# A Comprehensive Review on Theoretical Aspects of Nanofluids: Exact Solutions and Analysis

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## Abstract

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## 1. Introduction

## 2. Models for Nanofluids

## 3. Properties of Nanofluids

#### 3.1. Density

#### 3.2. Specific Heat Capacity

#### 3.3. Electrical Conductivity

#### 3.4. Coefficient of Thermal Expansion

#### 3.5. Coefficient of Mass Expansion

#### 3.6. Dynamic Viscosity

#### 3.7. Thermal Conductivity

## 4. Exact Solutions

## 5. Exact Solutions for the Flow of Nanofluids

## 6. Hybrid Nanofluid

_{3}O

_{4}is discussed by Sheikholislami et al. [129]. In this analysis, they considered the variable magnetic field in electric wires. By adding these hybrid nanoparticles, the heat transmission rate of the base fluid in the cavity is enhanced. Khan et al. studied the flow of a hybrid nanofluid in a channel with the analysis of entropy generation [130]. The flow is taken in a rotating system.

## 7. Conclusions

## 8. Future Suggestions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

MHD | Magnetohydrodynamics |

DAC | Direct Absorption Collector |

DNA | Deoxyribonucleic acid |

GO | Graphene oxide |

TiO_{2} | Titanium Oxide |

Al_{2}O_{3} | Aluminum Oxide |

CuO | Coper Oxide |

Cu | Copper |

CeO_{2} | Cerium Oxide |

Ag | Silver |

Nr | Radiation Parameter |

u | dimensional velocity |

$t$ | dimensional time |

$y$ | coordinate axis normal to the plate |

$g$ | acceleration due to gravity |

$\sigma $ | electrical conductivity |

$k$ | thermal conductivity of the fluid |

$U$ | amplitude of the plate oscillations |

${c}_{p}$ | specific heat at constant pressure |

$T$ | the temperature of the fluid |

${T}_{\infty}$ | ambient temperature |

${T}_{w}$ | wall temperature |

$C$ | concentration |

${C}_{\infty}$ | ambient concentration |

${C}_{w}$ | wall concentration |

H(t) | Unit step function |

ω | frequency of the plate oscillation |

ρ | density of the fluid |

$\mu $ | viscosity |

${\beta}_{T}$ | Thermal Expansion |

${\beta}_{C}$ | Mass Expansion |

D | Mass Diffusivity |

${\beta}_{1}$ | Brinkman parameter |

Re | Reynold’s number |

M | Hartmann number |

$Gr$ | Grashof number |

Pr | Prandtl number |

$Gm$ | Mass Grashof number |

$K$ | Porosity Parameter |

$Sc$ | Schmidt number |

$Nu$ | Nusselt number |

$\gamma $ | Chemical reaction parameter |

$\varnothing $ | the volume fraction of nanoparticles |

${d}_{p}$ | diameter of particle |

$H$ | diameter of the tube |

$\epsilon $ and $\eta $ | empirical coefficients of nanoparticles |

$d$ | diameter of the nanotube |

$L$ | length of the nanotube |

${a}_{k}$ | Kiptza’s Constant |

$\psi $ | is the sphericity factor |

$nf$ | subscript for nanofluid |

$hnf$ | subscript for hybrid nanofluid |

$f$ | subscript for fluid |

$bf$ | subscript for base fluid |

$CNT$ | subscript for carbon nanotube |

$s$ | subscript for solid |

$p$ | subscript for particle |

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**Figure 4.**Values of Nusselt number against volume fraction for five different water-based nanofluids; dotted curve for TiO

_{2}, the dot-dashed curve for Al

_{2}O

_{3}, the dashed curve for CuO, the thin curve for Cu, and thick curve for Ag. (

**a**) Nr = 0 and (

**b**) Nr = 1 [89].

**Figure 5.**Velocity profile for different nanoparticles [91].

**Figure 6.**The illustrative diagram of the problem [100].

**Figure 7.**Nanofluid-Based DAC [105].

**Figure 8.**Variation in Nusselt number for different nanoparticles [105].

**Figure 9.**Applications of gold nanoparticles in cancer therapy [111].

**Figure 10.**Direct absorption solar collector [117].

**Figure 11.**Variations in Nusselt number [117].

**Figure 12.**Different shapes of nanoparticles [122].

**Figure 13.**Hybrid nanofluid [128].

Model Name | Expression |
---|---|

Einstein model [30,31] | ${\mu}_{nf}=\left(2.5\varphi +1\right){\mu}_{f},\text{\hspace{0.17em}}0<\varphi <0.05$ |

Brownian model [31,32] | ${\mu}_{nf}={\mu}_{f}\left(1+2.5\varphi +6.17{\varphi}^{2}\right)$ |

Brinkman model [33] | ${\mu}_{nf}=\frac{{\mu}_{f}}{{\left(1-\varphi \right)}^{2.5}}$ |

Pak and Cho’s Correlation [25] | ${\mu}_{nf}={\mu}_{f}\left(1+39.11\varphi +533.9{\varphi}^{2}\right)$ |

Nguyen et al [34] model (altered by Abu-Nada [35]) | $\begin{array}{l}{\mu}_{CuO}=-0.6967+\frac{15.937}{T}T+1.238\varphi +\frac{1356.14}{{T}^{2}}-0.259{\varphi}^{2}-\frac{30.88\varphi}{T}\\ -\frac{19652.74}{{T}^{3}}+0.01593{\varphi}^{3}+\frac{4.38206{\varphi}^{2}}{T}+\frac{147.573\varphi}{{T}^{2}}\end{array}$ ${\mu}_{A{l}_{2}{\mathrm{O}}_{3}}=\mathrm{exp}\left(\begin{array}{l}3.003-0.04203T-0.5445\varphi \\ +0.0002553{T}^{2}+0.0524{\varphi}^{2}-1.622{\varphi}^{-1}\end{array}\right)$ |

Jang et al. model [36] | ${\mu}_{nf}=\left(2.5\varphi +1\right){\mu}_{f}\left[1+\eta {\left(\frac{{d}_{p}}{H}\right)}^{-2\epsilon}{\varphi}^{3/2}\left(\epsilon +1\right)\right]$ |

Koo and Kleinstreuer [37] | ${\mu}_{nf}=5\times {10}^{4}\beta {\rho}_{f}\varphi \sqrt{\frac{{k}_{B}T}{{d}_{p}{\rho}_{p}}}f\left(T,\varphi \right),\{\begin{array}{l}\beta =0.0137{\left(100\varphi \right)}^{-0.8229};\text{\hspace{0.17em}}\varphi <0.01\\ \beta =0.0011{\left(100\varphi \right)}^{-0.7272};\text{\hspace{0.17em}}\varphi >0.01\end{array}$ |

Maiga model [38] | ${\mu}_{nf}={\mu}_{f}\left(1+7.3\varphi +123{\varphi}^{2}\right)$ |

Nguyen model [39] | ${\mu}_{nf}={\mu}_{f}\left(1+0.025\varphi +0.015{\varphi}^{2}\right)$ |

Masoumi et al. [40] | ${\mu}_{nf}={\mu}_{f}+\frac{{\rho}_{p}{V}_{B}{d}_{p}^{2}}{72C\delta}$ |

Gherasim et al. [41] | ${\mu}_{nf}={\mu}_{f}0.904\mathrm{exp}\left(14.8\varphi \right)$ |

For Hybrid Nanofluid [42] | ${\mu}_{hnf}=\frac{{\mu}_{f}}{{\left(1-{\varphi}_{1}\right)}^{2.5}{\left(1-{\varphi}_{2}\right)}^{2.5}}$ |

Model Name | Expression |
---|---|

Maxwell model [43] | ${\kappa}_{nf}={\kappa}_{f}\frac{{\kappa}_{p}+2{\kappa}_{f}-2\varphi \left({\kappa}_{f}-{\kappa}_{p}\right)}{{\kappa}_{p}+2{\kappa}_{f}+\varphi \left({\kappa}_{f}-{\kappa}_{p}\right)}$ |

Hamilton and Crosser [44] | ${k}_{nf}=\frac{{k}_{s}+\left(s-1\right){k}_{f}+\left(s-1\right)\varphi \left({k}_{s}+{k}_{f}\right)}{{k}_{s}+\left(s-1\right){k}_{f}-\varphi \left({k}_{s}-{k}_{f}\right)}{k}_{f}$; $s=3/\psi $ |

Jang and Choi model [45] | ${\kappa}_{nf}={\kappa}_{f}\left[\left(1-\varphi \right)+B{\kappa}_{p}\varphi +18\times {10}^{6}\frac{3{d}_{f}}{{d}_{p}}\text{}{\kappa}_{f}\text{}{\mathrm{Re}}_{dp}^{2}\mathrm{Pr}\varphi \right]$ |

Bruggeman model [46] | ${\kappa}_{nf}=\frac{1}{4}{\kappa}_{f}\left(3\varphi -1\right)\frac{{\kappa}_{p}}{{\kappa}_{f}}+\left(3\left(1-\varphi \right)-1\right)+\sqrt{{\left(\begin{array}{l}\left(3\varphi -1\right)\frac{{\kappa}_{p}}{{\kappa}_{f}}\\ +\left(3\left(1-\varphi \right)-1\right)\end{array}\right)}^{2}+8\frac{{\kappa}_{p}}{{\kappa}_{f}}}$ |

Chon et al. model [47] | ${\kappa}_{nf}={\kappa}_{f}\left\{1+64.7{\varphi}^{0.7640}{\left(\frac{{d}_{f}}{{d}_{p}}\right)}^{0.369}{\left(\frac{{\kappa}_{p}}{{\kappa}_{f}}\right)}^{0:7476}{\mathrm{Pr}}_{T}^{0.9955}{\mathrm{Re}}^{1.2321}\right\}$ |

Koo and Kleinstreuer [37] | ${\kappa}_{nf}={\kappa}_{f}\frac{{\kappa}_{p}+2{\kappa}_{f}-2\varphi \left({\kappa}_{f}-{\kappa}_{p}\right)}{{\kappa}_{p}+2{\kappa}_{f}+\varphi \left({\kappa}_{f}-{\kappa}_{p}\right)}+50000\beta \varphi {\rho}_{f}{\left({c}_{p}\right)}_{f}\sqrt{\frac{{\kappa}_{b}T}{{d}_{p}{\rho}_{p}}}f\left(T,\varphi \right)$ |

Yimin, Li, and Hu [48] | ${\kappa}_{nf}={\kappa}_{f}\frac{{\kappa}_{p}+2{\kappa}_{f}-2\varphi \left({\kappa}_{f}-{\kappa}_{p}\right)}{{\kappa}_{p}+2{\kappa}_{f}+\varphi \left({\kappa}_{f}-{\kappa}_{p}\right)}+\frac{{\rho}_{p}\varphi {\left({c}_{p}\right)}_{f}}{2{\kappa}_{f}}\sqrt{\frac{{\kappa}_{b}T}{3\pi {r}_{c}\eta}}$ |

Charuyakorn et al. [49] | ${\kappa}_{nf}={\kappa}_{f}\left[\left(1+b\rho P{e}_{p}^{m}\right)\frac{{\kappa}_{p}+2{\kappa}_{f}-2\varphi \left({\kappa}_{f}-{\kappa}_{p}\right)}{{\kappa}_{p}+2{\kappa}_{f}+\varphi \left({\kappa}_{f}-{\kappa}_{p}\right)}\right]$ |

Stationary model [50] | ${\kappa}_{nf}={\kappa}_{f}\left[1+\frac{{\kappa}_{p}\varphi {d}_{f}}{{\kappa}_{f}{d}_{p}\left(1-\varphi \right)}\right]$ |

Yu and Choi [51] | ${\kappa}_{nf}={\kappa}_{f}\frac{{\kappa}_{p}+2{\kappa}_{f}-2\varphi \left({\kappa}_{f}-{\kappa}_{p}\right){\left(1+\eta \right)}^{3}}{{\kappa}_{p}+2{\kappa}_{f}+\varphi \left({\kappa}_{f}-{\kappa}_{p}\right){\left(1+\eta \right)}^{3}}$ |

Patel et al. [52] | ${\kappa}_{nf}={\kappa}_{f}\left[1+\frac{{\kappa}_{p}\varphi {d}_{f}}{{\kappa}_{f}{d}_{p}\left(1-\varphi \right)}\left(1+c\frac{2{\kappa}_{B}T{d}_{p}}{\pi {\alpha}_{f}{\mu}_{f}{d}_{p}^{2}}\right)\right]$ |

Mintsa et al. [53] | ${\kappa}_{nf}={\kappa}_{f}\left(1.72\varphi +1\right)$ |

Nan’s Model [54,55] | ${\kappa}_{nf}={\kappa}_{f}\frac{3+\varphi \left\{2\left(\frac{{\kappa}_{CNT}}{1+\left(\frac{2{a}_{k}}{d}\right)\left(\frac{{\kappa}_{CNT}}{{\kappa}_{f}}\right)}+{\kappa}_{f}\right)+\left(\frac{{\kappa}_{CNT}}{1+\left(\frac{2{a}_{k}}{L}\right)\left(\frac{{\kappa}_{CNT}}{{\kappa}_{f}}\right)}-1\right)\right\}}{3-2\varphi \left(\frac{{\kappa}_{CNT}}{1+\left(2{a}_{k}/d\right)\left({\kappa}_{CNT}/{\kappa}_{f}\right)}+{\kappa}_{f}\right)}$ |

Xue [56] | ${\kappa}_{nf}={\kappa}_{f}\frac{1-\varphi +2\varphi \frac{{\kappa}_{CNT}}{{\kappa}_{CNT}-{\kappa}_{f}}\mathrm{ln}\frac{{\kappa}_{CNT}+{\kappa}_{f}}{2{\kappa}_{f}}}{1-\varphi +2\varphi \frac{{\kappa}_{f}}{{\kappa}_{CNT}-{\kappa}_{f}}\mathrm{ln}\frac{{\kappa}_{CNT}+{\kappa}_{f}}{2{\kappa}_{f}}}$ |

Hybrid Nanofluid [42] | ${\kappa}_{hnf}={\kappa}_{bf}\frac{\frac{{\varphi}_{1}{\kappa}_{s1}+{\varphi}_{2}{\kappa}_{s2}}{{\varphi}_{1}+{\varphi}_{2}}+2{\kappa}_{bf}+2\left({\varphi}_{1}{\kappa}_{s1}+{\varphi}_{2}{\kappa}_{s2}\right)-2{\kappa}_{bf}\left({\varphi}_{1}+{\varphi}_{2}\right)}{\frac{{\varphi}_{1}{\kappa}_{s1}+{\varphi}_{2}{\kappa}_{s2}}{{\varphi}_{1}+{\varphi}_{2}}-2{\kappa}_{bf}+2\left({\varphi}_{1}{\kappa}_{s1}+{\varphi}_{2}{\kappa}_{s2}\right)+2{\kappa}_{bf}\left({\varphi}_{1}+{\varphi}_{2}\right)}$ |

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---|---|---|

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(Jordan et al. 2004) [64] https://doi.org/10.1016/j.ijnonlinmec.2003.12.003 | Stokes’ first problem, Maxwell fluids, Integral transform methods | Spatial sine transform (Fourier transform) |

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**Table 4.**Variations in skin friction for different parameters [28].

${\mathit{\beta}}_{1}$ | $\mathbf{Re}$ | $\mathit{M}$ | $\mathit{G}\mathit{r}$ | $\mathit{G}\mathit{m}$ | $\mathit{K}$ | $\mathbf{Pr}$ | $\mathit{N}\mathit{r}$ | $\mathit{S}\mathit{c}$ | $\mathit{\gamma}$ | $\mathit{t}$ | $\mathit{\varphi}$ | $\mathit{n}$ | ${\mathit{\tau}}_{0}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0.3 | 1 | 3 | 2 | 2 | 0.2 | 0.71 | 3 | 0.5 | 2 | 1 | 0.02 | 0 | 2.083 |

0.5 | 0 | 2.842 | |||||||||||

3 | 0 | 4.705 | |||||||||||

4 | 0 | 2.994 | |||||||||||

4 | 0 | 2.613 | |||||||||||

4 | 0 | 2.915 | |||||||||||

0.4 | 0 | 2.231 | |||||||||||

1 | 0 | 2.824 | |||||||||||

4 | 0 | 2.791 | |||||||||||

0.75 | 0 | 2.831 | |||||||||||

4 | 0 | 2.773 | |||||||||||

2 | 0 | 2.711 | |||||||||||

0 | 2.799 | ||||||||||||

0.02 | 1 | 2.802 |

**Table 5.**Impact of volume fraction on Nusselt number and percent enhancement [100].

$\mathit{\varphi}$ | $\mathit{N}\mathit{r}$ | $\mathbf{Pr}$ | $\mathit{t}$ | $\mathit{N}\mathit{u}$ | % Enhancement |
---|---|---|---|---|---|

0 | 0.2 | 1000 | 1 | 32.603 | - |

0.01 | 33.115 | 1.57 | |||

0.02 | 33.631 | 3.15 | |||

0.03 | 34.151 | 4.74 | |||

0.04 | 34.674 | 6.35 |

**Table 6.**Constants a and b empirical shape factors [122].

Model | Platelet | Blade | Cylinder | Brick |
---|---|---|---|---|

A | 37.1 | 14.6 | 13.5 | 1.9 |

B | 6 12.6 | 123.3 | 904.4 | 471.4 |

**Table 7.**Impact of volume fraction on heat transfer rate and percent enhancement [122].

$\mathit{\varphi}$ | $\mathit{N}\mathit{r}$ | $\mathit{\alpha}$ | $\mathit{t}$ | $\mathit{N}\mathit{u}$ | % |
---|---|---|---|---|---|

0 | 0.2 | 0.2 | 1 | 23.53 | - |

0.01 | 24.334 | 3.42 | |||

0.02 | 25.131 | 6.80 | |||

0.03 | 25.923 | 10.16 | |||

0.04 | 26.71 | 13.51 |

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Sheikh, N.A.; Chuan Ching, D.L.; Khan, I.
A Comprehensive Review on Theoretical Aspects of Nanofluids: Exact Solutions and Analysis. *Symmetry* **2020**, *12*, 725.
https://doi.org/10.3390/sym12050725

**AMA Style**

Sheikh NA, Chuan Ching DL, Khan I.
A Comprehensive Review on Theoretical Aspects of Nanofluids: Exact Solutions and Analysis. *Symmetry*. 2020; 12(5):725.
https://doi.org/10.3390/sym12050725

**Chicago/Turabian Style**

Sheikh, Nadeem Ahmad, Dennis Ling Chuan Ching, and Ilyas Khan.
2020. "A Comprehensive Review on Theoretical Aspects of Nanofluids: Exact Solutions and Analysis" *Symmetry* 12, no. 5: 725.
https://doi.org/10.3390/sym12050725