# Stability Analysis of Buoyancy Magneto Flow of Hybrid Nanofluid through a Stretchable/Shrinkable Vertical Sheet Induced by a Micropolar Fluid Subject to Nonlinear Heat Sink/Source

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## Abstract

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## 1. Introduction

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_{3}–Cu as a novel heat transfer fluid; (II) exploring a two-dimensional free time-dependent flow close to the stagnation point on a stretchable/shrinkable sheet using a single-phase model; (III) examining the magnetic radiation and irregular heat source/sink effects together; (IV) investigating the behavior of mixed convective or buoyancy forces; and (V) executing stability tests/assessments to check the stable solution in connecting to the dual solutions. To address these aims, the novelty of this research is to explore the impression of an irregular heat source/sink on the stagnation point buoyancy flow induced by micropolar hybrid nanofluids via a stretched/ shrinkable sheet along with magnetic and radiation effects. We believe that this problem has not yet been discussed. The flow problem was solved by using the bvp4c method, which is dependent on the finite difference approach. The impacts of the new parameters are explored numerically in the form of various tables, as well as being graphically represented. This research of buoyancy radiative flows induced by hybrid nanofluids with an irregular heat source/sink effect through a shrinking/stretching surface is particularly important in food processes, polymer processing, biomedicine, aerodynamic heating, etc.

## 2. Description and Background of the Model

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_{3}). Additionally, it is conceivable that the configuration of the fluid flow was also affected by the combined effects of thermal radiation and an irregular heat sink/source term. Another presumption is that the wall variable temperature and constant far-field temperature are signified by ${T}_{w}\left({x}_{s}\right)$ and ${T}_{\infty}$, respectively. The magnetic field measured to the sheet is assumed to have a constant strength ${B}_{0}$. Additionally, it is anticipated that the nanoparticles and water-based fluid would not slip and would be in thermal equilibrium. The single-phase method used in this hybrid nanofluid model assumes that the nanoparticles are homogenous in size and shape, and they do not interact with the surrounding fluid (see Sheremet et al. [36], Pang et al. [37], and Ebrahimi et al. [38]). This statement supports the adoption of the single-phase model in this study since it is practically applicable when the base fluid can be effectively disseminated and is thought to behave as a single fluid.

#### 2.1. The Engineering Quantities of Interest

#### 2.1.1. The Shear Stress Coefficient (SSC)

#### 2.1.2. The Couple-Stress Coefficient (CSC)

#### 2.1.3. The Heat Transfer Rate (HTR)

## 3. Stability Analysis

## 4. Multiple Solution Methodology and Authentication of the Code

## 5. Analysis of Results

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_{3}) hybrid nanofluid and heat transfer flow examination comprised several distinct distinguished parameters which are namely and symbolically denoted as ${M}_{c}$ for the magnetic factor, ${K}_{c}$ for the material factor, ${E}_{c}$ for the micro-inertia factor, ${N}_{r}$ for the radiation factor, ${f}_{w}$ for the mass suction/injection factor, ${\epsilon}_{c}$ for the stretched/shrinked factor, ${\gamma}_{c}$ for the mixed convection or buoyancy factor, and ${\phi}_{1}$ and ${\phi}_{2}$ for the solid nanoparticles volume fraction. The physical impact of these numerous notable influential parameters on the shear stress coefficient, couple-stress coefficient, heat transfer, and temperature profile was presented tabularly in Table 4, Table 5 and Table 6, but graphically illustrated in Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16, respectively. However, the eigenvalue ${\lambda}_{a}$ for the several/sundry values of ${\epsilon}_{c}$ is established in Figure 17 while the streamlines are presented in Figure 18 and Figure 19. For the computation purpose, we have executed the following ranges for the parameters such as $-5.0<{\epsilon}_{c}<5.0$, $-15.0<{\gamma}_{c}<15.0$, $-3.0<{f}_{w}<3.0$, $0.0<{M}_{c}<1.0$, $0.0<{E}_{c}<1.0$, ${m}_{s}=0.5$, $0.0<{N}_{r}<5.0$, $-1.0<{A}_{s}<1.0$, $0.0<{K}_{c}<1.0$, and $-1.0<{B}_{s}<1.0$. Furthermore, the first branch solution (FBS) and second branch solution (SBS) are highlighted by the complete filled solid and dashed black lines, respectively, while the black solid balls denoted the critical or bifurcation points.

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_{3}) hybrid nanofluid for the several values of ${f}_{w}$ and ${M}_{c}$. It was found that multiple or double solutions (FB and SB results) happened for Equations (20)–(22) subject to the BCs (23) in the region of $\epsilon >{\epsilon}_{c}$, in which ${\epsilon}_{c}$ represents the bifurcation value of $\epsilon $. Note that a unique solution exists for $\epsilon ={\epsilon}_{c}$, while no solutions exist when $\epsilon <{\epsilon}_{c}$. Additionally, from these graphs, it is clear that the output values of $\left|{\epsilon}_{c}\right|$ upsurge as the requisite parameters ${f}_{w}$ and ${M}_{c}$ upsurge, indicating that these factors/parameters expand the range in which dual/double solutions can occur. Further evidence supports the idea that the inclusion of hybrid nanoparticles, the suction effect, the magnetic field, and the stretching sheet could slow the boundary layer separation (BLS), but the presence of the material parameter and the shrinking sheet could speed up the BLS. Also, for the FBS, the values of the HTR and CSC are consistently positive owing to the HT from the precise hot surface of the stretched/contracted sheet to the requisite cold fluid. The opposite pattern is noticed in the phenomenon of the SBS, i.e., the heat transfer rate and the couple stress coefficient become unavailable as ${\epsilon}_{c}\to {0.55}^{+}$ and ${\epsilon}_{c}\to {0.55}^{-}$.

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_{3}) hybrid nanofluid for the FB and SB solutions is presented in Figure 2, Figure 3 and Figure 4, respectively. The outcomes display that the gradients (SSC, CSC, and HTR) boosted up for the FBS due to the higher values of ${f}_{w}$ while it declines for the branch of second solutions. In addition, it is noticeable that the SSC values of the FBS are slightly higher and better than the values of CSC with higher impacts of the mass suction parameter as seen graphically in Figure 2 and Figure 3. This behavior is due to the fact that the influence of the mass suction at the surface boundary of the stretched/shrinked surface slows down the hybrid nanofluid motion and escalates the shear stress and couple-stress coefficients at the surface of the vertical sheet. On the other hand, the rate of heat transmission developed as the magnitude of the mass transpiration parameter was boosted (see Figure 4). According to this fact, the thermal boundary layer thickness decelerates with larger impacts of the mass suction parameter, and as a response, the temperature distribution gradient of the sheet uplifts.

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_{3}) hybrid nanofluid for the FB and SB solutions, respectively. In these graphs, it is initiated that the CSC and HTR improve in the FBS but decay in the SBS owing to the larger impacts of ${M}_{c}$ while the SSC behaves increasingly for the FB solutions and a monotonic kind of behavior or a changeable behavior was followed in the same branch, moving away from the critical values for the larger effects of the magnetic field. Physically, by enlarging the values of the magnetic parameter, a force is produced which slows down the motion of fluid along the stretching/shrinking sheet and increases the convection of thermal energy by boosting the interactivity of the fluid particles which is known as the Lorentz force. According to this force, the speed of the hybrid nanoparticles of the fluid slows down. As a result, the shear stress and couple-stress coefficients are enlarged. Additionally, the gap among the curves for the FB is compared to the SB solutions as seen in Figure 5 and Figure 6, and vice versa, for the heat transfer rate as graphically depicted in Figure 7. Furthermore, when the effects of magnetic field strength grow, the rate of heat transmission increases. This tendency results from the decreasing thermal boundary layer thickness caused by a growing magnetic field, which causes an intensification in the posited temperature gradient at the vertical sheet’s surface.

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_{3}) hybrid nanofluid for the FB and SB solutions due to larger values of the material parameter ${K}_{c}$. Notably, it is evident from the Figure 11 and Figure 12 that the shear stress initially declines and then upsurges for the FBS with a higher impact of material parameters, but the couple shear stress continuously decreases for the FBS. Whereas, for the branch of second solutions, the gradient of micro-rotation escalates due to larger values of ${K}_{c}$, while shear stress decelerates and behaves vice versa. On the other hand, the heat transfer decreases and increases for the branch of the first and second solutions, respectively, owing to the superior hammering of the parameter ${K}_{c}$ (see Figure 13). In all three graphs, the gap was slightly lesser in both solution branches, therefore, we have to zoom in on the specific part where both solutions can meet or merge at a single point, called a bifurcation point ${\gamma}_{c}$. The bifurcation values for the distinct choices of ${K}_{c}$ are shown by the small solid black balls and also numerally highlighted in the suggested plots. Additionally, the absolute values of $\left|{\gamma}_{c}\right|$ are smaller for the superior values of ${K}_{c}$. In this regard, the pattern of the outcomes indicates that the larger values of the material parameter speed up the level of the separation of the boundary layer.

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_{3}) hybrid nanofluid for both branches of solutions is graphically exemplified in Figure 14. From the graph, it is seen that both solution branches and the thickness of the TTBL boosted up as the value of the ${N}_{r}$ was augmented. Moreover, the double branch (FB and SB) solutions asymptotically hold the boundary Conditions (23). Also, the gap in both solution curves is reasonable/understandable for the higher impacts of the radiation parameter. Physically, the thermal conductivity is improved by bigger values of the radiation parameter, which can lead to an increase in temperature distribution profiles as well as in the thickness of the thermal boundary layer.

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_{3}) hybrid nanoparticles, respectively. In both graphs, the results were constructed/prepared for the branch of the first and second solutions. In addition, the outcome of both plots demonstrates that the profile of the temperature and the TTBL augment with the superior impacts of the internal heat source factor for the FBS as well as for the SBS, while it is reducing continuously with the internal heat sink factor. Generally, the reason this happens is because the heat source factor causes the system to absorb more energy as a result of heat, and, as a response, the temperature enriches (see Figure 15). On the other hand (see Figure 16), the system did not receive a better level of heat (in the form of energy) owing to the requisite heat sink factor and, as a consequence, the temperature profile decelerated. Furthermore, the gap in both solution cases is looking similar to the rising value of the internal heat source/sink factors.

## 6. Conclusions

- The mass suction parameter ${f}_{w}$ and the magnetic parameter ${M}_{c}$ contributes to enhancing the SSC and the CSC, as well as the heat transfer performance of the HN.
- Besides enhancing the skin friction and the couple stress coefficients, the added nanoparticles volume fraction ${\phi}_{1}$ and ${\phi}_{2}$ also improves the Nusselt number. This behavior is expected since the nanoparticles improve the thermal conductivity of the base fluid due to their synergic effect.
- Moreover, the material parameter ${K}_{c}$ lowers the couple stress coefficient and heat transfer performance of the hybrid nanofluid, but the SSC is slightly increased with this parameter.
- The domain of the stretching/shrinking parameter ${\epsilon}_{c}$ is expanded for the larger mass suction parameter ${f}_{w}$ and the magnetic parameter ${M}_{c}$. These behaviors are proven by looking at the critical points of the parameter where they are moving on the left side of the shrinking regions. Similar behavior is observed for the buoyancy parameter ${\gamma}_{c}$ for some values of the material parameter ${K}_{c}$.
- The heat source parameter boosts the temperature profiles, and the opposite behavior is shown by the heat sink parameter for both the first and second solutions.
- According to the stability analysis, the eigenvalue obtained for the first solution is in positive values, while the negative values of the eigenvalues are shown in the second solution. These behaviors prove that the FBS is stable in the long run and vice versa.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Variation of the skin friction coefficient with ${\epsilon}_{c}$ for the several values of the ${f}_{w}$.

**Figure 3.**Variation of the couple stress coefficient with ${\epsilon}_{c}$ for the several values of ${f}_{w}$.

**Figure 8.**Variation of the skin friction coefficient with ${\gamma}_{c}$ for the several values of ${\phi}_{1}$ and ${\phi}_{2}$.

**Figure 9.**Variation of the couple stress coefficient with ${\gamma}_{c}$ for the several values of ${\phi}_{1}$ and ${\phi}_{2}$.

**Figure 10.**Variation of the heat transfer with ${\gamma}_{c}$ for the several values of ${\phi}_{1}$ and ${\phi}_{2}$.

**Figure 11.**Variation of the skin friction coefficient with ${\gamma}_{c}$ for the several values of ${K}_{c}$.

**Figure 12.**Variation of the couple stress coefficient with ${\gamma}_{c}$ for the several values of ${K}_{c}$.

Properties | $\mathit{\rho}\left(\mathit{k}\mathit{g}/{\mathit{m}}^{3}\right)$ | ${\mathit{c}}_{\mathit{p}}\left(\mathit{J}/\mathit{k}\mathit{g}\mathit{K}\right)$ | ${\mathit{\beta}}_{\mathit{T}}\times {10}^{-5}$$\left(1/\mathit{K}\right)$ | $\mathit{\sigma}\left(\mathit{S}/\mathit{m}\right)$ | $\mathit{k}\left(\mathit{W}/\mathit{m}\mathit{k}\right)$ | Pr |
---|---|---|---|---|---|---|

Water | 997.1 | 4179 | 21 | $5.5\times {10}^{-6}$ | 0.613 | 6.2 |

Alumina | 3970 | 765 | 0.85 | $35\times {10}^{6}$ | 40 | |

Copper | 8933 | 385 | 1.67 | $59.6\times {10}^{6}$ | 400 | --- |

**Table 2.**Values of shear stress for the several values of ${\gamma}_{c}$ and ${K}_{c}$ when ${\epsilon}_{c}=0$, ${f}_{w}=0$, ${\phi}_{1}=0$, ${\phi}_{2}=0$, ${E}_{c}=1$, ${M}_{c}=0$, ${m}_{s}=0$, and $\mathrm{Pr}=0.7$.

${\mathit{\gamma}}_{\mathit{c}}$ | ${\mathit{K}}_{\mathit{c}}$ | Lok et al. [50] | Present Solution | ||
---|---|---|---|---|---|

FBS | SBS | FBS | SBS | ||

−1.1 | 0.0 | 0.631500 | −0.350112 | 0.631412 | −0.350102 |

−1.4 | - | 0.440161 | −0.494103 | 0.440116 | −0.494124 |

−1.7 | - | 0.225110 | −0.574153 | 0.225075 | −0.574135 |

−2.0 | - | −0.039513 | −0.578523 | −0.039541 | −0.578493 |

−1.1 | 3.0 | 0.338030 | - | 0.337967 | - |

−1.4 | - | 0.272370 | −0.232634 | 0.272212 | −0.232728 |

−1.7 | - | 0.202475 | −0.273943 | 0.202345 | −0.273934 |

−2.0 | - | 0.126644 | −0.030171 | 0.126452 | −0.030198 |

**Table 3.**Values of couple-stress coefficient for the several values of ${\gamma}_{c}$ and ${K}_{c}$ when ${\epsilon}_{c}=0$, ${f}_{w}=0$, ${\phi}_{1}=0$, ${\phi}_{2}=0$, ${E}_{c}=1$, ${M}_{c}=0$, ${m}_{s}=0$, and $\mathrm{Pr}=0.7$.

${\mathit{\gamma}}_{\mathit{c}}$ | ${\mathit{K}}_{\mathit{c}}$ | Lok et al. [50] | Present Solution | ||
---|---|---|---|---|---|

FBS | SBS | FBS | SBS | ||

−1.1 | 0.0 | 0.623635 | −0.174194 | 0.623650 | −0.174172 |

−1.4 | - | 0.590886 | −0.044680 | 0.590846 | −0.044371 |

−1.7 | - | 0.549045 | 0.073812 | 0.590150 | 0.073804 |

−2.0 | - | 0.486596 | 0.198590 | 0.486586 | 0.198590 |

−1.1 | 3.0 | 0.541705 | - | 0.541039 | - |

−1.4 | - | 0.523084 | −0.149594 | 0.523015 | −0.149688 |

−1.7 | - | 0.502440 | −0.059305 | 0.502333 | −0.059348 |

−2.0 | - | 0.477867 | 0.020600 | 0.477732 | 0.020533 |

**Table 4.**Quantitative values of SSC for the several distinct parameters when ${\epsilon}_{c}=-1.5$, ${\gamma}_{c}=-1.0$, ${E}_{c}=0.5$, ${m}_{s}=0.5$, ${N}_{r}=2.0$, ${A}_{s}=0.1$ and ${B}_{s}=0.1$.

${\mathit{\phi}}_{1},{\mathit{\phi}}_{2}$ | ${\mathit{K}}_{\mathit{c}}$ | ${\mathit{M}}_{\mathit{c}}$ | ${\mathit{f}}_{\mathit{w}}$ | FBS | SBS |
---|---|---|---|---|---|

0.025 | 0.25 | 0.20 | 1.0 | 3.2811 | 1.2167 |

0.030 | - | - | - | 3.4592 | 1.2056 |

0.035 | - | - | - | 3.6352 | 1.1979 |

0.025 | 0.25 | 0.20 | 1.0 | 3.2811 | 1.2167 |

- | 0.30 | - | - | 3.2775 | 1.2558 |

- | 0.35 | - | - | 3.2733 | 1.2952 |

0.025 | 0.25 | 0.20 | 1.0 | 3.2811 | 1.2167 |

- | - | 0.25 | - | 3.4014 | 1.1981 |

- | - | 0.30 | - | 3.5140 | 1.1882 |

0.025 | 0.25 | 0.20 | 0.90 | 2.8005 | 1.4393 |

- | - | - | 0.95 | 3.0547 | 1.3151 |

- | - | - | 1.0 | 3.2811 | 1.2167 |

**Table 5.**Quantitative values of CSC for the several distinct parameters when ${\epsilon}_{c}=-1.5$, ${\gamma}_{c}=-1.0$, ${f}_{w}=1.0$, ${M}_{c}=0.20$, ${m}_{s}=0.5$, ${N}_{r}=2.0$, ${A}_{s}=0.1$, and ${B}_{s}=0.1$.

${\mathit{\phi}}_{1},{\mathit{\phi}}_{2}$ | ${\mathit{K}}_{\mathit{c}}$ | ${\mathit{E}}_{\mathit{c}}$ | FBS | SBS |
---|---|---|---|---|

0.025 | 0.25 | 0.5 | 0.8057 | −0.3938 |

0.030 | - | - | 0.8956 | −0.4363 |

0.035 | - | - | 0.9843 | −0.4786 |

0.025 | 0.25 | 0.5 | 0.8057 | −0.3938 |

- | 0.30 | - | 0.7696 | −0.3929 |

- | 0.35 | - | 0.7344 | −0.3909 |

0.025 | 0.25 | 0.1 | 0.8144 | −0.3519 |

- | - | 0.3 | 0.8099 | −0.3745 |

- | - | 0.5 | 0.8057 | −0.3938 |

**Table 6.**Quantitative values of the HTR for the several distinct parameters when ${\epsilon}_{c}=-1.5$, ${\gamma}_{c}=-1.0$, ${f}_{w}=1.0$, ${M}_{c}=0.20$, ${E}_{c}=0.5$, ${m}_{s}=0.5$, ${N}_{r}=2.0$, ${A}_{s}=0.1$, and ${B}_{s}=0.1$.

${\mathit{\phi}}_{1},{\mathit{\phi}}_{2}$ | ${\mathit{N}}_{\mathit{r}}$ | ${\mathit{A}}_{\mathit{s}},{\mathit{B}}_{\mathit{s}}$ | FBS | SBS |
---|---|---|---|---|

0.025 | 2.0 | 0.1 | 2.1035 | −6.8374 |

0.030 | - | - | 2.2358 | −7.3791 |

0.035 | - | - | 2.3492 | −7.9004 |

0.025 | 1.5 | 0.1 | 2.0141 | −7.1554 |

- | 2.0 | - | 2.1035 | −6.8374 |

- | 2.5 | - | 2.2270 | −6.5757 |

0.025 | 2.0 | 0.1 | 2.1035 | −6.8374 |

- | - | 0.2 | 1.8660 | −6.9680 |

- | - | 0.3 | 1.6205 | −7.0865 |

- | −0.1 | 2.5575 | −6.5426 | |

- | −0.2 | 2.7750 | −6.3797 | |

- | −0.3 | 2.9868 | −6.2073 |

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## Share and Cite

**MDPI and ACS Style**

Khan, U.; Zaib, A.; Ishak, A.; Alotaibi, A.M.; Eldin, S.M.; Akkurt, N.; Waini, I.; Madhukesh, J.K.
Stability Analysis of Buoyancy Magneto Flow of Hybrid Nanofluid through a Stretchable/Shrinkable Vertical Sheet Induced by a Micropolar Fluid Subject to Nonlinear Heat Sink/Source. *Magnetochemistry* **2022**, *8*, 188.
https://doi.org/10.3390/magnetochemistry8120188

**AMA Style**

Khan U, Zaib A, Ishak A, Alotaibi AM, Eldin SM, Akkurt N, Waini I, Madhukesh JK.
Stability Analysis of Buoyancy Magneto Flow of Hybrid Nanofluid through a Stretchable/Shrinkable Vertical Sheet Induced by a Micropolar Fluid Subject to Nonlinear Heat Sink/Source. *Magnetochemistry*. 2022; 8(12):188.
https://doi.org/10.3390/magnetochemistry8120188

**Chicago/Turabian Style**

Khan, Umair, Aurang Zaib, Anuar Ishak, Abeer M. Alotaibi, Sayed M. Eldin, Nevzat Akkurt, Iskandar Waini, and Javali Kotresh Madhukesh.
2022. "Stability Analysis of Buoyancy Magneto Flow of Hybrid Nanofluid through a Stretchable/Shrinkable Vertical Sheet Induced by a Micropolar Fluid Subject to Nonlinear Heat Sink/Source" *Magnetochemistry* 8, no. 12: 188.
https://doi.org/10.3390/magnetochemistry8120188