Cavity with a Bottom Trapezoidal Body: Two-Phase Nanofluid Model. Energies
**2018**, 11, 2943. [Google Scholar] [CrossRef] - Das, P.K.; Mahmud, S.; Humaira Tasnim, S.; Sadrul Islam AK, M. Effect of surface waviness and aspect ratio on heat transfer inside a wavy enclosure. Int. J. Numer. Methods Heat Fluid Flow
**2003**, 13, 1097–1122. [Google Scholar] [CrossRef] [Green Version] - Abbas, M.A.; Faraz, N.; Bai, Y.Q.; Khan, Y. Analytical study of the non-orthogonal stagnation point flow of a micro polar fluid. J. King Saud Univ. Sci.
**2017**, 29, 126–132. [Google Scholar] [CrossRef] - Umavathi, J.C.; Chamkha, A.J.; Sridhar KS, R. Generalized plain Couette flow and heat transfer in a composite channel. Transp. Porous Media
**2010**, 85, 157–169. [Google Scholar] [CrossRef] - Bhatti, M.M.; Abbas, M.A. Simultaneous effects of slip and MHD on peristaltic blood flow of Jeffrey fluid model through a porous medium. Alex. Eng. J.
**2016**, 55, 1017–1023. [Google Scholar] [CrossRef] [Green Version] - Abbas, M.A.; Bai, Y.Q.; Bhatti, M.M.; Rashidi, M.M. Three-dimensional peristaltic flow of hyperbolic tangent fluid in non-uniform channel having flexible walls. Alex. Eng. J.
**2016**, 55, 653–662. [Google Scholar] [CrossRef] - Chamkha, A.J.; Al-Subaie, M.A. Hydromagnetic buoyancy-induced flow of a particulate suspension through a vertical pipe with heat generation or absorption effects. Turk. J. Eng. Environ. Sci.
**2010**, 33, 127–134. [Google Scholar] - Takhar, H.S.; Chamkha, A.J.; Nath, G. Unsteady flow and heat transfer on a semi-infinite flat plate with an aligned magnetic field. Int. J. Eng. Sci.
**1999**, 37, 1723–1736. [Google Scholar] [CrossRef] - Chamkha, A.J.; Khaled, A.R.A. Hydromagnetic combined heat and mass transfer by natural convection from a permeable surface embedded in a fluid-saturated porous medium. Int. J. Numer. Methods Heat Fluid Flow
**2000**, 10, 455–477. [Google Scholar] [CrossRef] - Chamkha, A.J. Flow of two-immiscible fluids in porous and nonporous channels. J. Fluids Eng.
**2000**, 122, 117–124. [Google Scholar] [CrossRef] - Chamkha, A. Unsteady flow of a dusty conducting fluid through a pipe. Mech. Res. Commun.
**1994**, 21, 281–288. [Google Scholar] [CrossRef] - Chamkha, A.J. Non-Darcy fully developed mixed convection in a porous medium channel with heat generation/absorption and hydromagnetic effects. Numer. Heat Transf. Part A Appl.
**1997**, 32, 653–675. [Google Scholar] [CrossRef] - Umavathi, J.C.; Chamkha, A.J.; Mateen, A.; Al-Mudhaf, A. Unsteady oscillatory flow and heat transfer in a horizontal composite porous medium channel. Nonlinear Anal. Model. Control
**2009**, 14, 397–415. [Google Scholar] - Umavathi, J.C.; Chamkha, A.J.; Mateen, A.; Al-Mudhaf, A. Unsteady two-fluid flow and heat transfer in a horizontal channel. Heat Mass Transf.
**2005**, 42, 81. [Google Scholar] [CrossRef] - Chamkha, A.J. Unsteady laminar hydromagnetic flow and heat transfer in porous channels with temperature-dependent properties. Int. J. Numer. Methods Heat Fluid Flow
**2001**, 11, 430–448. [Google Scholar] [CrossRef] - Chamkha, A.J. Hydromagnetic two-phase flow in a channel. Int. J. Eng. Sci.
**1995**, 33, 437–446. [Google Scholar] [CrossRef] [Green Version] - Ismael, M.A.; Pop, I.; Chamkha, A.J. Mixed convection in a lid-driven square cavity with partial slip. Int. J. Therm. Sci.
**2014**, 82, 47–61. [Google Scholar] [CrossRef] - Parvin, S.; Nasrin, R.; Alim, M.A.; Hossain, N.F.; Chamkha, A.J. Thermal conductivity variation on natural convection flow of water–alumina nanofluid in an annulus. Int. J. Heat Mass Transf.
**2012**, 55, 5268–5274. [Google Scholar] [CrossRef] - Ghalambaz, M.; Doostani, A.; Izadpanahi, E.; Chamkha, A.J. Phase-change heat transfer in a cavity heated from below: The effect of utilizing single or hybrid nanoparticles as additives. J. Taiwan Inst. Chem. Eng.
**2017**, 72, 104–115. [Google Scholar] [CrossRef] - Shapiro, A.H.; Jaffrin, M.Y.; Weinberg, S.L. Peristaltic pumping with long wavelengths at low Reynolds number. J. Fluid Mech.
**1969**, 37, 799–825. [Google Scholar] [CrossRef] - Zaraki, A.; Ghalambaz, M.; Chamkha, A.J.; Ghalambaz, M.; De Rossi, D. Theoretical analysis of natural convection boundary layer heat and mass transfer of nanofluids: Effects of size, shape and type of nanoparticles, type of base fluid and working temperature. Adv. Powder Technol.
**2015**, 26, 935–946. [Google Scholar] [CrossRef] - Gupta, B.B.; Seshadri, V. Peristaltic pumping in non-uniform tubes. J. Biomech.
**1976**, 9, 105–109. [Google Scholar] [CrossRef] - Mekheimer, K.S. Peristaltic flow of blood under effect of a magnetic field in a non-uniform channel. Appl. Math. Comput.
**2004**, 153, 763–777. [Google Scholar] [CrossRef] - Rashidi, M.M.; Bhatti, M.M.; Abbas, M.A.; Ali ME, S. Entropy Generation on MHD Blood Flow of Nanofluid Due to Peristaltic Waves. Entropy
**2016**, 18, 117. [Google Scholar] [CrossRef] - Torabi, M.; Zhang, K.; Karimi, N.; Peterson, G.P. Entropy generation in thermal systems with solid structures—A concise review. Int. J. Heat Mass Transf.
**2016**, 97, 917–931. [Google Scholar] [CrossRef] - Biswal, P.; Basak, T. Entropy generation vs energy efficiency for natural convection based energy flow in enclosures and various applications: A review. Renew. Sustain. Energy Rev.
**2017**, 80, 1412–1457. [Google Scholar] [CrossRef] - Rashidi, M.M.; Abbas, M.A. Effect of Slip Conditions and Entropy Generation Analysis with an Effective Prandtl Number Model on a Nanofluid Flow through a Stretching Sheet. Entropy
**2017**, 18, 414. [Google Scholar] [CrossRef] - Qing, J.; Bhatti, M.M.; Abbas, M.A.; Rashidi, M.M.; Ali, M.E.S. Entropy Generation on MHD Casson Nanofluid Flow over a Porous Stretching/Shrinking Surface. Entropy
**2016**, 18, 123. [Google Scholar] [CrossRef] - Bhatti, M.M.; Abbas, T.; Rashidi, M.M. Numerical study of entropy generation with nonlinear thermal radiation on magnetohydrodynamics non-newtonian nanofluid through a porous shrinking sheet. J. Magn.
**2016**, 21, 468–475. [Google Scholar] [CrossRef] - Abbas, M.A.; Bai, Y.; Rashidi, M.M.; Bhatti, M.M. Analysis of Entropy Generation in the Flow of Peristaltic Nanofluids in Channels with Compliant Walls. Entropy
**2016**, 18, 90. [Google Scholar] [CrossRef] - Mahian, O.; Kianifar, A.; Kleinstreuer, C.; Moh’d, A.A.N.; Pop, I.; Sahin, A.Z.; Wongwises, S. A review of entropy generation in Nanofluid flow. Int. J. Heat Mass Transf.
**2013**, 65, 514–532. [Google Scholar] [CrossRef]

**Figure 5.**Pressure distribution for various values of $M$ and friction force profile for various values of ${G}_{rf}$.

Model | R | R Square | Adjusted R Square | Standard Error of the Estimate |
---|---|---|---|---|

1 | 0.900 ^{a} | 0.809 | 0.799 | 0.7558884 |

2 | 0.999 ^{a} | 0.998 | 0.998 | 0.0550427 |

3 | 0.635 ^{a} | 0.403 | 0.370 | 4.6675041 |

4 | 1.000 ^{a} | 1.000 | 1.000 | 0.19437519 |

^{a}Predictors: (Constant), M.

Model | Unstandardized Coefficients | Standardized Coefficients | T | Significant | ||
---|---|---|---|---|---|---|

B | Standard Error | Beta | ||||

1 | (Constant) | 74.223 | 0.351 | 211.381 | 0.000 | |

M | −2.562 | 0.293 | −0.900 | −8.739 | 0.000 | |

2 | (Constant) | 29.097 | 0.026 | 1137.975 | 0.000 | |

M | 2.049 | 0.021 | 0.999 | 95.977 | 0.000 | |

3 | (Constant) | 65.565 | 2.168 | 30.239 | 0.000 | |

M | 6.307 | 1.810 | 0.635 | 3.485 | 0.003 | |

4 | (Constant) | 1.359 | 0.090 | 15.056 | 0.000 | |

M | 68.492 | 0.075 | 1.000 | 908.676 | 0.000 |

Entropy and Parameters | ${\mathit{N}}_{\mathit{S}}\text{}{\mathit{V}}_{\mathit{S}}\text{}{\mathbf{B}}_{\mathbf{r}}$ | ${\mathit{N}}_{\mathit{S}}\text{}{\mathit{V}}_{\mathit{S}}\text{}{\mathit{N}}_{\mathit{t}}$ | ${\mathit{N}}_{\mathit{S}}\text{}{\mathit{V}}_{\mathit{S}}\text{}{\mathit{N}}_{\mathit{b}}$ | ${\mathit{N}}_{\mathit{S}}\text{}{\mathit{V}}_{\mathit{S}}\text{}\mathit{M}$ |
---|---|---|---|---|

Values range | 0.1 to 2.0 | 0.1 to 2.0 | 0.1 to 2.0 | 0.1 to 2.0 |

N | 20 | 20 | 20 | 20 |

Pearson Correlation Siganificant (2-tailed) | 1.000 ** 0.000 | 0.635 ** 0.003 | 0.999 ** 0.000 | 0.900 ** 0.000 |

Remarks | Perfect Relation | Strong Positive Relation | Strong Positive Relation | Very Strong Negative Relations |

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**MDPI and ACS Style**

Abbas, M.A.; Hussain, I.
Statistical Analysis of the Mathematical Model of Entropy Generation of Magnetized Nanofluid. *Inventions* **2019**, *4*, 32.
https://doi.org/10.3390/inventions4020032

**AMA Style**

Abbas MA, Hussain I.
Statistical Analysis of the Mathematical Model of Entropy Generation of Magnetized Nanofluid. *Inventions*. 2019; 4(2):32.
https://doi.org/10.3390/inventions4020032

**Chicago/Turabian Style**

Abbas, Munawwar Ali, and Ibrahim Hussain.
2019. "Statistical Analysis of the Mathematical Model of Entropy Generation of Magnetized Nanofluid" *Inventions* 4, no. 2: 32.
https://doi.org/10.3390/inventions4020032