# Symmetry and Asymmetry in the Thermo-Magnetic Convection of Silver Nanofluid

^{*}

## Abstract

**:**

## 1. Introduction

_{2}O

_{3}and TiO

_{2}as solid particles) influenced by a magnetic field was discussed in [19]. One of the observations coming from this report was that flow with one dominating cell could be changed by increasing the strength of the magnetic field to symmetrical three rolls flow, one above the other. Examination of aluminum oxide–water nanofluid free convection within cubic porous cavity was conducted in [20]. Magnetic force was responsible for decreasing velocity component values (presented by x and z components); however, it did not change the symmetrical distribution of those parameters for Al

_{2}O

_{3}nanoparticle concentration. The effect of silver nanoparticles’ volume fraction and magnetic field on convective heat transfer processes in cylindrical annulus enclosure was studied in [21]. The velocity components’ values increased as a function of increasing nanoparticle concentrations. On the other hand, the pattern of isotherms was affected strongly and convection was suppressed by increasing the magnetic field. Investigations on the influence of magnetic and thermal gradients’ mutual orientations (at a different angle) on ferrofluid flow was studied in [22]. Unicellular flow, rotating counter-clockwise, changed to the two-cell mode. Then, a single clockwise cell appeared for a varied angle, which showed that in such a case, the resultant force is strong enough to reverse the fluid flow. The phenomena of reverse flow was occurring when the thermo-magnetic convection force was dominant over the thermo-gravity convection—force names are as in the referenced article.

## 2. Considered System

#### Magnetic Field

**H**is a vector of the magnetic field strength [A/m],

**B**is a vector of the magnetic induction [T],

**J**is a vector of electric current density, which is a source of the magnetic field [A/m

^{2}],

**A**is a vector of magnetic field potential [Wb/m] and μ is the medium magnetic permeability [H/m], resulting in the final equation:

**A**, when the electrical current density is known. The boundary conditions of the magnetic field potential equal to zero was set at the large shell (4 m diameter) covering the electrical solenoid. Calculations were conducted in Comsol software.

## 3. Mathematical Modelling of Nanofluid Flow

- carrier phase:$$\nabla \cdot ((1-\varphi ){\rho}_{\mathrm{c}}u+\varphi \text{\hspace{0.17em}}{\rho}_{\mathrm{d}}v)=0$$
- dispersed phase:$$\nabla \cdot (\varphi \text{\hspace{0.17em}}{\rho}_{\mathrm{d}}v)=0$$
**u**,**v**are the velocity vectors of the carrier and dispersed phases, respectively [m/s]; ${\rho}_{\mathrm{c}},{\rho}_{\mathrm{d}}$ are the density of the carrier and dispersed phases, respectively, [kg/m^{3}].

- carrier phase:$${\rho}_{\mathrm{c}}\frac{\partial u}{\partial t}+{\rho}_{\mathrm{c}}(u\cdot \nabla )u=-\nabla \cdot (pI)+\nabla \cdot {\tau}_{\mathrm{c}}+{F}_{\mathrm{c}}$$
- dispersed phase:$${\rho}_{\mathrm{d}}\frac{\partial v}{\partial t}+{\rho}_{\mathrm{d}}(v\cdot \nabla )v=-\nabla \cdot (pI)+\nabla \cdot {\tau}_{\mathrm{d}}+{F}_{\mathrm{d}}$$
^{2}];**F**and_{c}**F**are the sum of body forces acting on the carrier phase and dispersed phase, respectively. Definitions of these forces are written as:_{d}$${F}_{\mathrm{c}}={F}_{\mathrm{g}}+{F}_{\mathrm{m}}+{F}_{\mathrm{d},\mathrm{c}}/\left(1-\varphi \right)$$$${F}_{\mathrm{d}}={F}_{\mathrm{g}}+{F}_{\mathrm{m}}+{F}_{\mathrm{d},\mathrm{d}}/\varphi $$**F**_{g}and**F**_{m}are the vectors of volumetric forces (gravitational and magnetic, [N/m^{3}]):$${F}_{\mathrm{g}}=\rho \cdot \beta \left(T-{T}_{0}\right)g$$^{2}]; ρ is the density, [kg/m^{3}]; β is the thermal expansion coefficient, [1/K]; T is the temperature, [K]; T_{0}is the reference temperature, which in this case was average value of hot and cold walls’ temperature, [K]. The symbols describing phase properties in Equation (7) represent the carrier phase, while in Equation (8) represent the dispersed phase,$${F}_{\mathrm{m}}=-\frac{{\chi}_{\mathrm{m}}\cdot \rho \cdot \beta \left(T-{T}_{0}\right)}{2{\mu}_{\mathrm{m}}}\nabla {{\rm B}}^{2}$$**F**_{m}is the magnetic buoyancy force [N/m^{3}]; χ_{m}is the mass magnetic susceptibility, [m^{3}/kg]; μ_{m}is the vacuum magnetic permeability, 4π × 10^{−7}, [H/m] = [N/A^{2}]; $\nabla $**B**^{2}is the gradient of magnetic induction square, [T^{2}/m], ${F}_{\mathrm{d},\mathrm{c}},{F}_{\mathrm{d},\mathrm{d}}$ are the vectors of volumetric force exerted by one phase on the second phase (as a result of an exchange of momentum between the phases) [N/m^{3}], defined as:$${F}_{\mathrm{d},\mathrm{c}}=-{F}_{\mathrm{d},\mathrm{d}}=\epsilon (v-u)$$^{3}s)]. The collisions between the particles were not considered, and their constant concentration and uniform spatial distribution were assumed.

^{3}]; ${C}_{\mathrm{p},\mathrm{nf}}$ is the specific heat of the nanofluid, [J/(kgK)]; ${k}_{\mathrm{nf}}$ is the thermal conductivity coefficient of the nanofluid, [W/(mK)].

## 4. Numerical Model

^{6}and the Prandtl number about 6.5—the analyzed cases were in the range of laminar and transition flows. Therefore, the authors decided to treat the flow as laminar one. Moreover, the mesh characteristics, especially in a vicinity of the walls, enable the correct solution of some turbulence features, which can appear in the transition flow regime. Additionally, the calculations discussed in [29] and utilizing the realizable k-ε turbulence model [30], led to the results being similar to the presented ones in the case of natural convection. In the results section, it will be shown that magnetic field stabilized the flow, which is why in the authors’ opinion the laminar approach was the right selection.

## 5. Results

**F**

_{c}) acting on the fluid (carrier phase) and the particles (dispersed phase) also at 0 and 6T of magnetic induction. Attention should be payed to the fact that, in the majority of cases, an individual scale is shown. This was done on purpose to visualize the flow structure. When a uniform scale was used, the fine flow pattern disappeared.

_{T}is the thermal Rayleigh number represented by:

^{2}]; β

_{nf}is the thermal expansion coefficient of the nanofluid [1/K] [27]; ρ

_{nf}is the density of the nanofuid [kg/m

^{3}]; C

_{p,nf}is the specific heat of nanofluid [J/(kgK)]; μ

_{nf}is the dynamic viscosity of the nanofluid [kg/(ms)]; k

_{nf}is the thermal conductivity of the nanofluid [W/(mK)]; d is the characteristic dimension [m]; ΔT is the temperature difference [K] and ${\chi}_{m}$ is the magnetic susceptibility determined experimentally [m

^{3}/kg].

## 6. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Maxwell, J.C. A treatise on Electricity and Magnetism; Dover Publications, Inc.: New York, NY, USA, 1954. [Google Scholar]
- Eastman, J.A.; Choi, U.S.; Li, S.; Thompson, L.J.; Lee, S. Enhanced thermal conductivity through the development of nanofluids. MRS Proc.
**1996**, 457, 3–11. [Google Scholar] [CrossRef] [Green Version] - Xie, H.; Lee, H.; Youn, W.; Choi, M. Nanofluids containing multiwalled carbon nanotubes and their enhanced thermal conductivities. J. Appl. Phys.
**2003**, 94, 4967. [Google Scholar] [CrossRef] - Eastman, J.A.; Choi, S.U.S.; Li, S.; Yu, W.; Thompson, L.J. Anomalously increased effective thermal conductivities of ethylene glycol-based nanofluids containing copper nanoparticles. Appl. Phys. Lett.
**2001**, 78, 718. [Google Scholar] [CrossRef] - Keblinski, P.; Prasher, R.; Eapen, J. Thermal conductance of nanofluids: Is the controversy over? J. Nanoparticle Res.
**2008**, 10, 1089–1097. [Google Scholar] [CrossRef] - Lee, S.; Taylor, R.A.; Dai, L.; Prasher, R.; Phelan, P.E. The effective latent heat of aqueous nanofluids. Mater. Res. Express
**2015**, 2, 65004. [Google Scholar] [CrossRef] - Ternik, P. Conduction and convection heat transfer characteristics of water–Au nanofluid in a cubic enclosure with differentially heated side walls. Int. J. Heat Mass Transf.
**2015**, 80, 368–375. [Google Scholar] [CrossRef] - Fornalik, E.; Filar, P.; Tagawa, T.; Ozoe, H.; Szmyd, J.S. Experimental study on the magnetic convection in a vertical cylinder. Exp. Therm. Fluid Sci.
**2005**, 29, 971–980. [Google Scholar] [CrossRef] - Fornalik-Wajs, E.; Filar, P.; Wajs, J.; Roszko, A.; Pleskacz, Ł.; Ozoe, H. Flow structure, heat transfer and scaling analysis in the case of thermo-magnetic convection in a differentially heated cylindrical enclosure. J. Phys. Conf. Ser.
**2014**, 503. [Google Scholar] [CrossRef] [Green Version] - Kenjereš, S.; Fornalik-Wajs, E.; Wrobel, W.; Szmyd, J.S. Inversion of flow and heat transfer of the paramagnetic fluid in a differentially heated cube. Int. J. Heat Mass Transf.
**2020**, 151, 119407. [Google Scholar] [CrossRef] - Zhang, T.; Che, D. Double MRT thermal lattice Boltzmann simulation for MHD natural convection of nanofluids in an inclined cavity with four square heat sources. Int. J. Heat Mass Transf.
**2016**, 94, 87–100. [Google Scholar] [CrossRef] - Nemati, H.; Farhadi, M.; Sedighi, K.; Ashorynejad, H.R.; Fattahi, E. Magnetic field effects on natural convection flow of nanofluid in a rectangular cavity using the Lattice Boltzmann model. Sci. Iran.
**2012**, 19, 303–310. [Google Scholar] [CrossRef] [Green Version] - Mliki, B.; Abbassi, M.A.; Omri, A.; Zeghmati, B. Effects of nanoparticles Brownian motion in a linearly/sinusoidally heated cavity with MHD natural convection in the presence of uniform heat generation/absorption. Powder Technol.
**2016**, 295, 69–83. [Google Scholar] [CrossRef] - Sheikholeslami, M.; Ganji, D.D. Entropy generation of nanofluid in presence of magnetic field using Lattice Boltzmann Method. Phys. A Stat. Mech. Appl.
**2015**, 417, 273–286. [Google Scholar] [CrossRef] - Sheremet, M.A.; Oztop, H.F.; Pop, I. MHD natural convection in an inclined wavy cavity with corner heater filled with a nanofluid. J. Magn. Magn. Mater.
**2016**, 416, 37–47. [Google Scholar] [CrossRef] - Al Kalbani, K.S.; Rahman, M.M.; Alam, S.; Al-Salti, N.; Eltayeb, I.A. Buoyancy induced heat transfer flow inside a tilted square enclosure filled with nanofluids in the presence of oriented magnetic field. Heat Transf. Eng.
**2018**, 39, 511–525. [Google Scholar] [CrossRef] - Pordanjani, A.H.; Jahanbakhshi, A.; Ahmadi Nadooshan, A.; Afrand, M. Effect of two isothermal obstacles on the natural convection of nanofluid in the presence of magnetic field inside an enclosure with sinusoidal wall temperature distribution. Int. J. Heat Mass Transf.
**2018**, 121, 565–578. [Google Scholar] [CrossRef] - Sheikholeslami, M.; Hayat, T.; Alsaedi, A. MHD free convection of Al2O3-water nanofluid considering thermal radiation: A numerical study. Int. J. Heat Mass Transf.
**2016**, 96, 513–524. [Google Scholar] [CrossRef] - Muthtamilselvan, M.; Doh, D.-H. Magnetic field effect on mixed convection in a lid-driven square cavity filled with nanofluids. J. Mech. Sci. Technol.
**2014**, 28, 137–143. [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] [Green Version] - Ashorynejad, H.R.; Mohamad, A.A.; Sheikholeslami, M. Magnetic field effects on natural convection flow of a nanofluid in a horizontal cylindrical annulus using Lattice Boltzmann method. Int. J. Therm. Sci.
**2013**, 64, 240–250. [Google Scholar] [CrossRef] - Bouhrour, A.; Kalache, D. Thermomagnetic convection of a magnetic nanofluid influenced by a magnetic field. Therm. Sci.
**2017**, 21, 1261–1274. [Google Scholar] [CrossRef] [Green Version] - Roszko, A.; Fornalik-Wajs, E. Extend of magnetic field interference in the natural convection of diamagnetic nanofluid. Heat Mass Transf.
**2018**, 54, 2243–2254. [Google Scholar] [CrossRef] - Parametthanuwat, T.; Bhuwakietkumjohn, N.; Rittidech, S.; Ding, Y. Experimental investigation on thermal properties of silver nanofluids. Int. J. Heat Fluid Flow
**2015**, 56, 80–90. [Google Scholar] [CrossRef] - Sunil, J.; Dhayanithi Pooja, M.; Ginil, R.; Alex, S.N.; Ajith Pravin, A. Thermal conductivity and dynamic viscosity of aqueous-silver nanoparticle dispersion. Mater. Today Proc.
**2020**. [Google Scholar] [CrossRef] - Minkowycz, W.J.; Sparrow, E.M.; Murthy, J.Y. Handbook of Numerical Heat Transfer; Wiley: Hoboken, NJ, USA, 2006. [Google Scholar]
- Xuan, Y.; Roetzel, W. Conceptions for heat transfer correlation of nanofluids. Int. J. Heat Mass Transf.
**2000**, 43, 3701–3707. [Google Scholar] [CrossRef] - Mueller, G. Convection and Inhomogenities in Crystal Growth from Melt; Springer: Berlin, Germany, 1998; Volume 12. [Google Scholar]
- Sipa, J. Numerical analysis of the base fluid influence on the natural convection heat transfer of the nanofluids with the silver particles. Master’s Thesis, AGH University of Science and Technology, Krakow, Poland, 2019. (In Polish). [Google Scholar]
- ANSYS Fluent Theory Guide, 15.0. Available online: http://www.pmt.usp.br/academic/martoran/notasmodelosgrad/ANSYS%20Fluent%20Theory%20Guide%2015.pdf (accessed on 1 October 2020).
- Fornalik-Wajs, E.; Roszko, A.; Donizak, J. Nanofluid flow driven by thermal and magnetic forces–Experimental and numerical studies. Energy
**2020**, 201, 117658. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) Magnetic induction distribution (the values are in [T]) with the magnetic field lines and (

**b**) magnetic induction square gradient distribution [T

^{2}/m].

**Figure 2.**Velocity magnitude, temperature and resultant force distributions at 0 and 6 T of magnetic induction for water.

**Figure 3.**Velocity magnitude, temperature and resultant force distributions at 0 and 6 T of magnetic induction for Ag0.1.

**Figure 4.**Velocity magnitude, temperature and resultant force distributions at 0 and 6 T of magnetic induction for Ag0.25.

**Figure 5.**Velocity magnitude, temperature and resultant force distributions at 0 and 6 T of magnetic induction for Ag0.5.

**Figure 6.**Location of the temperature sensors on (

**a**) plane “a”, (

**b**) plane “b” and (

**c**) distance between them (in mm).

**Figure 7.**Temperature local values in selected points and their Fast Fourier Transform (FFT) spectrum at plane “a”—left column and at plane “b”—right column, at 0T—upper part and at 6T of magnetic induction—lower part for Ag0.1.

**Figure 8.**Temperature local values in selected points and their FFT spectra at plane “a”—left column and at plane “b”—right column, at 0T—upper part and at 6T of magnetic induction—lower part for Ag0.25.

**Figure 9.**Temperature local values in selected points and their FFT spectra at plane “a”—left column and at plane “b”—right column, at 0T—upper part and at 6T of magnetic induction—lower part for Ag0.5.

**Figure 10.**The average Nusselt number dependence on thermo-magnetic Rayleigh number for all investigated fluids, left side—numerical analysis, right side—experimental analysis.

Property | Symbol and Unit | Water * | Silver |
---|---|---|---|

Thermal conductivity | k, W/(mK) | 0.6 | 429 |

Density | ρ, kg/m^{3} | 998 | 10,500 |

Specific heat | c_{p}, J/(kgK) | 4183 | 235 |

Thermal expansion coefficient | β, 1/K | 18.5 × 10^{−5} | - |

Dynamic viscosity | μ, kg/(ms) | 10.6 × 10^{−4} | - |

Mass magnetic susceptibility | χ_{m}, kg/m^{3} | −8.9 × 10^{−9} | −2.27 × 10^{−9} |

Volume magnetic susceptibility | χ_{v}, - | −8.8 × 10^{−6} | −2.38 × 10^{−5} |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 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**

Fornalik-Wajs, E.; Roszko, A.; Donizak, J.
Symmetry and Asymmetry in the Thermo-Magnetic Convection of Silver Nanofluid. *Symmetry* **2020**, *12*, 1891.
https://doi.org/10.3390/sym12111891

**AMA Style**

Fornalik-Wajs E, Roszko A, Donizak J.
Symmetry and Asymmetry in the Thermo-Magnetic Convection of Silver Nanofluid. *Symmetry*. 2020; 12(11):1891.
https://doi.org/10.3390/sym12111891

**Chicago/Turabian Style**

Fornalik-Wajs, Elzbieta, Aleksandra Roszko, and Janusz Donizak.
2020. "Symmetry and Asymmetry in the Thermo-Magnetic Convection of Silver Nanofluid" *Symmetry* 12, no. 11: 1891.
https://doi.org/10.3390/sym12111891