# Numerical Simulations through PCM for the Dynamics of Thermal Enhancement in Ternary MHD Hybrid Nanofluid Flow over Plane Sheet, Cone, and Wedge

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{*}

## Abstract

**:**

_{2}), cobalt ferrite (CoFe

_{2}O

_{4}) and magnesium oxide (MgO) nanoparticles (NPs) through wedge, cone, and plate surfaces is reported in the present study. TiO

_{2}, CoFe

_{2}O

_{4}, and MgO NPs were dispersed in water to synthesize a trihybrid nanofluid. For this purpose, a mathematical model was calculated to augment the energy transport rate and efficiency for variety of commercial and medical functions. The consequences of heat source/sink, activation energy, and the magnetic field are also analyzed. Such problems mostly occur in symmetrical phenomena and are applicable to engineering, physics, and applied mathematics. The phenomena were formulated in the form of a nonlinear system of PDEs, which are simplified to the system of dimensionless ODEs through similarity replacement (obtained from symmetry analysis). The obtained set of differential equations is resolved through a parametric continuation approach (PCM). Graphical depictions are used to evaluate and address the impact of significant factors on energy, mass, and flow exchange rates. The velocity and energy propagation rates over a cone surface were greater than those of a wedge and plate versus the variation of Grashof number, porosity effect, and heat source, while the mass transfer ratio under the impact of a chemical reaction and activation energy over a wedge surface was higher than that of a plate.

## 1. Introduction

_{2}, CoFe

_{2}O

_{4}and MgO. TiO

_{2}is an inorganic chemical that has been utilised for over a decade in a number of applications. It is reliable because of its phosphorescence, and nontoxic and nonreactive characteristics. It is the world’s brightest and frostiest substance, with reflecting properties and a UV light absorption capability that can protect from skin cancer [22,23,24,25]. MgO is a stain-resistant material that occurs naturally and serves as a magnesium source. Its overall structure comprises Mg

^{2+}and O

^{2−}ion connections. Bilal et al. [26] investigated the upshots of electromagnetic interaction on energy transference through water-based hybrid nanocomposites via twin turning discs. Ullah et al. [27,28,29] examined the influence of Darcy–Forchheimer and Coriolis force on nanofluid flow consisting of CNTs in ethylene glycol across a circling edge. Krishna et al. [30] mathematically investigated the effects of ion slip and Hall on an unstable laminar MHD convection revolving flow of second-grade fluid across a semi-infinite upward sliding porous medium. Arif et al. [31] revealed the comportment of ternary hybrid NF in Al

_{2}O

_{3}, Graphene and CNTs. The trihybrid nanoliquid boosted the energy transmission ratio up to 33.67%, as compared to the nano and hybrid nanoliquids. Sahoo et al. [32] used CNTs, Al

_{2}O

_{3}, and graphene ternary hybrid NF to reduce heat transmission in a condenser. Fattahi and Karimi [33] used a ternary hybrid nanofluid to conduct and test solar-panel efficiency with the use of the hybrid nanofluid. Some related works and uses of CoFe

_{2}O

_{4}and Cu NPs in solvent for biological and production purposes can be found in [34,35,36,37].

- To mathematically model the Darcy–Forchheimer ternary hybrid nanofluid flow via a porous wedge, cone, and plate.
- A trihybrid nanofluid is prepared by dispersing TiO
_{2}, CoFe_{2}O_{4}, and MgO NPs in base liquid water. - The Lorentz and gravitational effects are considered to see the variations in hybrid nanofluid motion.
- A mathematical model is obtained with the objective of optimizing energy transmission rates and productivity for a variety of commercial and medical applications. This research looks at the effects of heat source/sink, activation energy, and magnetic field.
- To solve the obtained system of ODEs through the PCM technique.

## 2. Mathematical Formulation

_{0}, K* and $F={C}_{b}/r{K}^{\ast 1/2}$ are the activation energy, heat source term, porosity term and nonuniform inertia term, respectively. u, v signifies the velocity along the x and y directions, respectively, ${\beta}_{T}$ is the volumetric thermal expansion term, g is gravity acceleration, ${\left(T/{T}_{\infty}\right)}^{n}{e}^{-\frac{Ea}{KT}}$ is the modified Arrhenius constraint, and D

_{B}is the Brownian diffusion. The slip condition was considered for the fluid velocity to be $u={U}_{w}+L\frac{\partial u}{\partial y}$. The boundary conditions are:

Viscosity | $\frac{{\mu}_{Thnf}}{{\mu}_{f}}=\frac{1}{{(1-{\varphi}_{MgO})}^{2.5}{(1-{\varphi}_{Ti{O}_{2}})}^{2.5}{(1-{\varphi}_{CoF{e}_{2}{O}_{4}})}^{2.5}},$ |

Density | $\frac{{\rho}_{Thnf}}{{\rho}_{f}}=\left(1-{\varphi}_{Ti{O}_{2}}\right)\left[\left(1-{\varphi}_{Ti{O}_{2}}\right)\left\{\left(1-{\varphi}_{CoF{e}_{2}{O}_{4}}\right)+{\varphi}_{CoF{e}_{2}{O}_{4}}\frac{{\rho}_{CoF{e}_{2}{O}_{4}}}{{\rho}_{f}}\right\}+{\varphi}_{Ti{O}_{2}}\frac{{\rho}_{Ti{O}_{2}}}{{\rho}_{f}}\right]+{\varphi}_{MgO}\frac{{\rho}_{MgO}}{{\rho}_{f}},$ |

Specific heat | $\frac{{(\rho cp)}_{Thnf}}{{\left(\rho cp\right)}_{f}}={\varphi}_{MgO}\frac{{\left(\rho cp\right)}_{MgO}}{{\left(\rho cp\right)}_{f}}+\left(1-{\varphi}_{MgO}\right)\left[\left(1-{\varphi}_{Ti{O}_{2}}\right)\left\{\left(1-{\varphi}_{CoF{e}_{2}{O}_{4}}\right)+{\varphi}_{CoF{e}_{2}{O}_{4}}\frac{{\left(\rho cp\right)}_{CoF{e}_{2}{O}_{4}}}{{\left(\rho cp\right)}_{f}}\right\}+{\varphi}_{Ti{O}_{2}}\frac{{\left(\rho cp\right)}_{Ti{O}_{2}}}{{\left(\rho cp\right)}_{f}}\right]\}$ |

Thermal conduction | $\begin{array}{l}\frac{{k}_{Thnf}}{{k}_{hnf}}=\left(\frac{{k}_{CoF{e}_{2}{O}_{4}}+2{k}_{hnf}-2{\varphi}_{CoF{e}_{2}{O}_{4}}\left({k}_{hnf}-{k}_{CoF{e}_{2}{O}_{4}}\right)}{{k}_{CoF{e}_{2}{O}_{4}}+2{k}_{hnf}+{\varphi}_{CoF{e}_{2}{O}_{4}}\left({k}_{hnf}-{k}_{CoF{e}_{2}{O}_{4}}\right)}\right),\frac{{k}_{hnf}}{{k}_{nf}}=\left(\frac{{k}_{Ti{O}_{2}}+2{k}_{nf}-2{\varphi}_{Ti{O}_{2}}\left({k}_{nf}-{k}_{Ti{O}_{2}}\right)}{{k}_{Ti{O}_{2}}+2{k}_{nf}+{\varphi}_{Ti{O}_{2}}\left({k}_{nf}-{k}_{Ti{O}_{2}}\right)}\right),\\ \frac{{k}_{nf}}{{k}_{f}}=\left(\frac{{k}_{MgO}+2{k}_{f}-2{\varphi}_{MgO}\left({k}_{f}-{k}_{MgO}\right)}{{k}_{MgO}+2{k}_{f}+{\varphi}_{MgO}\left({k}_{f}-{k}_{MgO}\right)}\right),\end{array}\}$ |

Electrical conductivity | $\frac{{\sigma}_{Thnf}}{{\sigma}_{hnf}}\left(1+\frac{3\left(\frac{{\sigma}_{CoF{e}_{2}{O}_{4}}}{{\sigma}_{hnf}}-1\right){\varphi}_{CoF{e}_{2}{O}_{4}}}{\left(\frac{{\sigma}_{CoF{e}_{2}{O}_{4}}}{{\sigma}_{hnf}}+2\right)-\left(\frac{{\sigma}_{CoF{e}_{2}{O}_{4}}}{{\sigma}_{hnf}}-1\right){\varphi}_{CoF{e}_{2}{O}_{4}}}\right)\begin{array}{l},\hspace{0.17em}\frac{{\sigma}_{hnf}}{{\sigma}_{nf}}=\left(1+\frac{3\left(\frac{{\sigma}_{Ti{O}_{2}}}{{\sigma}_{nf}}-1\right){\varphi}_{Ti{O}_{2}}}{\left(\frac{{\sigma}_{Ti{O}_{2}}}{{\sigma}_{nf}}+2\right)-\left(\frac{{\sigma}_{Ti{O}_{2}}}{{\sigma}_{nf}}-1\right){\varphi}_{Ti{O}_{2}}}\right)\\ ,\frac{{\sigma}_{nf}}{{\sigma}_{f}}=\left(1+\frac{3\left(\frac{{\sigma}_{MgO}}{{\sigma}_{f}}-1\right){\varphi}_{MgO}}{\left(\frac{{\sigma}_{MgO}}{{\sigma}_{f}}+2\right)-\left(\frac{{\sigma}_{MgO}}{{\sigma}_{f}}-1\right){\varphi}_{MgO}}\hspace{0.17em}\right)\end{array}\}.$ |

- Case 1: Wedge $\to {n}_{2}=0$ and $\gamma \ne 0;$
- Case 2: Cone $\to {n}_{2}=1$ and $\gamma \ne 0;$
- Case 3: Plate $\to {n}_{2}=0$ and $\gamma =0$.

_{1}is the slip parameter of velocity defined as follows:

- 4.
- Numerical Solution

**Step 1:**Simplifying the BVP to the 1st order

**Step 2: Introducing parameter p:**

**Step 3: Applying Cauchy Principal and Discretized Equations (19)–(21):**

- 5.
- Results and Discussion

**:**

_{,}and nanoparticle volume friction $\varphi =\left({\varphi}_{1},\hspace{0.17em}{\varphi}_{2},\hspace{0.17em}{\varphi}_{3}\right)$, respectively. The velocity field was dramatically reduced by the effect of the magnetic field, Darcy–Forchheimer term, porosity term, and NP volume fraction, while augments with the positive variation of thermal Grashof number. Physically, the resistive force (Lorentz force) opposes the flow field which causes the decline of velocity contour. That repellant force is generated due to the effects of the magnetic flux as shown in Figure 2. Similarly, the rising values of Darcy–Forchheimer and porosity constraints enhance the surface permeability, which results in the deceleration of velocity outlines ${f}^{\prime}\left(\eta \right)$, as displayed in Figure 3 and Figure 4, respectively. The inclusion of ternary NPs to the base fluid magnified its viscosity and density, and created a hurdle in the flow field, as shown in Figure 5. The variation in the thermal Grashof number reduces the stretching velocity of cone, wedge, and plate, and diminishes the kinetic viscosity, which provides a suitable platform for flow field ${f}^{\prime}\left(\eta \right)$ to move fast, as elaborated in Figure 6. Figure 7 shows the velocity outlines of ternary nanoliquid drops with the rising effect of velocity slip parameter. Figure 8 shows a relative comparison of the published literature (Rekha et al. [43]) with the present outcomes. The present results are accurate and reliable.

_{2}, CoFe

_{2}O

_{4}, and MgO) improved the viscosity of the trihybrid nanoliquid, which also improved the heat-absorbing capacity of the fluid; such a scenario was noticed in the energy field. Because the nanofluid absorbed more heat, the fluid temperature was kept normal. This property of the ternary nanomaterials renders them more efficient for industrial and biomedical applications. Figure 12 expresses the relative comparison of the published literature (Rekha et al. [39]) with the present outcomes for accuracy and validity purposes.

_{2}, CoFe

_{2}O

_{4}, and MgO. Table 2 reports the arithmetic valuation of the present work with the published literature to confirm the authenticity of the current study. Table 3 and Table 4 show the statistical valuations of ternary hybrid NF for skin friction ${f}^{\u2033}(0)$, energy transmission ${\theta}^{\prime}(0)$

_{,}and mass transfer rate ${\phi}^{\prime}\left(0\right)$ over cone, wedge, and plate, respectively. The velocity and energy transmission over the cone were more effective than those over the wedge and plate.

## 3. Conclusions

_{2}, CoFe

_{2}O

_{4}, and MgO NPs through a wedge, cone, and plate. A mathematical model was created with the objective to optimize the energy and mass transfer rates, and efficiency for a variety of commercial and medical functions. The phenomena were expressed as a nonlinear system of PDEs, which were reduced to a system of dimensionless ODEs through similarity replacement. The obtained set of differential equations was solved using the PCM technique. The following are the main findings from the above assessment:

- The velocity field was dramatically reduced due to the influence of the magnetic field, the Darcy–Forchheimer term, porosity term, and NPs volume fraction, while it was augmented with the positive variation of thermal Grashof number.
- The heat energy profile was boosted under the effects of a magnetic field and heat source.
- The addition of nanoparticles (TiO
_{2}, CoFe_{2}O_{4}and MgO) to the water reduced the energy distribution. - The mass transfer $\phi \left(\eta \right)$ profile was reduced with the upshot of the chemical reaction rate and Schmidt number, while it was boosted with the increment of activation energy.
- The velocity and energy propagation rates over a cone surface were greater than those of the wedge and plate versus the variation in Grashof number, porosity effect, and heat source.
- The mass transfer ratio under the impact of chemical reaction and activation over a wedge surface was higher than that of a plate.
- The inclusion of ternary nanoparticles to the base fluid is significantly efficient for industrial and biomedical applications.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Magnetic parameter M effect on velocity ${f}^{\prime}\left(\eta \right)$, where $Fr=0.5,$ $Gr=0.1,$ $\lambda =0.5,$ $M=1.0,$ ${L}_{1}=0,$ $Hs=0.1,$ $\varphi =0.01,$ $Rc=0.4$ and $Sc=0.1$.

**Figure 3.**Darcy–Forchheimer parameter Fr effect on velocity ${f}^{\prime}\left(\eta \right)$, where $M=1.0,$ $Gr=0.1,$ $\lambda =0.5,$ ${L}_{1}=0,$ $M=1.0,$ $Hs=0.1,$ $\varphi =0.01,$ $Rc=0.4$ and $Sc=0.1$.

**Figure 4.**Thermal Grashof number Gr effect on velocity ${f}^{\prime}\left(\eta \right)$, where $Fr=0.5,$ $M=1.0,$ $\lambda =0.5,$ $M=1.0,$ ${L}_{1}=0,$ $Hs=0.1,$ $\varphi =0.01,$ $Rc=0.4$ and $Sc=0.1$.

**Figure 5.**Porosity variable $\lambda $ effect on velocity ${f}^{\prime}\left(\eta \right)$, where $Fr=0.5,$ $M=1.0,$ $Gr=0.1,$ $M=1.0,$ $Hs=0.1,$ $\varphi =0.01,$ ${L}_{1}=0,$ $Rc=0.4$ and $Sc=0.1$.

**Figure 6.**Nanoparticle volume friction $\varphi $ effect on velocity ${f}^{\prime}\left(\eta \right)$, where $Fr=0.5,$ $M=1.0,$ $Gr=0.1,$ $\lambda =0.5,$ ${L}_{1}=0,$ $M=1.0,$ $Hs=0.1,$ $Rc=0.4$ and $Sc=0.1$.

**Figure 7.**Velocity slip parameter ${L}_{1}$ effect on velocity ${f}^{\prime}\left(\eta \right)$ profile, where $Fr=0.5,$ $M=1.0,$ $Gr=0.1,$ $\lambda =0.5,$ $M=1.0,$ $Hs=0.1,$ $Rc=0.4$ and $Sc=0.1$.

**Figure 8.**Comparison of published work [39] with the current results.

**Figure 9.**Magnetic term M upshot on energy contour $\theta \left(\eta \right),$ where $Fr=0.5,$ $Gr=0.1,$ $\lambda =0.5,$ $M=1.0,$ $Hs=0.1,$ $\varphi =0.01,$ $Rc=0.4$ and $Sc=0.1$.

**Figure 10.**Heat source variable M upshot on temperature $\theta \left(\eta \right),$ where $Fr=0.5,$ $M=1.0,$ $Gr=0.1,$ $\lambda =0.5,$ $M=1.0,$ $\varphi =0.01,$ $Rc=0.4$ and $Sc=0.1$.

**Figure 11.**Volume friction of the nanoparticle $\varphi $ effect on temperature $\theta \left(\eta \right),$ where $Fr=0.5,$ $M=1.0,$ $Gr=0.1,$ $\lambda =0.5,$ $M=1.0,$ $Hs=0.1,$ $Rc=0.4$ and $Sc=0.1$.

**Figure 12.**Comparison of published work [39] with the current results.

**Figure 13.**Chemical reaction rate Rc effect on concentration $\phi \left(\eta \right),$ where $Fr=0.5,$ $M=1.0,$ $Gr=0.1,$ $\lambda =0.5,$ $M=1.0,$ $Hs=0.1,$ $\varphi =0.01,$ and $Sc=0.1$.

**Figure 14.**Schmidt number Sc effect on concentration $\phi \left(\eta \right)$, where $Fr=0.5,$ $M=1.0,$ $Gr=0.1,$ $\lambda =0.5,$ $M=1.0,$ $Hs=0.1,$ $\varphi =0.01,$ and $Rc=0.4$.

**Figure 15.**Activation energy parameter E effect on concentration $\phi \left(\eta \right)$, where $Fr=0.5$, $M=1.0,$ $Gr=0.1,$ $\lambda =0.5,$ $M=1.0,$ $Hs=0.1,$ $\varphi =0.01,$ $Rc=0.4$ and $Sc=0.1$.

Base Fluid and Nanoparticles $\mathit{\varphi}=\left({\mathit{\varphi}}_{\mathbf{1}}={\mathit{\varphi}}_{\mathbf{2}}={\mathit{\varphi}}_{\mathbf{3}}\right)$ | $\mathit{\rho}(\mathbf{k}\mathbf{g}/{\mathbf{m}}^{\mathbf{3}})$ | $\mathit{k}(\mathbf{W}/\mathbf{m}\mathbf{K})$ | $\mathit{C}\mathit{p}(\mathit{j}/\mathit{k}\mathit{g}\mathbf{K})$ | $\mathit{\sigma}(\mathit{S}/\mathbf{m})$ |
---|---|---|---|---|

Pure water (H_{2}O) | 997.1 | 0.613 | 4179 | 0.05 |

$Cobalt\text{}ferrite\hspace{0.17em}{\varphi}_{1}={\varphi}_{CoF{e}_{2}{O}_{4}}$ | 4907 | 3.7 | 700 | $5.51\times {10}^{9}$ |

$Titanium\text{}dioxide\hspace{0.17em}{\varphi}_{2}={\varphi}_{Ti{O}_{2}}$ | 4250 | 8.9538 | 686.2 | $2.38\times {10}^{6}$ |

$Magnesium\text{}oxide\hspace{0.17em}{\varphi}_{2}={\varphi}_{MgO}$ | 3560 | 45 | 955 | $1.42\times {10}^{-3}$ |

**Table 2.**Statistical comparison with the existing literature for numerical outputs of $-{f}^{\u2033}(0)$.

Parameter | Kameswaran et al. [48] | Rekha et al. [39] | Present Work | |
---|---|---|---|---|

$\lambda $ | Analytical | Numerical | RKF-45 | PCM |

0.5 | 1.22464487 | 1.22464487 | 1.224657521 | 1.224758432 |

1.0 | 1.41411356 | 1.41411356 | 1.414116330 | 1.414217254 |

1.5 | 1.58103883 | 1.58103883 | 1.581038786 | 1.591139677 |

2.0 | 1.73215081 | 1.73215081 | 1.732150762 | 1.812052855 |

5.0 | 2.44938974 | 2.44938974 | 2.449389673 | 2.559489884 |

**Table 3.**Numerical outputs for ${f}^{\u2033}(0)$ and ${\theta}^{\prime}(0)$ using numerous constraints for the cone.

Parameters | Cone | Wedge | Plate | |||||
---|---|---|---|---|---|---|---|---|

$\mathit{G}\mathit{r}$ | $\mathit{\lambda}$ | $\mathit{H}\mathit{s}$ | ${\mathit{f}}^{\u2033}(0)$ | ${\mathit{\theta}}^{\prime}(0)$ | ${\mathit{f}}^{\u2033}(0)$ | ${\mathit{\theta}}^{\prime}(0)$ | ${\mathit{f}}^{\u2033}(0)$ | |

1.0 | 1.0 | 1.0 | 1.310429 | 1.762376 | 0.575218 | 1.183103 | 1.190889 | 1.110755 |

5.0 | 0.675524 | 1.832668 | 0.138028 | 1.273024 | 0.138028 | 1.273024 | ||

10 | 0.365352 | 2.104684 | 1.138079 | 1.080946 | 1.017223 | 1.401200 | ||

1.0 | 1.310429 | 1.762376 | 1.301688 | 1.038794 | 1.190889 | 1.110755 | ||

1.5 | 1.462260 | 1.727821 | 1.450276 | 1.199999 | 1.156682 | 1.070569 | ||

2.0 | 1.602108 | 1.695784 | 1.196383 | 1.862916 | 1.307562 | 1.033601 | ||

0.3 | 1.342571 | 2.321712 | 1.176035 | 1.247111 | 1.075516 | 1.872300 | ||

0.0 | 1.332838 | 1.859068 | 1.133634 | 0.259555 | 1.036729 | 1.267148 | ||

0.3 | 1.318364 | 1.275103 | 1.138079 | 1.080946 | 1.160646 | 0.332229 |

**Table 4.**Numerical outputs for ${\phi}^{\prime}\left(0\right)$ using numerous constraints for the wedge and plate.

Parameters | Wedge | Plate | ||||||
---|---|---|---|---|---|---|---|---|

${\mathit{\phi}}^{\prime}\left(\mathbf{0}\right)$ | ${\mathit{\phi}}^{\prime}\left(\mathbf{0}\right)$ | |||||||

$\mathit{E}$ | $\mathit{R}\mathit{c}$ | $\mathit{\delta}$ | $\begin{array}{l}{\mathit{\varphi}}_{\mathbf{1}}=\mathbf{0.01}\\ {\mathit{\varphi}}_{\mathbf{2}}={\mathit{\varphi}}_{\mathbf{3}}=\mathbf{0}\end{array}$ | $\begin{array}{l}{\mathit{\varphi}}_{2}=0.01\\ {\mathit{\varphi}}_{1}={\mathit{\varphi}}_{3}=0\end{array}$ | $\begin{array}{l}{\mathit{\varphi}}_{3}=0.01\\ {\mathit{\varphi}}_{1}={\mathit{\varphi}}_{2}=0\end{array}$ | $\begin{array}{l}{\mathit{\varphi}}_{1}=0.01\\ {\mathit{\varphi}}_{2}={\mathit{\varphi}}_{3}=0\end{array}$ | $\begin{array}{l}{\mathit{\varphi}}_{2}=0.01\\ {\mathit{\varphi}}_{1}={\mathit{\varphi}}_{3}=0\end{array}$ | $\begin{array}{l}{\mathit{\varphi}}_{3}=0.01\\ {\mathit{\varphi}}_{1}={\mathit{\varphi}}_{2}=0\end{array}$ |

0.5 | 0.1 | 0.1 | 0.713197 | 0.795709 | 0.563422 | 0.562321 | 0.572317 | 0.571158 |

1.0 | 0.632772 | 1.001334 | 0.470259 | 0.468917 | 0.480954 | 0.479526 | ||

1.5 | 0.573798 | 1.167276 | 0.400317 | 0.398767 | 0.412233 | 0.410574 | ||

0.1 | 0.554349 | 0.552038 | 0.376497 | 0.374928 | 0.386797 | 0.385122 | ||

0.3 | 0.692406 | 0.690695 | 0.539073 | 0.437966 | 0.547308 | 0.546136 | ||

0.5 | 0.797092 | 0.795682 | 0.658107 | 0.657217 | 0.665194 | 0.864259 | ||

0.1 | 0.797118 | 0.795709 | 0.658127 | 0.657238 | 0.665218 | 0.864283 | ||

0.2 | 0.797702 | 0.796272 | 0.659114 | 0.658198 | 0.666725 | 0.865766 | ||

0.3 | 0.798183 | 0.796734 | 0.659967 | 0.659027 | 0.668102 | 0.867122 |

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## Share and Cite

**MDPI and ACS Style**

Bilal, M.; Ullah, I.; Alam, M.M.; Weera, W.; Galal, A.M.
Numerical Simulations through PCM for the Dynamics of Thermal Enhancement in Ternary MHD Hybrid Nanofluid Flow over Plane Sheet, Cone, and Wedge. *Symmetry* **2022**, *14*, 2419.
https://doi.org/10.3390/sym14112419

**AMA Style**

Bilal M, Ullah I, Alam MM, Weera W, Galal AM.
Numerical Simulations through PCM for the Dynamics of Thermal Enhancement in Ternary MHD Hybrid Nanofluid Flow over Plane Sheet, Cone, and Wedge. *Symmetry*. 2022; 14(11):2419.
https://doi.org/10.3390/sym14112419

**Chicago/Turabian Style**

Bilal, Muhammad, Ikram Ullah, Mohammad Mahtab Alam, Wajaree Weera, and Ahmed M. Galal.
2022. "Numerical Simulations through PCM for the Dynamics of Thermal Enhancement in Ternary MHD Hybrid Nanofluid Flow over Plane Sheet, Cone, and Wedge" *Symmetry* 14, no. 11: 2419.
https://doi.org/10.3390/sym14112419