# Dual Solutions and Stability Analysis of a Hybrid Nanofluid over a Stretching/Shrinking Sheet Executing MHD Flow

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## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

## 3. Stability Analysis

## 4. Result and Discussion

**Table 1.**Thermophysical properties of the hybrid nanofluid [46].

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

Dynamic viscosity | ${\mu}_{hnf}=\frac{{\mu}_{f}}{{\left(1-{\varphi}_{A{l}_{2}{O}_{3}}\right)}^{2.5}{\left(1-{\varphi}_{Cu}\right)}^{2.5}}$ |

Density | ${\rho}_{hnf}=\left(1-{\varphi}_{Cu}\right)\left[\left(1-{\varphi}_{A{l}_{2}{O}_{3}}\right){\rho}_{f}+{\varphi}_{A{l}_{2}{O}_{3}}{\rho}_{A{l}_{2}{O}_{3}}\right]+{\varphi}_{Cu}{\rho}_{Cu}$ |

Thermal conductivity | ${k}_{hnf}=\frac{{k}_{Cu}+2{k}_{nf}-2{\varphi}_{Cu}\left({k}_{nf}-{k}_{Cu}\right)}{{k}_{Cu}+2{k}_{nf}+{\varphi}_{Cu}\left({k}_{nf}-{k}_{Cu}\right)}\times \left({k}_{nf}\right)$ where ${k}_{nf}=\frac{{k}_{A{l}_{2}{O}_{3}}+2{k}_{f}-2{\varphi}_{A{l}_{2}{O}_{3}}\left({k}_{f}-{k}_{A{l}_{2}{O}_{3}}\right)}{{k}_{A{l}_{2}{O}_{3}}+2{k}_{f}+{\varphi}_{A{l}_{2}{O}_{3}}\left({k}_{f}-{k}_{A{l}_{2}{O}_{3}}\right)}\times \left({k}_{f}\right)$ |

Heat capacity | ${\left(\rho {c}_{p}\right)}_{hnf}=\left(1-{\varphi}_{Cu}\right)\left[\left(1-{\varphi}_{A{l}_{2}{O}_{3}}\right){\left(\rho {c}_{p}\right)}_{f}+{\varphi}_{A{l}_{2}{O}_{3}}{\left(\rho {c}_{p}\right)}_{A{l}_{2}{O}_{3}}\right]+{\varphi}_{Cu}{\left(\rho {c}_{p}\right)}_{Cu}$ |

Fluids | $\mathit{\rho}$ | ${\mathit{c}}_{\mathit{p}}(\mathbf{J}/\mathbf{kg}\text{}\mathbf{K})$ | k (W/m K) |
---|---|---|---|

Alumina ($A{l}_{2}{O}_{3}$) | 3970 | 765 | 40 |

Copper (Cu) | 8933 | 385 | 400 |

Water (${H}_{2}O$) | 997.1 | 4179 | 0.613 |

**Table 3.**Values of ${f}^{\u2033}\left(0\right)$ and ${\theta}^{\prime}\left(0\right)$ for the Cu–water nanofluid (${\varphi}_{Cu}=0.2$) with various values of $A$ when $M=Rd=0,S=2.1$, $Pr=6.2,$ and $\lambda =-1$.

$\mathit{A}$ | ${\mathit{\varphi}}_{\mathit{A}{\mathit{l}}_{2}{\mathit{O}}_{3}}$ | ${\mathit{f}}^{\u2033}\left(0\right)$ | $-{\mathit{\theta}}^{\prime}\left(0\right)$ | ||||||
---|---|---|---|---|---|---|---|---|---|

Waini | [31] | Present | Results | Waini | [31] | Present | Results | ||

1st Soln | 2nd Soln | 1st Soln | 2nd Soln | 1st Soln | 2nd Soln | 1st Soln | 2nd Soln | ||

−1 | 0 | 2.194247 | −1.491281 | 2.19424658 | −1.49128119 | 7.073680 | 6.884548 | 7.0736798 | 6.88454793 |

−1 | 0.1 | -- | -- | 1.60888878 | −0.69818494 | -- | -- | 5.2238150 | 5.0737595 |

−3 | 0 | 1.521197 | −4.144746 | 1.52119229 | −4.14474572 | 7.497151 | 7.296176 | 7.4971508 | 7.2961762 |

−3 | 0.1 | -- | -- | 0.83697819 | −2.53135605 | -- | -- | 5.6316794 | 5.4925121 |

−5 | 0 | 0.844435 | −6.431507 | 0.84443506 | −6.43150738 | 7.858446 | 7.657801 | 7.8584457 | 7.6578012 |

−5 | 0.01 | -- | -- | 0.77597819 | −6.20435575 | -- | -- | 7.6375695 | 7.4426329 |

−9 | 0 | −0.517287 | −10.58983 | −0.51728551 | −10.5898303 | 8.473316 | 8.277676 | 8.4733163 | 8.27767616 |

−9 | 0.01 | -- | -- | −0.60021181 | −10.2417356 | -- | -- | 8.24709639 | 8.05819003 |

## 5. Conclusions

## Abbreviation

Nomenclature | |||

${T}_{0}$ | a constant | ${\varphi}_{A{l}_{2}{O}_{3}}$ | nanoparticle volume fraction of the iron oxide. |

${T}_{\infty}$ | ambient temperature | M | magnetic parameter |

$\prime $ | differentiation with respect to $\eta $ | ${K}_{1}$ | Porous parameter |

$Rd$ | Thermal radiation | c | constant |

${\rho}_{hnf}$ | effective density of hybrid nanofluid | $Pr$ | Prandtl number |

${\rho}_{nf}$ | effective density of nanofluid | ${C}_{f}$ | skin friction coefficient |

${\mu}_{hnf}$ | effective dynamic viscosity of hybrid nanofluid | ${\gamma}_{1}$ | smallest eigen value |

${\mu}_{nf}$ | effective dynamic viscosity of nanofluid | $\tau $ | Stability transformed variable |

${\sigma}^{*}$ | electrical conductivity | ${v}_{w}$ | suction/injection velocity |

$f$ | fluid fraction | T | Temperature |

M | Hartmann/magnetic number | ${k}_{hnf}$ | thermal conductivity of the hybrid nanofluid |

${\left(\rho {c}_{p}\right)}_{hnf}$ | heat capacitance of the hybrid nanofluid | ${k}_{nf}$ | thermal conductivity of the nanofluid |

${\left(\rho {c}_{p}\right)}_{nf}$ | heat capacitance of the nanofluid | t | time |

hnf | Hybrid nanofluid | $\eta $ | transformed variable |

${N}_{u}$ | local Nusselt number | $A$ | Unsteadiness parameter |

$Re$ | local Reynolds number | ${T}_{w}$ | variable temperature at the sheet |

B | Magnetic field | u, v | velocity components |

$nf$ | nanofluid fraction | $\lambda $ | shrinking/stretching parameter |

${k}^{*}$ | mean absorption coefficient | ${\sigma}_{1}$ | Stefan–Boltzmann constant |

${\varphi}_{Cu}$ | nanoparticle volume fraction of the copper | $S$ | $S<0$ for suction parameter and $S>0$ for blowing parameter |

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Choi, S.U.; Eastman, J.A. Enhancing Thermal Conductivity of Fluids with Nanoparticles; No. ANL/MSD/CP-84938; CONF-951135-29; Argonne National Lab: Lemont, IL, USA, 1995. [Google Scholar]
- Tiwari, R.K.; Das, M.K. Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids. Int. J. Heat Mass Transf.
**2007**, 50, 2002–2018. [Google Scholar] [CrossRef] - Benzema, M.; Benkahla, Y.K.; Labsi, N.; Ouyahia, S.E.; El Ganaoui, M. Second law analysis of MHD mixed convection heat transfer in a vented irregular cavity filled with Ag-MgO/water hybrid nanofluid. J. Therm. Anal. Calorim.
**2019**, 137, 1113–1132. [Google Scholar] [CrossRef] - Dero, S.; Rohni, A.M.; Saaban, A. The dual solutions and stability analysis of nanofluid flow using tiwari-das modelover a permeable exponentially shrinking surface with partial slip conditions. J. Eng. Appl. Sci.
**2019**, 14, 4569–4582. [Google Scholar] [CrossRef] [Green Version] - Lund, L.A.; Omar, Z.; Khan, I.; Dero, S. Multiple solutions of Cu-C
_{6}H_{9}NaO_{7}and Ag-C_{6}H_{9}NaO_{7}nanofluids flow over nonlinear shrinking surface. J. Cent. South Univ.**2019**, 26, 1283–1293. [Google Scholar] [CrossRef] - 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] - Dogonchi, A.S.; Chamkha, A.J.; Hashemi-Tilehnoee, M.; Seyyedi, S.M.; Ganji, D.D. Effects of homogeneous-heterogeneous reactions and thermal radiation on magneto-hydrodynamic Cu-water nanofluid flow over an expanding flat plate with non-uniform heat source. J. Cent. South Univ.
**2019**, 26, 1161–1171. [Google Scholar] [CrossRef] - Dogonchi, A.S.; Ismael, M.A.; Chamkha, A.J.; Ganji, D.D. Numerical analysis of natural convection of Cu-water nanofluid filling triangular cavity with semicircular bottom wall. J. Therm. Anal. Calorim.
**2019**, 135, 3485–3497. [Google Scholar] [CrossRef] - Amini, Y.; Akhavan, S.; Izadpanah, E. Vortex-induced vibration of a cylinder in pulsating nanofluid flow. J. Therm. Anal. Calorim.
**2019**. [Google Scholar] [CrossRef] - Zaib, A.; Khan, M.; Shafie, S. Boundary-layer flow of a cu-water nanofluid over a permeable shrinking cylinder with homogenous-hetrogenous reactions: Dual solutions. Therm. Sci.
**2019**, 23, 295–306. [Google Scholar] [CrossRef] [Green Version] - Raza, J.; Rohni, A.M.; Omar, Z.; Awais, M. Heat and mass transfer analysis of MHD nanofluid flow in a rotating channel with slip effects. J. Mol. Liq.
**2016**, 219, 703–708. [Google Scholar] [CrossRef] - Rasool, G.; Zhang, T.; Shafiq, A. Marangoni effect in second grade forced convective flow of water based nanofluid. J. Adv. Nanotechnol.
**2019**, 1, 50. [Google Scholar] [CrossRef] - Rasool, G.; Shafiq, A.; Khalique, C.M.; Zhang, T. Magneto-hydrodynamic Darcy-Forchheimer nanofluid flow over nonlinear stretching sheet. Phys. Scr.
**2019**, 94, 10. [Google Scholar] [CrossRef] - Dinarvand, S. Nodal/saddle stagnation-point boundary layer flow of CuO-Ag/water hybrid nanofluid: A novel hybridity model. Microsyst. Technol.
**2019**, 25, 2609–2623. [Google Scholar] [CrossRef] - Roşca, N.C.; Roşca, A.V.; Pop, I. Unsteady separated stagnation-point flow and heat transfer past a stretching/shrinking sheet in a copper-water nanofluid. Int. J. Numer. Methods Heat Fluid Flow
**2019**. [Google Scholar] [CrossRef] - Ahmed, N.; Saba, F.; Khan, U.; Mohyud-Din, S.T.; Sherif, E.S.M.; Khan, I. Nonlinear thermal radiation and chemical reaction effects on a Cu-CuO/NaAlg hybrid nanofluid flow past a stretching curved surface. Processes
**2019**, 7, 962. [Google Scholar] [CrossRef] [Green Version] - Devi, S.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] - Suresh, S.; Venkitaraj, K.P.; Selvakumar, P.; Chandrasekar, M. Synthesis of Al
_{2}O_{3}-Cu/water hybrid nanofluids using two step method and its thermo physical properties. Colloids Surf. A Physicochem. Eng. Asp.**2011**, 388, 41–48. [Google Scholar] [CrossRef] - Toghraie, D.; Chaharsoghi, V.A.; Afrand, M. Measurement of thermal conductivity of ZnO-TiO
_{2}/EG hybrid nanofluid. J. Therm. Anal. Calorim.**2016**, 125, 527–535. [Google Scholar] [CrossRef] - Moghadassi, A.; Ghomi, E.; Parvizian, F. A numerical study of water based Al
_{2}O_{3}and Al_{2}O_{3}-Cu hybrid nanofluid effect on forced convective heat transfer. Int. J. Therm. Sci.**2015**, 92, 50–57. [Google Scholar] [CrossRef] - Hayat, T.; Nadeem, S.; Khan, A.U. Rotating flow of Ag-CuO/H
_{2}O hybrid nanofluid with radiation and partial slip boundary effects. Eur. Phys. J. E**2018**, 41, 75. [Google Scholar] [CrossRef] - Saba, F.; Ahmed, N.; Khan, U.; Waheed, A.; Rafiq, M.; Mohyud-Din, S. Thermophysical analysis of water based (Cu-Al
_{2}O_{3}) hybrid nanofluid in an asymmetric channel with dilating/squeezing walls considering different shapes of nanoparticles. Appl. Sci.**2018**, 8, 1549. [Google Scholar] [CrossRef] [Green Version] - Acharya, N.; Bag, R.; Kundu, P.K. Influence of hall current on radiative nanofluid flow over a spinning disk: A hybrid approach. Phys. E Low-Dimens. Syst. Nanostruct.
**2019**, 111, 103–112. [Google Scholar] [CrossRef] - Afridi, M.I.; Alkanhal, T.A.; Qasim, M.; Tlili, I. Entropy Generation in Cu-Al
_{2}O_{3}-H_{2}O Hybrid Nanofluid Flow over a Curved Surface with Thermal Dissipation. Entropy**2019**, 21, 941. [Google Scholar] [CrossRef] [Green Version] - Shafiq, A.; Khan, I.; Rasool, G.; Sherif, E.S.M.; Sheikh, A.H. Influence of single-and multi-wall carbon nanotubes on magnetohydrodynamic stagnation point nanofluid flow over variable thicker surface with concave and convex effects. Mathematics
**2020**, 8, 104. [Google Scholar] [CrossRef] [Green Version] - Shafiq, A.; Zari, I.; Rasool, G.; Tlili, I.; Khan, T.S. On the MHD casson axisymmetric marangoni forced convective flow of nanofluids. Mathematics
**2019**, 7, 1087. [Google Scholar] [CrossRef] [Green Version] - Manh, T.D.; Nam, N.D.; Abdulrahman, G.K.; Moradi, R.; Babazadeh, H. Impact of MHD on hybrid nanomaterial free convective flow within a permeable region. J. Therm. Anal. Calorim.
**2019**. [Google Scholar] [CrossRef] - Khashi’ie, N.S.; Arifin, N.M.; Nazar, R.; Hafidzuddin, E.H.; Wahi, N.; Pop, I. Magnetohydrodynamics (MHD) axisymmetric flow and heat transfer of a hybrid nanofluid past a radially permeable stretching/shrinking sheet with joule heating. Chin. J. Phys.
**2019**. [Google Scholar] [CrossRef] - Lund, L.A.; Omar, Z.; Khan, I.; Seikh, A.H.; Sherif, E.S.M.; Nisar, K.S. Stability analysis and multiple solution of Cu-Al
_{2}O_{3}/H_{2}O nanofluid contains hybrid nanomaterials over a shrinking surface in the presence of viscous dissipation. J. Mater. Res. Technol.**2019**, 9, 421–432. [Google Scholar] [CrossRef] - Waini, I.; Ishak, A.; Pop, I. Hybrid nanofluid flow and heat transfer past a vertical thin needle with prescribed surface heat flux. Int. J. Numer. Methods Heat Fluid Flow
**2019**. [Google Scholar] [CrossRef] - Waini, I.; Ishak, A.; Pop, I. Unsteady flow and heat transfer past a stretching/shrinking sheet in a hybrid nanofluid. Int. J. Heat Mass Transf.
**2019**, 136, 288–297. [Google Scholar] [CrossRef] - Merkin, J.H. On dual solutions occurring in mixed convection in a porous medium. J. Eng. Math.
**1986**, 20, 171–179. [Google Scholar] [CrossRef] - Dero, S.; Rohni, A.M.; Saaban, A.; Khan, I. Dual solutions and stability analysis of micropolar nanofluid flow with slip effect on stretching/shrinking surfaces. Energies
**2019**, 12, 4529. [Google Scholar] [CrossRef] [Green Version] - Dero, S.; Uddin, M.J.; Rohni, A.M. Stefan blowing and slip effects on unsteady nanofluid transport past a shrinking sheet: Multiple solutions. Heat Transf. Asian Res.
**2019**, 6, 2047–2066. [Google Scholar] [CrossRef] - Lund, L.A.; Omar, Z.; Khan, I. Mathematical analysis of magnetohydrodynamic (MHD) flow of micropolar nanofluid under buoyancy effects past a vertical shrinking surface: Dual solutions. Heliyon
**2019**, 5, e02432. [Google Scholar] [CrossRef] [Green Version] - Lund, L.A.; Omar, Z.; Dero, S.; Khan, I. Linear stability analysis of MHD flow of micropolar fluid with thermal radiation and convective boundary condition: Exact solution. Heat Transf. Asian Res.
**2019**. [Google Scholar] [CrossRef] - Lund, L.A.; Omar, Z.; Khan, I.; Kadry, S.; Rho, S.; Mari, I.A.; Nisar, K.S. Effect of viscous dissipation in heat transfer of MHD flow of micropolar fluid partial slip conditions: Dual solutions and stability analysis. Energies
**2019**, 12, 4617. [Google Scholar] [CrossRef] [Green Version] - Lund, L.A.; Omar, Z.; Khan, I.; Raza, J.; Sherif, E.S.M.; Seikh, A.H. Magnetohydrodynamic (MHD) flow of micropolar fluid with effects of viscous dissipation and joule heating over an exponential shrinking sheet: Triple solutions and stability analysis. Symmetry
**2020**, 12, 142. [Google Scholar] [CrossRef] [Green Version] - Khashi’ie, N.S.; Arifin, N.M.; Rashidi, M.M.; Hafidzuddin, E.H.; Wahi, N. Magnetohydrodynamics (MHD) stagnation point flow past a shrinking/stretching surface with double stratification effect in a porous medium. J. Therm. Anal. Calorim.
**2019**. [Google Scholar] [CrossRef] - Waini, I.; Ishak, A.; Pop, I. Transpiration effects on hybrid nanofluid flow and heat transfer over a stretching/shrinking sheet with uniform shear flow. Alex. Eng. J.
**2019**. [Google Scholar] [CrossRef] - Asma, M.; Othman, W.A.M.; Muhammad, T. Numerical study for darcy-forchheimer flow of nanofluid due to a rotating disk with binary chemical reaction and arrhenius activation energy. Mathematics
**2019**, 7, 921. [Google Scholar] [CrossRef] [Green Version] - Alarifi, I.M.; Abokhalil, A.G.; Osman, M.; Lund, L.A.; Ayed, M.B.; Belmabrouk, H.; Tlili, I. MHD flow and heat transfer over vertical stretching sheet with heat sink or source effect. Symmetry
**2019**, 11, 297. [Google Scholar] [CrossRef] [Green Version] - Basir, M.F.M.; Kumar, R.; Ismail, A.I.M.; Sarojamma, G.; Narayana, P.S.; Raza, J.; Mahmood, A. Exploration of thermal-diffusion and diffusion-thermal effects on the motion of temperature-dependent viscous fluid conveying microorganism. Arab. J. Sci. Eng.
**2019**, 44, 8023–8033. [Google Scholar] [CrossRef] - Wang, Q. Optimal strokes of low Reynolds number linked-sphere swimmers. Appl. Sci.
**2019**, 9, 4023. [Google Scholar] [CrossRef] [Green Version] - Iqbal, Z.; Akbar, N.S.; Azhar, E.; Maraj, E.N. Performance of hybrid nanofluid (Cu-CuO/water) on MHD rotating transport in oscillating vertical channel inspired by Hall current and thermal radiation. Alex. Eng. J.
**2018**, 57, 1943–1954. [Google Scholar] [CrossRef] - Khashi’ie, N.S.; Arifin, N.M.; Hafidzuddin, E.H.; Wahi, N. Thermally stratified flow of Cu-Al2O3/water hybrid nanofluid past a permeable stretching/shrinking circular cylinder. J. Adv. Res. Fluid Mech. Therm. Sci.
**2019**, 63, 154–163. [Google Scholar] - Rahman, A.N.H.; Bachok, N.; Rosali, H. Numerical solutions of MHD stagnation-point flow over an exponentially stretching/shrinking sheet in a nanofluid. In Journal of Physics: Conference Series; IOP Publishing: Bristol, UK, 2019; Volume 1366. [Google Scholar]
- Zaib, A.; Khan, U.; Shah, Z.; Kumam, P.; Thounthong, P. Optimization of entropy generation in flow of micropolar mixed convective magnetite (Fe
_{3}O_{4}) ferroparticle over a vertical plate. Alex. Eng. J.**2019**, 58, 1461–1470. [Google Scholar] [CrossRef]

**Figure 2.**Comparison with the 6th figure of Waini et al. [31].

**Figure 3.**Variation of ${f}^{\u2033}\left(0\right)$ with $A$ for various values of ${\varphi}_{Cu}$.

**Figure 4.**Variation of $-{\theta}^{\prime}\left(0\right)$ with $A$ for various values of ${\varphi}_{Cu}$.

**Figure 5.**Variation of ${f}^{\u2033}\left(0\right)$ with $M$ for various values of ${\varphi}_{Cu}$.

**Figure 6.**Variation of $-{\theta}^{\prime}\left(0\right)$ with $M$ for various values of ${\varphi}_{Cu}$.

**Figure 9.**Variation of ${f}^{\u2033}\left(0\right)$ with $S$ for various values of ${\varphi}_{Cu}$.

**Figure 10.**Variation of $-{\theta}^{\prime}\left(0\right)$ with $S$ for various values of ${\varphi}_{Cu}$.

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Lund, L.A.; Omar, Z.; Khan, I.; Sherif, E.-S.M.
Dual Solutions and Stability Analysis of a Hybrid Nanofluid over a Stretching/Shrinking Sheet Executing MHD Flow. *Symmetry* **2020**, *12*, 276.
https://doi.org/10.3390/sym12020276

**AMA Style**

Lund LA, Omar Z, Khan I, Sherif E-SM.
Dual Solutions and Stability Analysis of a Hybrid Nanofluid over a Stretching/Shrinking Sheet Executing MHD Flow. *Symmetry*. 2020; 12(2):276.
https://doi.org/10.3390/sym12020276

**Chicago/Turabian Style**

Lund, Liaquat Ali, Zurni Omar, Ilyas Khan, and El-Sayed M. Sherif.
2020. "Dual Solutions and Stability Analysis of a Hybrid Nanofluid over a Stretching/Shrinking Sheet Executing MHD Flow" *Symmetry* 12, no. 2: 276.
https://doi.org/10.3390/sym12020276