#
Mixed Convective Stagnation Point Flow towards a Vertical Riga Plate in Hybrid Cu-Al_{2}O_{3}/Water Nanofluid

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## Abstract

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## 1. Introduction

## 2. Mathematical Formulation

- The base fluid and nanoparticles are maintained in a thermal equilibrium state.
- The nanofluid is assumed to be stable; hence, the effect of nanoparticle aggregation and sedimentation is omitted.
- The nanoparticles are uniform with a spherical shape.
- The wall temperature is ${T}_{w}\left(x\right)={T}_{\infty}+{T}_{0}\left(x/\phantom{xL}\phantom{\rule{0.0pt}{0ex}}L\right);$${T}_{0}>0\left({T}_{w}>{T}_{\infty}\right)$ specified for a heated sheet (assisting flow) while ${T}_{0}<0\left({T}_{w}<{T}_{\infty}\right)$ for a cooled sheet (opposing flow) (see Figure 2).
- The constant ambient temperature ${T}_{\infty}$ is assumed for the case of unstratified fluid.
- The classical Hartmann term $\left(\frac{{\sigma}_{hnf}{{B}_{0}}^{2}u}{{\rho}_{hnf}}\right)$ is used to represent the magnetohydrodynamics (MHD) but the Grinberg term $\left(\frac{\pi {j}_{0}{M}_{0}}{8{\rho}_{hnf}}{e}^{-\pi y/\phantom{-\pi yp}\phantom{\rule{0.0pt}{0ex}}p}\right)$ is used for the Riga plate in the momentum equation uncoupled with the flow velocity.

Properties | Nanofluid |
---|---|

Density | ${\rho}_{nf}=\left(1-\varphi \right){\rho}_{f}+\varphi {\rho}_{s}$ |

Heat Capacity | ${\left(\rho {C}_{p}\right)}_{nf}=\left(1-\varphi \right){\left(\rho {C}_{p}\right)}_{f}+\varphi {\left(\rho {C}_{p}\right)}_{s}$ |

Dynamic Viscosity | $\frac{{\mu}_{nf}}{{\mu}_{f}}=\frac{1}{{\left(1-\varphi \right)}^{2.5}}$ |

Thermal Conductivity | $\frac{{k}_{nf}}{{k}_{f}}=\left[\frac{{k}_{s}+2{k}_{f}-2\varphi \left({k}_{f}-{k}_{s}\right)}{{k}_{s}+2{k}_{f}+\varphi \left({k}_{f}-{k}_{s}\right)}\right]$ |

Thermal Expansion | ${\left(\rho {\beta}_{T}\right)}_{nf}=\left(1-\varphi \right)\left(1-\varphi \right){\left(\rho {\beta}_{T}\right)}_{f}+\varphi {\left(\rho {\beta}_{T}\right)}_{s}$ |

Thermophysical Properties | Pure Water | Alumina | Copper |
---|---|---|---|

$\rho (\frac{kg}{{m}^{3}})$ | 997.1 | 3970 | 8933 |

${C}_{p}(\frac{J}{kgK})$ | 4179 | 765 | 385 |

$k(\frac{W}{mK})$ | 0.6130 | 40 | 400 |

${\beta}_{T}({K}^{-1})$ | $21\times {10}^{-5}$ | $0.85\times {10}^{-5}$ | $1.67\times {10}^{-5}$ |

- Consider either a moving plate or stretching/shrinking plate $\left(u\left(x,0\right)={u}_{w}\left(x\right)\right)$. In the present work, the Riga plate is static $\left(u\left(x,0\right)=0\right)$.
- Consider the permeable Riga plate $\left(v\left(x,0\right)={v}_{w}\right)$ as studied by Ahmad et al. [44]. In the present work, the Riga plate is impermeable $\left(v\left(x,0\right)=0\right)$.
- Consider hybrid SiO${}_{2}$-Al${}_{2}$O${}_{3}$/water nanofluid as studied by Rostami et al. [34] and compare the heat transfer performance of both hybrid nanofluids. Equations (6)–(8) are reduced to Rostami et al. [34] when $Z=0$ and hybrid SiO${}_{2}$-Al${}_{2}$O${}_{3}$/water nanofluids are considered.

## 3. Temporal Flow Stability

## 4. Results and Discussion

## 5. Conclusions

- Dual solutions were obtained for both assisting and opposing flow cases within a specific range of the buoyancy parameter. The separation point was located in the opposing flow region.
- The stability analysis proved that the upper branch/first solution were stable whereas the lower branch/second solution were not stable.
- Hybrid Cu-Al${}_{2}$O${}_{3}$/water nanofluid has a greater skin friction coefficient and heat transfer rate than the alumina-water nanofluid and pure water.
- The reduced skin friction coefficient and heat transfer rate was greater for the assisting flow case than for the opposing flow case.
- An upsurge of copper volumetric concentration and EMHD parameters can hold the boundary layer separation.
- An upsurge of the magnet and electrode width reduced the heat transfer rate, while the accretion of the copper volumetric concentration and EMHD parameters boosted the heat transfer rate for both the assisting and opposing buoyancy flows.
- The heat transfer rate approximately increased up to an average of 10.216% when the copper volumetric concentration increased from 0.005 $(0.5\%)$ to 0.03 $(3\%)$.
- Both the velocity and temperature profiles increased with the enhancement of the copper volumetric concentration ${\varphi}_{2}$.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Abbreviations

L | characteristic length of the Riga plate |

${T}_{0}$ | reference temperature |

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

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

${M}_{0}$ | magnetization of the magnets |

Pr | Prandtl number |

$R{e}_{x}$ | local Reynolds number |

T | fluid temperature |

g | gravitational acceleration |

${j}_{0}$ | applied current density in the electrodes |

k | thermal conductivity |

p | magnets and electrodes width |

${u}_{e}$ | free stream velocity |

t | time ($\mathrm{s})$ |

$u,v$ | velocities along the x-, y- directions, respectively |

$\alpha $ | thermal diffusivity of the fluid |

${\beta}_{T}$ | thermal expansion |

$\theta $ | dimensionless temperature |

$\lambda $ | mixed convection parameter |

$\mu $ | dynamic viscosity |

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

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

$\rho {C}_{p}$ | heat capacity of fluid |

$\gamma $ | unknown eigenvalue |

${\gamma}_{1}$ | smallest eigenvalue |

$\tau $ | dimensionless time variable |

${\varphi}_{1},{\varphi}_{2}$ | dimensionless nanoparticles volume fraction/concentration for alumina and copper, respectively |

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**Figure 3.**${{\mathrm{Re}}_{x}}^{1/2}{C}_{f}$ towards $\lambda $ for hybrid nanofluid $\left({\varphi}_{1}=0.1,{\varphi}_{2}=0.005\right)$, alumina-water nanofluid $\left({\varphi}_{1}=0.1,{\varphi}_{2}=0\right)$, and pure water $\left({\varphi}_{1}={\varphi}_{2}=0\right)$ when $Z=d=0.5$.

**Figure 4.**${{\mathrm{Re}}_{x}}^{-1/2}N{u}_{x}$ towards $\lambda $ for hybrid nanofluid $\left({\varphi}_{1}=0.1,{\varphi}_{2}=0.005\right)$, alumina-water nanofluid $\left({\varphi}_{1}=0.1,{\varphi}_{2}=0\right)$, and pure water $\left({\varphi}_{1}={\varphi}_{2}=0\right)$ when $Z=d=0.5$.

**Figure 5.**${{\mathrm{Re}}_{x}}^{1/2}{C}_{f}$ towards $\lambda $ when ${\varphi}_{2}=0.005$ and $d=0.5$ with various Z.

**Figure 6.**${{\mathrm{Re}}_{x}}^{-1/2}N{u}_{x}$ towards $\lambda $ when ${\varphi}_{2}=0.005$ and $d=0.5$ with various Z.

Properties | Hybrid Nanofluid |
---|---|

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

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

Thermal Conductivity | $\begin{array}{c}{\displaystyle \frac{{k}_{hnf}}{{k}_{bf}}=\left[\frac{{k}_{s2}+2{k}_{bf}-2{\varphi}_{2}\left({k}_{bf}-{k}_{s2}\right)}{{k}_{s2}+2{k}_{bf}+{\varphi}_{2}\left({k}_{bf}-{k}_{s2}\right)}\right]}\hfill \\ \mathrm{where}\hfill \\ {\displaystyle \frac{{k}_{bf}}{{k}_{f}}=\left[\frac{{k}_{s1}+2{k}_{f}-2{\varphi}_{1}\left({k}_{f}-{k}_{s1}\right)}{{k}_{s1}+2{k}_{f}+{\varphi}_{1}\left({k}_{f}-{k}_{s1}\right)}\right]}\hfill \end{array}$ |

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

**Table 4.**Numerical values of ${f}^{\u2033}\left(0\right)$ for $\lambda ={\varphi}_{1}={\varphi}_{2}=0$, $Z=d=0.5$, $Pr=5$, and ${\eta}_{\infty}=20$.

Present (bvp4c Solution) | Ahmad et al. [45] (bvp4c Solution) | Ahmad et al. [45] (Shooting Method) | |
---|---|---|---|

${f}^{\u2033}\left(0\right)$ | 1.539473230 | 1.5394732 | 1.5394682 |

(0.77 s) | (1.4 s) | (1.4 s) |

**Table 5.**${f}^{\u2033}\left(0\right)$ when ${\varphi}_{1}={\varphi}_{2}=Z=d=0$ and $\lambda =1$ for various values of Pr.

Pr | Present | Rostami et al. [34] | Ishak et al. [33] | |||
---|---|---|---|---|---|---|

First Sol. | Second Sol. | First Sol. | Second Sol. | First Sol. | Second Sol. | |

$0.7$ | 1.7063 | 1.2387 | 1.7063 | 1.2344 | 1.7063 | 1.2387 |

1 | 1.6754 | 1.1332 | 1.6754 | 1.1296 | 1.6754 | 1.1332 |

7 | 1.5179 | 0.5824 | 1.5179 | 0.5815 | 1.5179 | 0.5824 |

10 | 1.4928 | 0.4958 | 1.4928 | 0.4956 | 1.4928 | 0.4958 |

20 | 1.4485 | 0.3436 | 1.4485 | 0.3436 | 1.4485 | 0.3436 |

**Table 6.**$-{\theta}^{\prime}\left(0\right)$ when ${\varphi}_{1}={\varphi}_{2}=Z=d=0$ and $\lambda =1$ for various values of Pr.

Pr | Present | Rostami et al. [34] | Ishak et al. [33] | |||
---|---|---|---|---|---|---|

First Sol. | Second Sol. | First Sol. | Second Sol. | First Sol. | Second Sol. | |

$0.7$ | 0.7641 | 1.0226 | 0.7641 | 1.0235 | 0.7641 | 1.0226 |

1 | 0.8708 | 1.1691 | 0.8708 | 1.1706 | 0.8708 | 1.1691 |

7 | 1.7224 | 2.2192 | 1.7224 | 2.2203 | 1.7224 | 2.2192 |

10 | 1.9446 | 2.4940 | 1.9446 | 2.4943 | 1.9446 | 2.4940 |

20 | 2.4576 | 3.1647 | 2.4576 | 3.1647 | 2.4576 | 3.1646 |

$\mathit{\varphi}$ | Present | Bachok et al. [76] | Yacob et al. [77] | |||
---|---|---|---|---|---|---|

Cu-Water | Al${}_{2}$O${}_{3}$-Water | Cu-Water | Al${}_{2}$O${}_{3}$-Water | Cu-Water | Al${}_{2}$O${}_{3}$-Water | |

$0.05$ | 1.5538 | 1.4088 | - | - | - | - |

$0.10$ | 1.8843 | 1.6020 | 1.8843 | 1.6019 | 1.8843 | 1.6019 |

$0.15$ | 2.2369 | 1.8168 | - | - | - | - |

$0.20$ | 2.6227 | 2.0583 | 2.6226 | 2.0584 | 2.6226 | 2.0584 |

$\mathit{\varphi}$ | Present | Bachok et al. [76] | Yacob et al. [77] | |||
---|---|---|---|---|---|---|

Cu-Water | Al${}_{2}$O${}_{3}$-Water | Cu-Water | Al${}_{2}$O${}_{3}$-Water | Cu-Water | Al${}_{2}$O${}_{3}$-Water | |

$0.05$ | 1.7758 | 1.7169 | - | - | - | - |

$0.10$ | 1.9692 | 1.8603 | - | - | - | - |

$0.15$ | 2.1593 | 2.0045 | - | - | - | - |

$0.20$ | 2.3494 | 2.1502 | - | - | - | - |

**Table 9.**Critical values ${\lambda}_{c}$ for selected values of ${\varphi}_{1}$, ${\varphi}_{2}$, and Z when $Pr=6.2$ and $d=0.5$.

${\mathit{\varphi}}_{1}$ | ${\mathit{\varphi}}_{2}$ | Z | ${\mathit{\lambda}}_{\mathit{c}}$ |
---|---|---|---|

0 | 0 | 0.5 | −4.61663 |

0.1 | 0 | 0.5 | −5.73469 |

0.1 | 0.005 | 0.5 | −5.83576 |

0.1 | 0.005 | 0 | −4.80459 |

0.1 | 0.005 | −0.5 | −3.81356 |

**Table 10.**The values of $R{e}_{x}^{-1/2}N{u}_{x}$ for the opposing and assisting flow cases when ${\varphi}_{1}=0.1$ and $Z=d=0.5$ ([] indicates the second solution).

$\mathit{\lambda}$ | ${\mathit{\varphi}}_{2}$ | ||
---|---|---|---|

0.005 | 0.01 | 0.03 | |

−1 | 1.91254 | 1.93350 | 2.01692 |

[−2.17399] | [−2.23545] | [−2.48587] | |

−0.5 | 1.94865 | 1.96932 | 2.05167 |

[−4.34279] | [−4.47482] | [−5.02319] | |

0.5 | 2.01462 | 2.03486 | 2.11565 |

[5.78122] | [5.95238] | [6.65825] | |

1 | 2.04501 | 2.06510 | 2.14530 |

[3.54998] | [3.64189] | [4.02065] |

© 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**

Khashi’ie, N.S.; Md Arifin, N.; Pop, I. Mixed Convective Stagnation Point Flow towards a Vertical Riga Plate in Hybrid Cu-Al_{2}O_{3}/Water Nanofluid. *Mathematics* **2020**, *8*, 912.
https://doi.org/10.3390/math8060912

**AMA Style**

Khashi’ie NS, Md Arifin N, Pop I. Mixed Convective Stagnation Point Flow towards a Vertical Riga Plate in Hybrid Cu-Al_{2}O_{3}/Water Nanofluid. *Mathematics*. 2020; 8(6):912.
https://doi.org/10.3390/math8060912

**Chicago/Turabian Style**

Khashi’ie, Najiyah Safwa, Norihan Md Arifin, and Ioan Pop. 2020. "Mixed Convective Stagnation Point Flow towards a Vertical Riga Plate in Hybrid Cu-Al_{2}O_{3}/Water Nanofluid" *Mathematics* 8, no. 6: 912.
https://doi.org/10.3390/math8060912