# Bioconvection in Cross Nano-Materials with Magnetic Dipole Impacted by Activation Energy, Thermal Radiation, and Second Order Slip

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## Abstract

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## 1. Introduction

- The cross nanofluid was inspected with related equations over a stretching cylinder with magnetic dipole aspects.
- Nanofluid features, like Brownian movement and thermophoresis, were inspected via the famous Buongiorno nanofluid model.
- A famous ferroliquid was inspected to gather whether its thermal physical features were like those of non-Newtonian nanofluids.
- Heat transfer mechanisms were characterized by thermal radiation, variation thermal conductivity, and heat absorption generation features.
- Second order slip effects were implemented near the surface to control the boundary layer of moving nanoparticles.
- The well posed modeled equations for current flow problem were numerically tackled by imposing a shooting technique.

## 2. Flow Model

#### 2.1. Interaction of Ferromagnetic Dipole

#### 2.2. Dimensionless Variables

#### 2.3. Physical Quantities

## 3. Numerical Approach

## 4. Physical Interpretation of Results

## 5. Conclusions

- For the viscous case, i.e., $We=0$ the fluid velocity was larger than that of $We=0.1,0.4,0.8,1.2$.
- When $\gamma =0$, the flow problem was reduced to a flat plate, while $\gamma =0.2$ represents the flow induced by the stretching cylinder.
- The effects of these parameters were more effective for the plate than the cylinder.
- The variation of slip factors, the buoyancy ratio constant, the Rayleigh number, and the ferrohydrodynamic interaction parameter decayed the velocity profile for the flat plate and the stretching cylinder.
- More improved temperature distribution was claimed with the interaction of slip factors, the Biot number, the ferrohydrodynamic interaction constant, and the curvature parameter. The change in temperature wasmore dominant for flow in the flat plate case than the stretching cylinder case.
- The increase of the temperature ratio parameter, the thermophoresis parameter, and the Brownian motion constant led to anenhanced temperature profile.
- The presence of activation energy and the thermophoresis parameter increased the concentration profile, while a decreasing trend was noticed with the Lewis number.
- The microorganism distribution decreased with thePeclet number and thebioconvection Lewis number.
- It was further found that the buoyancy ratio constant and the first order slip parameter canlead to increments in the microorganism distribution.
- The observations based on the reported results can be used to improve thermal extrusion processes, bio-technology, biofuels, enzymes, etc.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$\left(u,v\right)$ are the velocity components |

${D}_{B}$ is the Brownian diffusion coefficient |

${\lambda}_{0}$ is the magnetic permeability |

${\rho}_{f}$ is the nanofluid density |

${\beta}^{*}$ is volume suspension coefficient |

${g}^{*}$ is gravity |

$C$ is the nanoparticle concentration |

$T$ is temperature |

${\sigma}_{s}$ determines the Stefan Boltzmann constant |

${\rho}_{p}$ is the nanoparticle density |

$l$ is the strength of the magnetic dipole |

$K\left(T\right)$ is the thermal conductivity |

${k}^{\ast}$ is the mean absorption coefficient, |

${D}_{T}$ is the thermophoretic diffusion coefficient |

$\tau $ is the liquid heat capacity to nanoparticles heat capacity ratio |

${Q}^{\u2034}$ is the heat source sink |

$Kr$ is the chemical reaction, |

${E}_{a}$ is the activation energy |

$n$ is the Power law index |

$b$ is chemotaxis |

${W}_{c}$ is the maximum amount of swimmingcells |

${T}_{\infty}$ is the free stream temperature |

${C}_{\infty}$ is the free stream concentration |

${N}_{\infty}$ is the free stream microorganism density |

${K}_{n}$ is the Knudsen number |

${\alpha}^{**}$ represents the momentum coefficients |

${\beta}^{**}$ is the molecular mean free path |

${H}^{*}$ is the magnetic force |

$M$ is magnetization |

${K}^{*}$ is the paramagnetic constant |

${T}_{c}$ is the Curie temperature |

${f}^{\prime}$ is the dimensionless velocity profile |

$\theta $ is the dimensionless temperature profile |

$\varphi $ is the dimensionless concentration profile |

$\chi $ is the dimensionless microorganism profile |

$We$ is the Weissenberg number |

$\beta $ is the ferromagnetic parameter |

$\lambda $ is the mixed convection parameter |

$Nr$ is the buoyancy ratio parameter |

$Nc$ is the bioconvection Rayleigh number |

$Le$ is the Lewis number |

${N}_{R}$ is the radiation parameter |

$Nt$ is the thermophoresis number |

$Nb$ is Brownian motion |

$\mathrm{Pr}$ is the Prandtl number |

${\theta}_{n}$ is the temperature ratio parameter |

$E$ is activation energy |

$\sigma $ is the chemical reaction parameter |

$\delta $ is the heat source parameter |

${\mathrm{\Omega}}_{1}$ is the bioconvection constant |

$Lb$ is the bioconvection Lewis number |

$\mathrm{P}e$ is the Peclet number |

$S$ is the time-dependent parameter |

$\alpha $ is the first order velocity slip |

$\mathrm{\Lambda}$ is the second order velocity slip |

${C}_{f}$ is the wall shear stress |

$Nu$ is the local Nusselt number |

$Sh$ is the local Sherwood number |

$Nh$ is the density of motilemicroorganisms |

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**Figure 2.**Impact of the ferromagnetic parameter ($\beta $) on the variation of velocity (${f}^{\prime}$).

**Figure 20.**Impact of the bioconvection Lewis number ($Lb$) on microorganism concentration ($\chi $).

**Table 1.**Variation of ${f}^{\u2033}\left(0\right)$ for different values of $\beta ,S,We,Nr,Nc,\lambda $, and $n.$

Flow Parameters | ${\mathit{f}}^{\u2033}\left(0\right)$ | |||||||
---|---|---|---|---|---|---|---|---|

$\mathit{\beta}$ | $\mathit{S}$ | $\mathit{W}\mathit{e}$ | $\mathit{N}\mathit{r}$ | $\mathit{N}\mathit{c}$ | $\mathit{\lambda}$ | $\mathit{n}$ | $\mathit{\gamma}=0.0$ | $\mathit{\gamma}=0.2$ |

1.0 1.5 2.0 | 0.1 | 1.0 | 0.1 | 0.1 | 0.1 | 0.8 | 0.4713 0.5431 0.5988 | 0.5038 0.5656 0.6153 |

0.5 | 0.2 0.5 0.8 | 0.4110 0.4632 0.5020 | 0.4554 0.4999 0.5336 | |||||

2.0 3.0 4.0 | 0.3741 0.3573 0.3392 | 0.4121 0.3845 0.3544 | ||||||

0.5 1.0 1.5 | 0.3887 0.3878 0.3870 | 0.4359 0.4345 0.4330 | ||||||

0.2 0.3 0.4 | 0.3923 0.3952 0.3982 | 0.4426 0.4482 0.4539 | ||||||

0.2 0.3 0.4 | 0.3691 0.3500 0.3318 | 0.4372 0.4229 0.4092 | ||||||

1.0 2.0 3.0 | 0.3880 0.3810 0.3742 | 0.4530 0.4575 0.4620 |

**Table 2.**Numerical variation in the local Nusselt number $-{\theta}^{\prime}\left(0\right)$ for ${\delta}_{1},$ $\mathrm{Pr},$ $\beta ,$ $Nt,$ $S,$ $\lambda ,$ $\epsilon $, and ${\delta}_{1}.$ Pr: Prandtl number.

Flow Parameters | $-{\mathit{\theta}}^{\prime}\left(0\right)$ | |||||||
---|---|---|---|---|---|---|---|---|

$\mathbf{Pr}$ | $\mathit{\beta}$ | $\mathit{N}\mathit{t}$ | $\mathit{S}$ | $\mathit{\lambda}$ | $\mathit{\epsilon}$ | ${\mathit{\delta}}_{1}$ | $\mathit{\gamma}=0.0$ | $\mathit{\gamma}=0.2$ |

1.3 1.5 1.7 | 0.5 | 0.3 | 0.1 | 0.1 | 0.3 | 0.1 | 0.4704 0.4974 0.5212 | 0.5977 0.6170 0.6341 |

1.2 | 1.0 1.5 2.0 | 0.3952 0.3260 0.2629 | 0.5503 0.5098 0.4736 | |||||

0.4 0.7 1.0 | 0.4492 0.4300 0.4104 | 0.5795 0.5559 0.5319 | ||||||

0.2 0.5 0.8 | 0.4855 0.5678 0.6356 | 0.6109 0.6757 0.7302 | ||||||

0.2 0.3 0.4 | 0.4670 0.4771 0.4862 | 0.5872 0.5937 0.5997 | ||||||

0.4 0.6 0.8 | 0.4373 0.4006 0.3632 | 0.5748 0.5502 0.5261 | ||||||

0.4 0.6 0.8 | 0.3037 0.1918 0.0693 | 0.4795 0.4010 0.3164 |

**Table 3.**Numerical variation in the local Sherwood number $-{\varphi}^{\prime}\left(0\right)$ for $\mathrm{Pr},$ $\beta ,$ $Nb,$ $Le,$ $S,$ $\lambda ,$ $Nt$, and $\sigma .$

Flow Parameters | $-{\mathit{\varphi}}^{\prime}\left(0\right)$ | ||||||||
---|---|---|---|---|---|---|---|---|---|

$\mathbf{Pr}$ | $\mathit{\beta}$ | $\mathit{N}\mathit{b}$ | $\mathit{L}\mathit{e}$ | $\mathit{S}$ | $\mathit{\lambda}$ | $\mathit{N}\mathit{t}$ | $\mathit{\sigma}$ | $\mathit{\gamma}=0.0$ | $\mathit{\gamma}=0.2$ |

1.3 1.5 1.7 | 0.5 | 0.3 | 2.0 | 0.1 | 0.2 | 0.2 | 0.5 | 0.7056 0.7461 0.7818 | 0.8966 0.9255 0.9512 |

1.2 | 1.0 1.5 2.0 | 0.5929 0.4891 0.3944 | 0.8255 0.7647 0.7104 | ||||||

0.4 0.7 1.0 | 0.8983 1.5051 2.0521 | 1.1589 1.9458 2.6597 | |||||||

3.0 3.5 4.0 | 0.6779 0.6790 0.6943 | 0.8558 0.8639 0.8722 | |||||||

0.2 0.5 0.8 | 0.7283 0.8516 0.9534 | 0.9163 1.0135 1.0953 | |||||||

0.2 0.3 0.4 | 0.7005 0.7157 0.7294 | 0.8807 0.8906 0.8996 | |||||||

0.5 0.7 0.9 | 0.2733 0.1952 0.1518 | 0.3569 0.2563 0.2004 | |||||||

0.6 0.8 0.9 | 0.6902 0.6880 0.6870 | 0.8838 0.8817 0.8807 |

**Table 4.**Numerical variation in local motile density $-{\chi}^{\prime}\left(0\right)$ for $\mathrm{Pr},$ $Lb,$ $\delta ,$ $S,$ $\beta ,$ $Nr,$ $Nc$, and $\lambda .$

Flow Parameters | $-{\mathit{\chi}}^{\prime}\left(0\right)$ | ||||||||
---|---|---|---|---|---|---|---|---|---|

$\mathit{P}\mathit{e}$ | $\mathit{L}\mathit{b}$ | $\mathit{\delta}$ | $\mathit{S}$ | $\mathit{\beta}$ | $\mathit{N}\mathit{r}$ | $\mathit{N}\mathit{c}$ | $\mathit{\lambda}$ | $\mathit{\gamma}=0.0$ | $\mathit{\gamma}=0.2$ |

0.4 0.6 0.8 | 2.0 | 0.1 | 0.1 | 0.5 | 0.1 | 0.1 | 0.1 | 0.9507 1.0705 1.1920 | 0.7466 0.7787 0.8110 |

0.1 | 2.5 3.0 3.5 | 0.8774 0.9707 1.0561 | 0.8044 0.9000 0.9876 | ||||||

0.2 0.3 0.4 | 0.7789 0.7838 0.7887 | 0.6999 0.7012 0.7026 | |||||||

0.2 0.5 0.8 | 0.7467 0.6752 0.6164 | 0.6723 0.6038 0.5484 | |||||||

1.0 1.5 2.0 | 0.7740 0.7729 0.7686 | 0.6734 0.6723 0.5878 | |||||||

0.5 1.0 1.5 | 0.8364 0.8329 0.8286 | 0.7322 0.7311 0.7300 | |||||||

0.2 0.3 0.4 | 0.8283 0.8173 0.8058 | 0.7287 0.7244 0.7199 | |||||||

0.2 0.3 0.4 | 0.7980 0.8194 0.8387 | 0.7528 0.7709 0.7877 |

**Table 5.**The range of parameters and theirresponse tovelocity, temperature, concentration, and microorganism profiles.

Range of Parameter | Velocity Profile | Temperature Profile | Concentration Profile | Microorganism Profile |
---|---|---|---|---|

$0.1\le Nc\le 1.2$ | Decrease | Increase | Increase | Increase |

$1.0\le \alpha \le 4.0$ | Decrease | Increase | Increase | Increase |

$0.5\le \beta \le 2.0$ | Decrease | Increase | - | - |

$0.1\le We\le 1.2$ | Decrease | - | - | - |

$-1.0\le \mathrm{\Lambda}\le -4.0$ | Decrease | - | - | - |

$0.1\le \lambda \le 1.2$ | Increase | Decrease | Decrease | - |

$0.1\le Nr\le 1.2$ | Decrease | Increase | Increase | - |

$0.1\le Nt\le 1.5$ | - | Increase | Increase | - |

$0.1\le {\delta}_{1}\le 1.0$ | - | Increase | - | - |

$0.1\le Bi\le 1.0$ | - | Increase | - | - |

$1.5\le {\theta}_{n}\le 1.8$ | - | Increase | - | - |

$1.0\le E\le 2.0$ | - | Increase | - | - |

$1.0\le Le\le 2.0$ | - | - | Decreases | - |

$0.1\le Pe\le 1.0$ | - | - | - | Decrease |

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## Share and Cite

**MDPI and ACS Style**

Abdelmalek, Z.; Al-Khaled, K.; Waqas, H.; Aldabesh, A.; Khan, S.U.; Musmar, S.A.; Tlili, I.
Bioconvection in Cross Nano-Materials with Magnetic Dipole Impacted by Activation Energy, Thermal Radiation, and Second Order Slip. *Symmetry* **2020**, *12*, 1019.
https://doi.org/10.3390/sym12061019

**AMA Style**

Abdelmalek Z, Al-Khaled K, Waqas H, Aldabesh A, Khan SU, Musmar SA, Tlili I.
Bioconvection in Cross Nano-Materials with Magnetic Dipole Impacted by Activation Energy, Thermal Radiation, and Second Order Slip. *Symmetry*. 2020; 12(6):1019.
https://doi.org/10.3390/sym12061019

**Chicago/Turabian Style**

Abdelmalek, Zahra, Kamel Al-Khaled, Hassan Waqas, A. Aldabesh, Sami Ullah Khan, Sa’ed A. Musmar, and Iskander Tlili.
2020. "Bioconvection in Cross Nano-Materials with Magnetic Dipole Impacted by Activation Energy, Thermal Radiation, and Second Order Slip" *Symmetry* 12, no. 6: 1019.
https://doi.org/10.3390/sym12061019