# Double Solutions and Stability Analysis of Micropolar Hybrid Nanofluid with Thermal Radiation Impact on Unsteady Stagnation Point Flow

^{1}

^{2}

^{*}

## Abstract

**:**

_{2}O

_{3}/water nanofluid driven by a deformable sheet in stagnation region with thermal radiation effect has been explored numerically. To achieve the system of nonlinear ordinary differential equations (ODEs), we have employed some appropriate transformations and solved it numerically using MATLAB software (built-in solver called bvp4c). Influences of relevant parameters on fluid flow and heat transfer characteristic are discussed and presented in graphs. The findings expose that double solutions appear in shrinking sheet case in which eventually contributes to the analysis of stability. The stability analysis therefore confirms that merely the first solution is a stable solution. Addition of nanometer-sized particle (Cu) has been found to significantly strengthen the heat transfer rate of micropolar nanofluid. When the copper nanoparticle volume fraction increased from 0 to 0.01 (1%) in micropolar nanofluid, the heat transfer rate increased roughly to an average of 17.725%. The result also revealed that an upsurge in the unsteady and radiation parameters have been noticed to enhance the local Nusselt number of micropolar hybrid nanofluid. Meanwhile, the occurrence of material parameter conclusively decreases it.

## 1. Introduction

_{2}/water) in micropolar fluid in a porous medium past an exponentially stretching sheet and point out that the heat transfer rate for micropolar hybrid nanofluid is greater than micropolar nanofluid. Afterwards, by taking into attention the simultaneous effects of MHD and slip, Nadeem and Abbas [44] examined the micropolar hybrid nanofluid flow past a circular cylinder. In another study of Abbas et al. [45] and Al-Hanaya [46], a theoretical investigation of micropolar hybrid nanofluid using carbon nanotubes (SWCNT and MWCNT) as a nanoparticle over an exponentially stretching Riga plate and curved stretching sheet have been investigated. Apparently, the research related to micropolar hybrid nanofluids are limited in number. Hence, the principal goal of this investigation is to address the behavior of micropolar hybrid nanofluid in a deformable sheet, i.e., stretching and shrinking. It is important to note that deformable sheet is not a new crucial topic among the researchers in the fluid field since their applications are well recognized in processing industries especially in polymer processing, glass fiber production, cooling, and drying of paper and many others [47].

_{2}O

_{3}) as the new heat transfer fluid for the micropolar flow problem with the thermal radiation effect, has not been performed up to now. In addition, this analysis also comprises a novel era for scientists to discover the shrinking features of micropolar hybrid nanofluids. Furthermore, the novelty of this study can also be seen in the discovery of non-unique solutions and the execution of stability analysis. To the best of authors’ knowledge, the results of the present work is new and still not considered and published by any researchers. Therefore, current studies are expected to bring good benefits to researchers who are experimentally working on micropolar hybrid nanofluids, and these results are also expected to reduce the cost of experimental work in the future.

## 2. Mathematical Framework

#### 2.1. Basic Equations

_{2}O

_{3}/water nanofluid past a deformable sheet in the stagnation region with the influence of thermal radiation impact are investigated in this work as exemplified in Figure 1. The Cartesian coordinates used are $x$ and $y$, given that $x-$axis is considered along the sheet while $y-$axis normal to it, respectively, the sheet is located in the plane $y=0$ and the fluid fill the half space at $y\ge 0$. The temperature far from the surface (inviscid flow) and at the surface are represented by ${T}_{\infty}$ and ${T}_{w}\left(x,t\right)$. The sheet is stretch and shrunk along the $x-$axis with velocity ${u}_{w}\left(x,t\right)$ and the free stream velocity is denoted by ${u}_{e}\left(x,t\right)$.

_{2}O

_{3}/water.

#### 2.2. Thermophysical Traits of Hybrid Nanofluid

_{2}O

_{3}) is picked as the first nanoparticle volume fraction and water act as a base fluid. Table 2 displays the thermophysical traits of nanoparticles and base fluid. It is important to note that Al

_{2}O

_{3}is originally disseminated into the water to achieve the appropriated hybrid nanofluid, i.e., Cu-Al

_{2}O

_{3}/water, and then Cu is disseminated into the Al

_{2}O

_{3}/water nanofluid. Additionally, the volume fraction of Al

_{2}O

_{3}nanoparticle is set to 1% and Cu is fluctuated from 0 to 2%.

#### 2.3. Similarity Solutions

## 3. Stability of the Solutions

#### 3.1. New Similarity Transformation

#### 3.2. Introducing Linear Eigenvalue Equations

#### 3.3. Relaxation of Boundary Conditions

## 4. Numerical Solutions

_{2}O

_{3}/water. It is obvious that existence of material parameter $\left(K=1,\hspace{0.17em}2\right)$ give rises to the local skin friction ${C}_{f}{\mathrm{Re}}_{x}^{1/2}$ if compared to the absence of material parameter $\left(K=0\right)$, i.e., no vortex viscosity. However, different results are observed for the local Nusselt number $N{u}_{x}{\mathrm{Re}}_{x}^{-1/2}$ where the nonexistence of micropolar fluid $\left(K=0\right)$ cause an enhancement in comparison with the existence of material parameter $\left(K=1,\hspace{0.17em}2\right)$. This phenomenon reveals the fact that upsurge value of material parameter gives rise on the vortex viscosity in the fluid flow which consequently enhance the skin friction at the wall and decrease the rate of heat transfer at the wall. Additionally, we observed that an upsurge values of material parameter prompt the domain of similarity solutions to exist become narrow. For instance, the similarity solutions in the nonexistence of material parameter are noted in the range of $-1.31178\le \lambda \le 1,$ while in the existence of material parameter $\left(K=1,\hspace{0.17em}2\right)$, the range of solutions are observed to be $-1.31164\le \lambda \le -1$ and $-1.31162\le \lambda \le -1$. Furthermore, an upsurge values of material parameter $K$ causing the thickness of the momentum and thermal boundary layer to increase in first and second solutions. We can see from Figure 7b that the microrotation boundary layer thickness for first and second solutions near the sheet decreases when this parameter rises while the contrary trend is observed for the large $\eta $.

## 5. Conclusions

_{2}O

_{3}/water flow over a deformable sheet with thermal radiation effect has been examined numerically. The similarity solutions were produced by utilizing the bvp4c function from MATLAB software. The impact of emerging parameters has been examined and illustrated graphically. Thus, the conclusions can be outlined as follows:

- The presence of double solutions is noticeable for shrinking sheet whereas a unique solution is observed for stretching sheet.
- The stability analysis was carried out and the first solution has proven to be a stable solution, whereas the other solution is not a stable solution.
- A raise in Cu nanoparticle volume fraction ${\phi}_{2}$ in micropolar nanofluid has tendency to improve the local Nusselt number and local skin friction for all domain of stretching/shrinking parameter $\lambda $.
- The heat transfer rate increased roughly to an average of 17.725% when the copper nanoparticle volume fraction increased from 0 to 0.01 (1%) in a micropolar nanofluid.
- The rising of unsteady parameter $A$ and radiation parameter $Rd$ in micropolar hybrid nanofluid increase the local Nusselt number while the reverse trend is observed with an increase of material parameter $K$.
- The local skin friction enhances as the value of unsteady parameter $A$ and material parameter $K$ increase.
- The domains of the similarity solutions decrease with a raise in Cu nanoparticle volume fraction ${\phi}_{2}$ and material parameter $K$, therefore fastens the boundary layer separation. However, upsurge value of unsteady parameter $A$ delays the boundary layer separation.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$a,b,c$ | positive constants [s^{−1}] |

$A$ | unsteady parameter [-] |

${C}_{f}$ | skin friction coefficient [-] |

${C}_{p}$ | specific heat at constant pressure [Jkg^{−1}K^{−1}] |

$f$ | dimensionless stream function [-] |

$h$ | dimensionless angular velocity [-] |

$j$ | microinertia density [m^{2}] |

$k$ | thermal conductivity [Wm^{−1}K^{−1}] |

${k}^{\ast}$ | mean absorption coefficient [m^{−1}] |

$K$ | dimensionless material parameter [-] |

$n$ | positive constant [s^{−1}] |

$N$ | angular velocity [ms^{−1}] |

$N{u}_{x}$ | local Nusselt number [-] |

$\mathrm{Pr}$ | Prandtl number [-] |

${q}_{r}$ | radiative heat flux [Wm^{−2}] |

$Rd$ | radiation parameter [-] |

${\mathrm{Re}}_{x}$ | local Reynolds number [-] |

$t$ | time [s] |

$T$ | temperature [K] |

$u,v$ | velocities component in the $x-$ and $y-$ directions, respectively [ms^{−1}] |

${u}_{e}$ | velocity of inviscid flow [ms^{−1}] |

${u}_{w}$ | stretching/shrinking velocity [ms^{−1}] |

$x,y$ | cartesian coordinates along the surface and normal to it, respectively [m] |

Greek Symbols | |

${\phi}_{1}$ | nanoparticle volume fractions for Al_{2}O_{3} (alumina) [-] |

${\phi}_{2}$ | nanoparticle volume fractions for Cu (copper) [-] |

$\theta $ | dimensionless temperature [-] |

$\gamma $ | unknown eigenvalues [-] |

$\lambda $ | stretching/shrinking parameter [-] |

$\eta $ | similarity variable [-] |

$\mu $ | dynamic viscosity [N s m^{−2}] |

$\nu $ | kinematic viscosity [m^{2}s^{−1}] |

$\rho $ | density [kgm^{−3}] |

$\tau $ | dimensionless time variable [-] |

${\sigma}^{\ast}$ | Stefan–Boltzmann constant [Wm^{−2}K^{−4}] |

$\psi $ | stream function [-] |

$\rho {C}_{p}$ | heat capacity [JK^{−1}m^{−3}] |

$\varsigma $ | spin gradient viscosity [kg m s^{−1}] |

$\kappa $ | vortex viscosity [kg m^{−1}s^{−1}] |

Subscripts | |

$f$ | base fluid |

$hnf$ | hybrid nanofluid |

$s1$ | solid component for Al_{2}O_{3} (alumina) |

$s2$ | solid component for Cu (copper) |

$w$ | condition at the surface |

$\infty $ | ambient condition |

Superscript | |

$\u2019$ | differentiation with respect to $\eta $ |

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**Figure 2.**(

**a**) ${C}_{f}{\mathrm{Re}}_{x}^{1/2}$; (

**b**) $N{u}_{x}{\mathrm{Re}}_{x}^{-1/2}$ with $\lambda $ for various ${\phi}_{2}$.

**Figure 3.**(

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

**b**) $h\left(\eta \right)$; (

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

**Figure 4.**(

**a**) ${C}_{f}{\mathrm{Re}}_{x}^{1/2}$; (

**b**) $N{u}_{x}{\mathrm{Re}}_{x}^{-1/2}$ with $\lambda $ for various $A$.

**Figure 5.**(

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

**b**) $h\left(\eta \right)$; (

**c**) $\theta \left(\eta \right)$ for various $A$.

**Figure 6.**(

**a**) ${C}_{f}{\mathrm{Re}}_{x}^{1/2}$; (

**b**) $N{u}_{x}{\mathrm{Re}}_{x}^{-1/2}$ with $\lambda $ for various $K$.

**Figure 7.**(

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

**b**) $h\left(\eta \right)$; (

**c**) $\theta \left(\eta \right)$ for various $K$.

**Figure 8.**(

**a**) $N{u}_{x}{\mathrm{Re}}_{x}^{-1/2}$ with $\lambda $; (

**b**) $\theta \left(\eta \right)$ for various $Rd$.

**Figure 9.**Smallest eigenvalues $\gamma $ against $\lambda $ when ${\phi}_{1}={\phi}_{2}=0.01,\hspace{0.17em}A=Rd=0.1,\hspace{0.17em}K=1,\hspace{0.17em}n=0.5$.

**Table 1.**Physical traits of hybrid nanofluids (Devi and Devi [38]).

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

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

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

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

Thermal conductivity | $\frac{{k}_{hnf}}{{k}_{bf}}=\frac{{k}_{s2}+2{k}_{bf}-2{\phi}_{2}\left({k}_{bf}-{k}_{s2}\right)}{{k}_{s2}+2{k}_{bf}+{\phi}_{2}\left({k}_{bf}-{k}_{s2}\right)}$ where $\frac{{k}_{bf}}{{k}_{f}}=\frac{{k}_{s1}+2{k}_{f}-2{\phi}_{1}\left({k}_{f}-{k}_{s1}\right)}{{k}_{s1}+2{k}_{f}+{\phi}_{1}\left({k}_{f}-{k}_{s1}\right)}$ |

**Table 2.**Thermo physical properties (Oztop and Abu-Nada [61]).

Physical Properties | |||
---|---|---|---|

${\mathit{C}}_{\mathit{p}}\left(\mathbf{J}\hspace{0.17em}{\mathbf{kg}}^{-1}{\mathbf{K}}^{-1}\right)$ | $\mathbf{\rho}\left(\mathbf{kg}\hspace{0.17em}{\mathbf{m}}^{-3}\right)$ | $\mathbf{k}\left(\mathbf{W}\hspace{0.17em}{\mathbf{m}}^{-1}{\mathbf{K}}^{-1}\right)$ | |

water | 4179 | 997.1 | 0.613 |

Cu | 385 | 8933 | 400 |

Al_{2}O_{3} | 765 | 3970 | 40 |

**Table 3.**Comparison values of ${f}^{\u2033}\left(0\right)$ when $A=K=n=0$ and ${\phi}_{1}={\phi}_{2}=0$ for different $\lambda $ values.

$\mathit{\lambda}$ | Refs. [10] (Keller-Box Method) | Refs. [65] (Shooting Method) | Refs. [59] (bvp4c Solution) | Present Result (bvp4c Solution) |
---|---|---|---|---|

−0.25 | 1.402241 | 1.402242 | 1.402241 | 1.402241 |

−0.5 | 1.495670 | 1.495672 | 1.495670 | 1.495670 |

−0.75 | 1.489298 | 1.489296 | 1.489298 | 1.489298 |

−1 | 1.328817 [0] | 1.328819 [0] | 1.328817 [0] | 1.328817 [0] |

−1.1 | 1.186681 [0.049229] | 1.186680 [0.049229] | 1.186680 [0.049229] | 1.186680 [0.049229] |

−1.15 | 1.082231 [0.116702] | 1.082232 [0.116702] | 1.082231 [0.116702] | 1.082231 [0.116702] |

−1.2 | 0.932474 [0.233650] | 0.932470 [0.233648] | 0.932473 [0.233650] | 0.932473 [0.233650] |

−1.246 | - | 0.584374 [0.554215] | 0.609826 [0.529035] | 0.609826 [0.529035] |

−1.2465 | 0.584295 [0.554283] | - | - | 0.584282 [0.554296] |

**Table 4.**Smallest eigenvalues $\gamma $ for selected $A$ and $\lambda $ when ${\phi}_{1}={\phi}_{2}=0.01$, $Rd=0.1$, $K=1$ and $n=0.5$.

$\mathit{A}$ | $\mathit{\lambda}$ | 1st Solution | 2nd Solution |
---|---|---|---|

0 | −1.2462 | 0.0174 | −0.0583 |

−1.246 | 0.0323 | −0.0728 | |

−1.24 | 0.1893 | −0.2253 | |

0.1 | −1.3112 | 0.0036 | −0.1086 |

−1.311 | 0.0156 | −0.1204 | |

−1.31 | 0.0577 | −0.1616 | |

0.2 | −1.3785 | 0.0224 | −0.2193 |

−1.378 | 0.0385 | −0.2348 | |

−1.37 | 0.1950 | −0.3832 |

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**MDPI and ACS Style**

Anuar, N.S.; Bachok, N.
Double Solutions and Stability Analysis of Micropolar Hybrid Nanofluid with Thermal Radiation Impact on Unsteady Stagnation Point Flow. *Mathematics* **2021**, *9*, 276.
https://doi.org/10.3390/math9030276

**AMA Style**

Anuar NS, Bachok N.
Double Solutions and Stability Analysis of Micropolar Hybrid Nanofluid with Thermal Radiation Impact on Unsteady Stagnation Point Flow. *Mathematics*. 2021; 9(3):276.
https://doi.org/10.3390/math9030276

**Chicago/Turabian Style**

Anuar, Nur Syazana, and Norfifah Bachok.
2021. "Double Solutions and Stability Analysis of Micropolar Hybrid Nanofluid with Thermal Radiation Impact on Unsteady Stagnation Point Flow" *Mathematics* 9, no. 3: 276.
https://doi.org/10.3390/math9030276