# Computational Study of MHD Darcy–Forchheimer Hybrid Nanofluid Flow under the Influence of Chemical Reaction and Activation Energy over a Stretching Surface

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## Abstract

**:**

_{2}O

_{3}) nanoparticles (nps) over an extended surface has been reported. The thermal and velocity slip conditions are also considered. Such a type of physical problems mostly occurs in symmetrical phenomena and are applicable in physics, engineering, applied mathematics, and computer science. For desired outputs, the fluid flow has been studied under the consequences of the Darcy effect, thermophoresis diffusion and Brownian motion, heat absorption, viscous dissipation, and thermal radiation. An inclined magnetic field is applied to fluid flow to regulate the flow stream. Hybrid nanofluid is created by the dispensation of Cu and Al

_{2}O

_{3}nps in the base fluid (water). For this purpose, the flow dynamics have been designed as a system of nonlinear PDEs, which are simplified to a system of dimensionless ODEs through resemblance substitution. The parametric continuation method is used to resolve the obtained set of dimensionless differential equations. It has been noticed that the consequences of heat absorption and thermal radiation boost the energy transmission rate; however, the effect of suction constraint and Darcy–Forchhemier significantly diminished the heat transference rate of hybrid nanofluids. Furthermore, the dispersion of Cu and Al

_{2}O

_{3}nps in the base fluid remarkably magnifies the velocity and energy transmission rate.

## 1. Introduction

_{2}O

_{3}) nano particulates, where the viscosity and thermal conductivity vary non-uniformly with the volume percentage [20], studied the convective flow and energy transmission of various nanofluids. With a rise in Cu nanocomposites, the streams of Cu-C2H6O (copper-ethylene glycol) base nanoliquids showed the best improvement in the heat transference rate. Al-Mubaddel et al. [21] examined a hybrid ferrofluid flow through a heated, irregular, extensible cylinder at a 3D stagnation point with the impact of slip parameters and variable diameter. Alsallami et al. [22] have performed a numerical analysis of the nanoliquid flow under the effects of thermophoresis, Brownian motion, and thermal radiation on a warmed revolving surface. It was thought that the radiation and Prandtl number impact increased the frequency of heat transfer. Shahsavar et al. [23] considered the impact of magnetic flux and the Hall effect on hybrid nanoliquid flow on a gyrating disk’s top. Their goal was to upsurge the efficiency of energy transport for use in engineering and industry. Ashraf et al. [24] used the generalized differential quadrature methodology to quantitatively investigate the peristaltic transport of magnetite Fe

_{3}O

_{4}nanocrystals in blood-based ferrofluid.

_{2}O

_{3}nanoparticles over an extending vertical surface with thermal and partial slip conditions. No such study and analysis have been found in the literature. The fluid flow has been studied under the consequences of the Darcy effect, thermophoresis diffusion and Brownian motion, heat absorption, viscous dissipation, and thermal radiation. An inclined magnetic field is applied to fluid flow to regulate the flow stream. Hybrid nanofluid is synthesized by the dispensation of Cu and Al

_{2}O

_{3}nps in the blood. The PCM approach is used to resolve the obtained set of dimensionless differential equations.

## 2. Mathematical Formulation

_{2}O

_{3}nps are dispersed into the base fluid for hybrid composition. ${C}_{\lambda}$ and ${T}_{\lambda}$ present the mass and energy of nanofluid, while ${C}_{\infty}$ and ${T}_{\infty}$ show the surrounding mass and energy. The basic equations are written as [33]:

_{2}O

_{3,}and blood are described in Table 1. In Table 2, the bf and hnf characterize the base fluid and hybrid nanofluid. ${\varphi}_{A{l}_{2}{O}_{3}}\hspace{0.17em}\mathrm{and}\hspace{0.17em}{\varphi}_{Cu}$ represent the nano particulates volume fraction. ${\sigma}_{hnf}$ is the electrical conductivity, ${(\rho {C}_{p})}_{hnf}$ is the heat capacitance, and ${k}_{hnf}$ is the thermal conductivity of the hybrid nanoliquid.

## 3. Numerical Solution

**Step 1:**Convert BVP to first order

**Step 2:**Introducing parameter p:

**Step 3:**Apply Cauchy Principal and Discretized Equations (23)–(25).

## 4. Results and Discussion

- Velocity Profile $\left({f}^{\prime}\left(\eta \right)\right)$:

- Energy Profile $\Theta \left(\eta \right)$:

- Concentration Profile:

_{2}O

_{3}) and hybrid nanoliquid (Cu + Al

_{2}O

_{3}) for both energy and velocity profiles. It can be observed that as compared to nanofluids, the hybrid nanofluid has a remarkable capability of energy transmission.

## 5. Conclusions

_{2}O

_{3}nps over an extended surface. The thermal and velocity slip conditions are also considered. Hybrid nanofluid is created by the inclusion of Cu and Al

_{2}O

_{3}nps in the water. The key findings are:

- The velocity contour ${f}^{\prime}\left(\eta \right)$ of a hybrid nanofluid is diminished by enriching the magnetic field’s angle and effect while boosting the energy profile $\Theta \left(\eta \right).$
- The velocity and energy profiles enhance by the intensification of injection constraint, while the Darcy–Forchheimer impact and suction term have an opposite influence on both outlines.
- The thermal profile of hybrid nanofluid increased by rising the thermal effects, heat source, and viscous dissipation, while diminished by varying the thermal slip factor.
- Heat absorption and thermal radiation contributions boost the energy transmission rate, and the thermal slip factor increases the thermal profile.
- The dispersion of Cu and Al
_{2}O_{3}nps in the base fluid remarkably magnifies the velocity and energy transmission rate, which is the most tremendous property of Cu and Al_{2}O_{3}nps, used for biomedical and industrial applications.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

$x,\hspace{0.17em}y$ | Cartesian coordinate (m) | $u,\hspace{0.17em}v$ | Components of velocity |

T | Temperature (k) | ${\epsilon}_{1}\left(x\right)$ | Stretching velocity $\left(m{s}^{-1}\right)$ |

${T}_{1}\left(x\right)$ | Exponential temperature | $\theta $ | Angle $\left(rd\right)$ |

C | Slip velocity factor | $\delta $ | Thermal slip factor |

${H}_{0}$ | Heat source $\left(J\right)$ | ${\epsilon}_{2}\left(x\right)$ | Suction velocity |

$\nu $ | Kinematic viscosity $\left({m}^{2}{s}^{-1}\right)$ | $\mu $ | Dynamic viscosity $\left(kg{m}^{-1}{s}^{-1}\right)$ |

$\rho $ | Density $\left(kg/{m}^{3}\right)$ | $\sigma $ | Electrical conductivity $\left(S/{m}^{-1}\right)$ |

${B}_{0}$ | Magnetic field $\left(A/{m}^{-1}\right)$ | $\alpha $ | Thermal diffusivity $\left({m}^{2}{s}^{-1}\right)$ |

$\left(\rho c\right)$ | Heat capacity $\left(Jk{g}^{-1}{k}^{-1}\right)$ | $k$ | Thermal conductivity $\left(w{m}^{-1}{k}^{-1}\right)$ |

$\zeta $ | Porosity parameter | ${T}_{\infty}$ | Ambient temperature $\left(k\right)$ |

Fr | Forchhemier term | Gr | Grashof number |

S | Suction/injection | d | Thermal slip |

L | Velocity slip | Ec | Eckert number |

Pr | Prandtl number | Re | Reynold number |

M | Magnetic factor | Ra | Radiation factor |

Ha | Heat absorption | f | Velocity profile |

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**Figure 2.**Velocity profile versus the upshot of (

**a**) Forchhemier number Fr, (

**b**) magnetic field M, (

**c**) suction effect +S, and (

**d**) injection −S.

**Figure 3.**Energy profile versus the upshot of (

**a**) suction +S, (

**b**) injection −S, (

**c**) thermophoresis Nt, and (

**d**) Brownian motion Nb.

**Figure 4.**Energy profile versus the upshot of (

**a**) magnetic field, (

**b**) heat source term, (

**c**) Eckert number, and (

**d**) thermal slip factor d.

**Figure 5.**Mass profile versus the upshot of (

**a**) Schmidt number Sc, (

**b**) chemical reaction Kr, (

**c**) thermophoresis Nt, (

**d**) Brownian motion Nb, and (

**e**) activation energy E.

$\mathit{\rho}(\mathbf{k}\mathbf{g}/{\mathbf{m}}^{3})$ | ${\mathit{C}}_{\mathit{p}}(\mathbf{j}/\mathbf{k}\mathbf{g}\mathbf{K})$ | $\mathit{k}(\mathbf{W}/\mathbf{m}\mathbf{K})$ | $\mathit{\sigma}\left(\mathbf{S}/\mathbf{m}\right)$ | |
---|---|---|---|---|

Water | 997.1 | 4179 | 0.613 | - |

$\mathrm{C}\mathrm{u}$ | 8933 | 385 | 401 | $59.6\times {10}^{6}$ |

$\mathrm{A}{\mathrm{l}}_{2}{\mathrm{O}}_{3}$ | 4907 | 765 | 40 | $35\times {10}^{6}$ |

**Table 2.**The mathematical model for the hybrid nanoliquid $\left({\varphi}_{1}={\varphi}_{Cu},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\varphi}_{2}={\varphi}_{A{l}_{2}{O}_{3}}\right)$ [35].

Properties | Models |
---|---|

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

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

Thermal Capacity | $\frac{{(\rho {C}_{p})}_{hnf}}{{(\rho {C}_{p})}_{bf}}={\varphi}_{Cu}\left(\frac{{(\rho {C}_{p})}_{Cu}}{{(\rho {C}_{p})}_{bf}}\right)+{\varphi}_{A{l}_{2}{O}_{3}}\left(\frac{{(\rho {C}_{p})}_{A{l}_{2}{O}_{3}}}{{(\rho {C}_{p})}_{bf}}\right)+\left(1-{\varphi}_{Cu}-{\varphi}_{A{l}_{2}{O}_{3}}\right)$ |

Thermal Expansion | $\frac{{(\rho {\beta}_{T})}_{hnf}}{{(\rho {\beta}_{T})}_{bf}}={\varphi}_{Cu}\left(\frac{{(\rho {\beta}_{T})}_{Cu}}{{(\rho {\beta}_{T})}_{bf}}\right)+{\varphi}_{A{l}_{2}{O}_{3}}\left(\frac{{(\rho {\beta}_{T})}_{A{l}_{2}{O}_{3}}}{{(\rho {\beta}_{T})}_{bf}}\right)+\left(1-{\varphi}_{Cu}-{\varphi}_{A{l}_{2}{O}_{3}}\right)$ |

Thermal Conductivity | $\frac{{k}_{hnf}}{{k}_{bf}}=\left[\frac{\left(\frac{{\varphi}_{Cu}{k}_{Cu}+{\varphi}_{A{l}_{2}{O}_{3}}{k}_{A{l}_{2}{O}_{3}}}{{\varphi}_{Cu}+{\varphi}_{A{l}_{2}{O}_{3}}}\right)+2{k}_{bf}+2\left({\varphi}_{Cu}{k}_{Cu}+{\varphi}_{A{l}_{2}{O}_{3}}{k}_{A{l}_{2}{O}_{3}}\right)-2\left({\varphi}_{Cu}+{\varphi}_{A{l}_{2}{O}_{3}}\right){k}_{bf}}{\left(\frac{{\varphi}_{Cu}{k}_{Cu}+{\varphi}_{A{l}_{2}{O}_{3}}{k}_{A{l}_{2}{O}_{3}}}{{\varphi}_{Cu}+{\varphi}_{A{l}_{2}{O}_{3}}}\right)+2{k}_{bf}-2\left({k}_{Cu}{\varphi}_{Cu}+{k}_{A{l}_{2}{O}_{3}}{\varphi}_{A{l}_{2}{O}_{3}}\right)+2\left({\varphi}_{Cu}+{\varphi}_{A{l}_{2}{O}_{3}}\right){k}_{bf}}\right]$ |

Electrical Conductivity | $\frac{{\sigma}_{hnf}}{{\sigma}_{bf}}=\left[\frac{\left(\frac{{\varphi}_{Cu}{\sigma}_{Cu}+{\sigma}_{A{l}_{2}{O}_{3}\hspace{0.17em}}{\varphi}_{A{l}_{2}{O}_{3}}}{{\varphi}_{CoF{e}_{2}{O}_{4}}+{\varphi}_{Cu}}\right)+2{\sigma}_{bf}+2\left({\varphi}_{Cu}{\sigma}_{Cu}+{\varphi}_{A{l}_{2}{O}_{3}}{\sigma}_{A{l}_{2}{O}_{3}}\right)-2\left({\varphi}_{Cu}+{\varphi}_{A{l}_{2}{O}_{3}}\right){\sigma}_{bf}}{\left(\frac{{\varphi}_{Cu}{\sigma}_{Cu}+{\varphi}_{A{l}_{2}{O}_{3}}{\sigma}_{A{l}_{2}{O}_{3}}}{{\varphi}_{Cu}+{\varphi}_{A{l}_{2}{O}_{3}}}\right)+2{\sigma}_{bf}-\left({\varphi}_{Cu}{\sigma}_{Cu}+{\varphi}_{A{l}_{2}{O}_{3}}{\sigma}_{A{l}_{2}{O}_{3}}\right)+\left({\varphi}_{Cu}+{\varphi}_{A{l}_{2}{O}_{3}}\right){\sigma}_{bf}}\right]$ |

Parameters | $\mathit{S}{\mathit{f}}_{\mathit{x}}\mathbf{for}\mathit{L}=0$ | $\mathit{S}{\mathit{f}}_{\mathit{x}}\mathbf{for}\mathit{L}=0.4$ | ||
---|---|---|---|---|

M | Gr | $\mathrm{C}\mathrm{u}+\mathrm{A}{\mathrm{l}}_{2}{\mathrm{O}}_{3}$ | Cu | $\mathrm{C}\mathrm{u}+\mathrm{A}{\mathrm{l}}_{2}{\mathrm{O}}_{3}$ |

0.2 | 0.1 | 1.136841 | 1.004931 | 0.576080 |

0.4 | 1.189545 | 1.048340 | 0.649375 | |

0.6 | 1.173506 | 1.219533 | 0.714397 | |

0.8 | 1.395022 | 1.225452 | 0.832752 | |

0.2 | 0.1 | 1.136841 | 1.00493 | 0.576070 |

0.5 | 1.209421 | 1.135243 | 0.607352 | |

0.7 | 1.298434 | 1.207564 | 0.653641 | |

0.9 | 1.370320 | 1.319647 | 0.787528 |

**Table 4.**Numerical outputs of Nusselt number $N{u}_{x}$ for slip, $L=0.4$, and no-slip, $L=0$, conditions.

Parameters | $\mathit{N}{\mathit{u}}_{\mathit{x}}\mathbf{for}\mathit{L}=0$ | $\mathit{N}{\mathit{u}}_{\mathit{x}}\mathbf{for}\mathit{L}=0.4$ | ||
---|---|---|---|---|

Nt | Nb | $\mathrm{C}\mathrm{u}+\mathrm{A}{\mathrm{l}}_{2}{\mathrm{O}}_{3}$ | Cu | $\mathrm{C}\mathrm{u}+\mathrm{A}{\mathrm{l}}_{2}{\mathrm{O}}_{3}$ |

0.1 | 0.2 | 1.358318 | 1.327615 | 1.433756 |

0.2 | 1.285501 | 1.073557 | 1.386492 | |

0.3 | 0.784536 | 0.574744 | 0.874617 | |

0.4 | 0.567454 | 0.539545 | 0.612644 | |

0.1 | 0.2 | 1.358318 | 1.327615 | 1.433756 |

0.4 | 1.164950 | 1.009444 | 1.294688 | |

0.6 | 1.068312 | 1.007518 | 1.173574 | |

0.8 | 0.863593 | 0.719485 | 1.021634 |

**Table 5.**Comparative analysis of Sherwood number $\left(S{h}_{x}\right)$ for no-slip $\left(L=0\right)$ conditions between PCM and bvp4c package.

Parameters | bvp4c | PCM | |||
---|---|---|---|---|---|

Sc | Kr | $\mathrm{C}\mathrm{u}+\mathrm{A}{\mathrm{l}}_{2}{\mathrm{O}}_{3}$ | $\mathrm{C}\mathrm{u}$ | $\mathrm{C}\mathrm{u}+\mathrm{A}{\mathrm{l}}_{2}{\mathrm{O}}_{3}$ | $\mathrm{C}\mathrm{u}$ |

0.3 | 0.4 | 2.596400 | 2.395722 | 2.595512 | 2.396730 |

0.6 | 2.638758 | 2.485473 | 2.647455 | 2.486474 | |

0.9 | 2.784748 | 2.684288 | 2.774682 | 2.685287 | |

1.2 | 3.052195 | 2.806367 | 3.050756 | 2.807366 | |

0.3 | 0.4 | 2.596400 | 2.395722 | 2.595512 | 2.396721 |

0.8 | 2.707473 | 2.695367 | 2.735074 | 2.697366 | |

1.2 | 2.875394 | 2.799536 | 3.878335 | 2.887535 | |

1.6 | 3.357847 | 3.064456 | 3.346754 | 3.065455 |

Parameters | $-\mathit{S}{\mathit{f}}_{\mathit{x}}$ (Ref. [33]) | $-\mathit{S}{\mathit{f}}_{\mathit{x}}(\mathbf{Present}\mathbf{Work})$ | |||
---|---|---|---|---|---|

M | $\theta $ | $L=0$ | $L=0.5$ | $L=0$ | $L=0.5$ |

0.2 | $\pi /3$ | 0.984548 | 0.566347 | 0.984559 | 0.566368 |

1.2 | 1.117289 | 0.626290 | 1.117297 | 0.626381 | |

2.2 | 1.235613 | 0.674850 | 1.235622 | 0.674862 | |

3.2 | 1.343110 | 0.715361 | 1.343131 | 0.715373 | |

$\pi /6$ | 1.057414 | 0.599995 | 1.057437 | 0.599999 | |

$\pi /4$ | 1.150075 | 0.640188 | 1.150082 | 0.640194 | |

$\pi /3$ | 1.235613 | 0.674850 | 1.235634 | 0.674861 | |

$\pi /3$ | 1.315349 | 0.705207 | 1.315357 | 0.705226 |

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**MDPI and ACS Style**

Haq, I.; Yassen, M.F.; Ghoneim, M.E.; Bilal, M.; Ali, A.; Weera, W.
Computational Study of MHD Darcy–Forchheimer Hybrid Nanofluid Flow under the Influence of Chemical Reaction and Activation Energy over a Stretching Surface. *Symmetry* **2022**, *14*, 1759.
https://doi.org/10.3390/sym14091759

**AMA Style**

Haq I, Yassen MF, Ghoneim ME, Bilal M, Ali A, Weera W.
Computational Study of MHD Darcy–Forchheimer Hybrid Nanofluid Flow under the Influence of Chemical Reaction and Activation Energy over a Stretching Surface. *Symmetry*. 2022; 14(9):1759.
https://doi.org/10.3390/sym14091759

**Chicago/Turabian Style**

Haq, Izharul, Mansour F. Yassen, Mohamed E. Ghoneim, Muhammad Bilal, Aatif Ali, and Wajaree Weera.
2022. "Computational Study of MHD Darcy–Forchheimer Hybrid Nanofluid Flow under the Influence of Chemical Reaction and Activation Energy over a Stretching Surface" *Symmetry* 14, no. 9: 1759.
https://doi.org/10.3390/sym14091759