#
Boundary Layer Flow and Heat Transfer of Al_{2}O_{3}-TiO_{2}/Water Hybrid Nanofluid over a Permeable Moving Plate

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^{2}

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## Abstract

**:**

_{2}O

_{3}) and ${\varphi}_{2}$ (TiO

_{2}) are taken into account. Numerical results are graphically described for the skin friction coefficient, ${C}_{f}$, and local Nusselt number, $N{u}_{x}$, as well as velocity and temperature profiles. The results showed that duality occurs when the plate and the free stream travel in the opposite direction. The range of dual solutions expand widely for S and closely reduce for $\varphi $. Thus, a stability analysis is performed. The first solution is stable and realizable compared to the second solution. The ${C}_{f}$ and $N{u}_{x}$ increase with the increment of S. It is also noted that the increase of ${\varphi}_{2}$ leads to an increase in ${C}_{f}$ and decrease in $N{u}_{x}$.

## 1. Introduction

## 2. Description of Flow Problem

_{w}, and the value the for temperature in ambient fluid is T

_{∞}. Meanwhile, the ambient fluid velocity of the plate is expected to be ${U}_{w}=\lambda U$, where $\lambda $ is the parameter for plate velocity (Bachok et al. [36]). Additionally, we also took into account the effect of suction, S, as illustrated in Figure 1. We chose two different nanoparticles: Al

_{2}O

_{3}and TiO

_{2}with water base fluid. The nanoparticles are assumed to have a uniform spherical shape and size. The hybrid nanofluid’s thermophysical attributes are given in Table 1.

_{2}into 0.1 volume of Al

_{2}O

_{3}/water to form the appropriate hybrid nanofluid. In this study, a 0.1 volume of Al

_{2}O

_{3}$\left({\varphi}_{1}=0.1\right)$ is added constantly to the water throughout while various volumes of solid fraction of TiO

_{2}$\left({\varphi}_{2}\right)$ are added to produce Al

_{2}O

_{3}-TiO

_{2}/water.

_{2}O

_{3}and ${\varphi}_{2}$ represents TiO

_{2}. In addition, $\rho $ represents the density, ${C}_{p}$ is specific heat at constant pressure and $k$ is thermal conductivity, where s1 indicates Al

_{2}O

_{3}and s2 TiO

_{2}nanoparticles.

_{f}, and local Nusselt number, Nu

_{x}, which are respectively defined as:

## 3. Stability Solution

## 4. Analysis of Results

_{2}O

_{3}/TiO

_{2}and S are analyzed and further discussed. Pr is taken as 6.2 (water) and $\varphi $ ranges from 0 to 0.2, where $0\le {\varphi}_{1}\le 0.15$ and $0\le {\varphi}_{2}\le 0.2$. Table 3 displays the result on ${f}^{\u2033}\left(0\right)$, $-{\theta}^{\prime}\left(0\right),$ ${C}_{f}{\left(2R{e}_{x}\right)}^{1/2}$ and $N{u}_{x}{\left(R{e}_{x}/2\right)}^{-1/2}$ for $\lambda =0.2$ and $S=1.5$. It can be seen that as the ${\varphi}_{1}$ and ${\varphi}_{2}$ increase, ${C}_{f}{\left(2R{e}_{x}\right)}^{1/2}$ increases while $N{u}_{x}{\left(R{e}_{x}/2\right)}^{-1/2}$ decreases.

_{2}O

_{3}/water $\left({\varphi}_{1}=0.15,\text{}{\varphi}_{2}=0\right)$, and Al

_{2}O

_{3}-TiO

_{2}/water $\left({\varphi}_{1}=0.15,\text{}{\varphi}_{2}=0.15\right)$. It can be clearly observed that the hybrid nanofluid was lower compared to viscous fluid and nanofluid. The flow moved up until a critical point, ${\lambda}_{c}$, where ${\lambda}_{c}=-1.82696$ for viscous fluid, ${\lambda}_{c}=-1.78040$ for Al

_{2}O

_{3}/water and ${\lambda}_{c}=-1.62541$ for Al

_{2}O

_{3}-TiO

_{2}/water. In addition, the hybrid nanofluid was better in enhancing the separation of the boundary layer compared to viscous fluid and nanofluid. In our assumptions, the collision of two nanoparticles with different thermophysical properties is easily dissolved in the base fluid and consequently shortens the separation of the boundary layer. Furthermore, the thickness of the solution is widened for each of the fluids.

_{2}O

_{3}/TiO

_{2}causes the nanoparticle molecules to collide with each other, consequently reducing the velocity and hence increasing the skin friction. The increased of ${\varphi}_{2}$ in the base fluid results in lower thermal conductivity of the base fluid which reduces the heat enhancement capacity of the base fluid. This is due to the addition of nanoparticles, which raises the complex viscosity of the base fluid. It should also be stated that the thickness of the shear stress and thermal boundary layer also rises.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ODE | Ordinary Differential Equation |

Pr | Prandtl number |

T | Temperature |

U | Uniform free stream |

S | Suction/injection parameter |

${q}_{w}$ | Plate heat flux |

Nu_{x} | Local Nusselt number |

Re_{x} | Reynolds number |

${C}_{f}$ | Skin friction coefficient |

$w$ | Condition on plate |

∞ | Ambient condition |

hnf | Hybrid nanofluid |

nf | Nanofluid |

f | Fluid |

α | Thermal diffusivity |

μ | Dynamic viscosity |

ρ | Density |

ψ | Stream function |

η | Similarity variables |

θ | Dimensionless temperature |

(ρC_{p})_{f} | Specific heat for base fluid |

C_{p} | Specific heat at constant pressure |

τ | Dimensionless time |

k | Thermal conductivity |

$\varphi $ | Concentration of nanoparticles |

ϒ | Eigenvalues |

υ | Kinematic viscosity |

## References

- Busemann, A. Ludwig Prandtl. 1875–1953. Biogr. Mem. Fellows R. Soc.
**1960**, 5, 193–205. [Google Scholar] - Duwairi, H.M.; Tashtoush, B.; Damseh, R.A. On heat transfer effects of a viscous fluid squeezed and extended between two parallel plates. Heat Mass Transf.
**2004**, 41, 112–117. [Google Scholar] - Arifuzzaman, S.M.; Islam, M.M.; Haque, M.M. Combined heat and mass transfer steady flow of a viscous over a vertical plate with large suction. Int. J. Sci. Technol. Soc.
**2015**, 3, 236–242. [Google Scholar] [CrossRef] - Liu, X.; Zheng, L.; Chen, G.; Ma, L. Coupling effects of viscous sheet and ambient fluid on boundary layer flow and heat transfer in Power law fluids. J. Heat Transf.
**2019**, 141, 061701. [Google Scholar] [CrossRef] - Anyanwu, O.E.; Raymond, D.; Adamu, A.K.; Ogwumu, O.D. Viscous and Joule dissipation effects on heat transfer in MHD Poiseuille flow in the presence of radial magnetic field. FUW Trends Sci. Technol. J.
**2017**, 2, 171–186. [Google Scholar] - Venkateswarlu, M.; Kumar, M.P. Soret and heat source effects on MHD flow of a viscous fluid in a parallel porous plat channel in presence of slip condition. U.P.B Sci. Bull. Ser. D
**2017**, 79, 171–186. [Google Scholar] - Raju, K.C.; Reddy, D.B.S. Effect of chemical reaction, thermo-diffusion, hall effects on MHD convective heat and mass transfer flow of a viscous fluid in a vertical channel bounded by stretching and stationary walls. Int. J. Sci. Innov. Math. Res.
**2019**, 7, 12–24. [Google Scholar] - Kakaç, S.; Pramuanjaroenkij, A. Review of convective heat transfer enhancement with nanofluids. Int. J. Heat Mass Transf.
**2009**, 52, 3187–3196. [Google Scholar] [CrossRef] - Masuda, H.; Ebata, A.; Teramae, K.; Hishinuma, N. Alteration of thermal conductivity and viscosity of liquid by dispersing ultra-fine particles. Netsu Bussei
**1993**, 7, 227–233. [Google Scholar] [CrossRef] - Choi, S.U.S. Enhancing Thermal Conductivity of Fluids with Nanoparticles. In Developments and Applications of Non-Newtonian Flows; Siginer, D.A., Wang, H.P., Eds.; ASME: New York, NY, USA, 1995; Volume 66, pp. 99–105. [Google Scholar]
- Choi, S.U.S.; Zhang, Z.G.; Yu, W.; Lockwood, F.E.; Grulke, E.E. Anomalously thermal conductivity enhancement in nanotube suspension. Appl. Phys. Lett.
**2001**, 79, 2252–2254. [Google Scholar] [CrossRef] - Long, N.M.A.N.; Suali, M.; Ishak, A.; Bachok, N.; Arifin, N.M. Unsteady stagnation point flow and heat transfer over stretching/shrinking sheet. Appl. Sci.
**2011**, 11, 3520–3524. [Google Scholar] - Mohyud-Din, S.T.; Zaidi, Z.A.; Khan, U.; Ahmed, N. On heat and mass transfer analysis of a nanofluid between rotating plates. Aerosp. Sci. Technol.
**2015**, 46, 514–522. [Google Scholar] [CrossRef] - Zhu, J.; Chu, P.; Sui, J. Exact analytical nanofluid flow and heat transfer involving asymmetric wall heat fluxes with nonlinear velocity slip. Math. Probl. Eng.
**2018**, 2018. [Google Scholar] [CrossRef] - Bakar, N.A.A.; Bachok, N.; Arifin, N.M. Boundary layer stagnation-point flow over a stretching/shrinking cylinder in a nanofluid: A stability analysis. Indian J. Pure Appl. Phys.
**2019**, 57, 106–117. [Google Scholar] - Mishra, S.R.; Baag, S.; Bhatti, M.M. Study of heat and mass transfer on MHD Walters B nanofluid flow induced by stretching plate surface. Alex. Eng. J.
**2018**, 57, 2435–2443. [Google Scholar] [CrossRef] - Prasad, P.D.; Kumar, R.V.M.S.S.K.; Varma, S.V.K. Heat and mass transfer analysis for the MHD flow of a nanofluid with radiation absorption. Ain Shams Eng. J.
**2018**, 9, 801–813. [Google Scholar] [CrossRef] [Green Version] - Farooq, M.; Ahmad, S.; Javed, M.; Anjum, A. Melting heat transfer in squeezed nanofluid flow through Darcy Forchheimer medium. J. Heat Transf.
**2019**, 141, 012402. [Google Scholar] [CrossRef] - He, W.; Tongraie, D.; Lotfipour, A.; Purfattah, F.; Karimipour, A.; Afrand, M. Effect of twisted tape inserts and nanofluid on flow field and heat transfer characteristics in a tube. Commun. Heat Mass Transf.
**2020**, 110, 104440. [Google Scholar] [CrossRef] - Sakiadis, B.C. Boundary layer behavior on continuous solid surfaces II. The boundary layer on a continuous flat surface. AIChE J.
**1961**, 7, 221–225. [Google Scholar] - Blasius, H. Grenzschichten in Flüssigkeiten mit kleiner Reibung. Z. Angew. Math. Phys.
**1908**, 56, 1–37. [Google Scholar] - Tsou, F.K.; Sparrow, E.M.; Goldstein, R.J. Flow and heat transfer in the boundary layer on a continuous moving surface. Int. J. Heat Mass Transf.
**1967**, 10, 219–235. [Google Scholar] [CrossRef] - Tsai, S.-Y.; Hsu, T.H. Thermal transport of a continuous moving plate in a non-Newtonian fluid. Comput. Math. Appl.
**1995**, 29, 99–108. [Google Scholar] [CrossRef] [Green Version] - Bataller, R.C. Radiation Effects in the Blasius Flow. Appl. Math. Comput.
**2008**, 198, 333–338. [Google Scholar] - Haile, E.; Shankar, B. Boundary layer flow of nanofluids over a moving surface in the presence of thermal radiation, viscous dissipation and chemical reaction. Appl. Appl. Math.
**2015**, 10, 952–969. [Google Scholar] - Maliki, H.; Safaei, M.R.; Togun, H.; Dahari, M. Heat transfer and fluid flow of pseudo-plastic nanofluid over a moving permeable plate with viscous dissipation and heat absorption/generation. J. Therm. Anal. Calorim.
**2019**, 135, 1643–1654. [Google Scholar] [CrossRef] - Hartnett, J.P. Mass transfer cooling. In Handbook of Heat Transfer Applications, 2nd ed.; McGraw-Hill: New York, NY, USA, 1985; pp. 1–111. [Google Scholar]
- Masad, J.A.; Nayfeh, A.H. Effects of suction and wall shaping on the fundamental parametric resonance in boundary layers. Phys. Fluids A Fluid Dyn.
**1992**, 4, 963–974. [Google Scholar] [CrossRef] - Rosali, H.; Ishak, A.; Pop, I. Micropolar fluid flow towards stretching/shrinking sheet in a porous medium with suction. Int. Commun. Heat Mass Transf.
**2012**, 39, 826–829. [Google Scholar] [CrossRef] - Pandey, A.K.; Kumar, M. Effects of viscous dissipation and suction/injection on MHD nanofluid flow over a wedge with porous medium and slip. Alex. Eng. J.
**2016**, 55, 3115–3123. [Google Scholar] [CrossRef] [Green Version] - Subamowo, G.M.; Akinshilo, A.; Yinusa, A.A. Two-dimensional flow analysis of nanofluid through porous channel with suction/injection at slowly expanding/contracting walls using variation of parameter method. Transp. Phenom. Nano Micro Scales
**2019**, 7, 120–129. [Google Scholar] - Lund, L.A.; Omar, Z.; Khan, U.; Khan, I.; Baleanu, D.; Nisar, K.S. Stability Analysis and Dual Solutions of Micropolar Nanofluid over the Inclined Stretching/Shrinking Surface with Convective Boundary Condition. Symmetry
**2020**, 12, 74. [Google Scholar] [CrossRef] [Green Version] - Kausar, M.S.; Hussanan, A.; Mamat, A.; Ahmad, B. Boundary Layer Flow through Darcy–Brinkman Porous Medium in the Presence of Slip Effects and Porous Dissipation. Symmetry
**2019**, 11, 659. [Google Scholar] [CrossRef] [Green Version] - Olatundun, A.T.; Makinde, O.D. Analysis of Blasius flow of hybrid nanofluids over a convectively heated surface. Defect Diffus. Forum
**2017**, 377, 29–41. [Google Scholar] [CrossRef] - Waini, I.; Ishak, A.; Pop, I. Hybrid nanofluid flow and heat transfer over a nonlinear permeable stretching/shrinking surface. Int. J. Numer. Methods Heat Fluid Flow
**2019**, 29, 3110–3127. [Google Scholar] [CrossRef] - Bachok, N.; Ishak, A.; Pop, I. Boundary layer flow over a moving surface in nanofluid with suction/injection. Acta Mech. Sin.
**2012**, 28, 34–40. [Google Scholar] [CrossRef] [Green Version] - Ahmed, M.S.; Mimi, E.A. Effect of hybrid and single nanofluids on the performance characteristics of chilled water air conditioning system. Appl. Therm. Eng.
**2019**, 163, 114398. [Google Scholar] [CrossRef] - Oztop, H.F.; Abu-Nada, E. Numerical study of natural convection in partially heated rectangular enclosure filled with nanofluids. Int. J. Heat Fluid Flow
**2008**, 29, 1326–1336. [Google Scholar] [CrossRef] - Devi, S.P.A.; Devi, S.S.U. Numerical investigation of hydromagnetic hybrid Cu-Al
_{2}O_{3}/water nanofluid flow over a permeable stretching sheet with suction. Int. J. Nonlinear Sci. Numer. Simul.**2016**, 17, 249–257. [Google Scholar] [CrossRef] - Weidman, P.D.; Kubitschek, D.G. The effect of transpiration on self-similar boundary layer flow over moving surfaces. Int J. Eng. Sci.
**2006**, 44, 730–737. [Google Scholar] [CrossRef] - Merkin, J.H. On dual solutions occurring in mixed convection in a porous medium. J. Eng. Math.
**1985**, 20, 171–179. [Google Scholar] [CrossRef] - Harris, S.D.; Ingham, D.B.; Pop, I. Mixed convection boundary layer flow near the stagnation point on a vertical surface in porous medium: Brinkman model with slip. Transp. Porous Media
**2009**, 77, 267–285. [Google Scholar] [CrossRef] - Shampine, L.F.; Gladwell, I.; Thomson, S. Solving ODEs with MATLAB; Cambridge University Press: Cambridge, UK, 2003. [Google Scholar]

**Figure 2.**Effect of ${\varphi}_{2}$ on variation of (

**a**) ${f}^{\u2033}\left(0\right)$ and (

**b**) $-{\theta}^{\prime}\left(0\right)$ with $\lambda .$

**Figure 3.**Effect of different types of fluid on variation of (

**a**) ${f}^{\u2033}\left(0\right)$ and (

**b**) $-{\theta}^{\prime}\left(0\right)$ with $\lambda .$

**Figure 9.**(

**a**) ${f}^{\prime}\left(\eta \right)$ and (

**b**) $\theta \left(\eta \right)$ for various ${\varphi}_{2}$.

**Table 1.**Base fluid and nanoparticle numerical values for thermophysical properties (Ahmed and Mimi [37]).

Thermophysical Properties | $\mathit{k}\left(\frac{\mathit{W}}{\mathit{m}\mathit{K}}\right)$ | ${\mathit{C}}_{\mathit{p}}\left(\frac{\mathit{J}}{\mathit{k}\mathit{g}}\right)\mathit{K}$ | $\mathit{\rho}\left(\frac{\mathit{k}\mathit{g}}{{\mathit{m}}^{3}}\right)$ |
---|---|---|---|

Alumina Oxide (Al_{2}O_{3}) | 36 | 773 | 3880 |

Titanium Oxide (TiO_{2}) | 8.7 | 690 | 4010 |

Water | 0.613 | 4179 | 997.1 |

Thermophysical | Hybrid Nanofluids |
---|---|

Density | ${\rho}_{hnf}=\left(1-{\varphi}_{2}\right)\left[\left(1-{\varphi}_{1}\right){\rho}_{f}+{\varphi}_{1}{\rho}_{s1}\right]+{\varphi}_{2}{\rho}_{s2}$ |

Heat capacity | ${\left(\rho {C}_{p}\right)}_{hnf}=\left(1-{\varphi}_{2}\right)\left[\left(1-{\varphi}_{1}\right){\left(\rho {C}_{p}\right)}_{f}+{\varphi}_{1}{\left(\rho {C}_{p}\right)}_{s1}\right]+{\varphi}_{2}{\left(\rho {C}_{p}\right)}_{s2}$ |

Viscosity | ${\mu}_{hnf}=\frac{{\mu}_{f}}{{(1-{\varphi}_{1})}^{2.5}{(1-{\varphi}_{2})}^{2.5}}$ |

Thermal conductivity | $\begin{array}{l}{k}_{hnf}=\left[\frac{{k}_{s2}+2{k}_{nf}-2{\varphi}_{2}\left({k}_{nf}-{k}_{s2}\right)}{{k}_{s2}+2{k}_{nf}+{\varphi}_{2}\left({k}_{nf}-{k}_{s2}\right)}\right]{k}_{nf}\\ {k}_{nf}=\left[\frac{{k}_{s1}+2{k}_{f}-2{\varphi}_{1}\left({k}_{f}-{k}_{s1}\right)}{{k}_{s1}+2{k}_{f}+{\varphi}_{1}\left({k}_{f}-{k}_{s1}\right)}\right]{k}_{f}\end{array}$ |

**Table 3.**Computed values of ${f}^{\u2033}\left(0\right)$, $-{\theta}^{\prime}\left(0\right),$ ${C}_{f}{\left(2R{e}_{x}\right)}^{1/2}$ and $N{u}_{x}{\left(R{e}_{x}/2\right)}^{-1/2}$ for $\lambda =0.2$ and $S=1.5$.

${\mathit{\varphi}}_{\mathbf{2}}$ | ${\mathit{\varphi}}_{\mathbf{1}}$ | ${\mathit{f}}^{\mathbf{\u2033}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}$ | $\mathbf{-}{\mathit{\theta}}^{\mathbf{\prime}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}$ | ${\mathit{C}}_{\mathit{f}}{\mathbf{\left(}\mathbf{2}\mathit{R}{\mathit{e}}_{\mathit{x}}\mathbf{\right)}}^{\mathbf{1}\mathbf{/}\mathbf{2}}$ | $\mathit{N}{\mathit{u}}_{\mathit{x}}{\mathbf{\left(}\mathit{R}{\mathit{e}}_{\mathit{x}}\mathbf{/}\mathbf{2}\mathbf{\right)}}^{\mathbf{-}\mathbf{1}\mathbf{/}\mathbf{2}}$ |
---|---|---|---|---|---|

0 | 0 | 1.4217 | 9.5054 | 1.4217 | 9.5054 |

0.1 | 1.4099 | 7.0938 | 1.8348 | 9.3293 | |

0.2 | 1.3008 | 5.3806 | 2.2724 | 9.1699 | |

0.1 | 0 | 1.4225 | 6.8347 | 1.8511 | 8.6534 |

0.1 | 1.3248 | 5.1875 | 2.2435 | 8.5188 | |

0.2 | 1.1727 | 4.0088 | 2.6661 | 8.3972 | |

0.2 | 0 | 1.3195 | 4.9333 | 2.3051 | 7.8144 |

0.1 | 1.1799 | 3.8055 | 2.6826 | 7.7189 | |

0.2 | 1.0149 | 2.9931 | 3.0971 | 7.6317 |

${\mathit{\varphi}}_{\mathbf{2}}$ | λ | First Solution | Second Solution |
---|---|---|---|

0 | −1.81725 | 0.0068 | −0.0068 |

−1.815 | 0.0614 | −0.0580 | |

−1.8 | 0.1772 | −0.1510 | |

0.1 | −1.74653 | 0.0076 | −0.0076 |

−1.746 | 0.0299 | −0.2303 | |

−1.7 | 0.3015 | −0.2431 | |

0.2 | −1.62439 | 0.0054 | −0.0054 |

−1.624 | 0.0250 | −0.0243 | |

−1.6 | 0.2085 | −0.1708 |

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**MDPI and ACS Style**

Aladdin, N.A.L.; Bachok, N.
Boundary Layer Flow and Heat Transfer of Al_{2}O_{3}-TiO_{2}/Water Hybrid Nanofluid over a Permeable Moving Plate. *Symmetry* **2020**, *12*, 1064.
https://doi.org/10.3390/sym12071064

**AMA Style**

Aladdin NAL, Bachok N.
Boundary Layer Flow and Heat Transfer of Al_{2}O_{3}-TiO_{2}/Water Hybrid Nanofluid over a Permeable Moving Plate. *Symmetry*. 2020; 12(7):1064.
https://doi.org/10.3390/sym12071064

**Chicago/Turabian Style**

Aladdin, Nur Adilah Liyana, and Norfifah Bachok.
2020. "Boundary Layer Flow and Heat Transfer of Al_{2}O_{3}-TiO_{2}/Water Hybrid Nanofluid over a Permeable Moving Plate" *Symmetry* 12, no. 7: 1064.
https://doi.org/10.3390/sym12071064