# Spectral Relaxation Methodology for Chemical and Bioconvection Processes for Cross Nanofluid Flowing around an Oblique Cylinder with a Slanted Magnetic Field Effect

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## Abstract

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## 1. Introduction

_{2}O

_{3}+ Cu nanoparticles. Other recent studies related to chemical process with different facts such as activation energy and bi-convection Carreau nanofluid flow, stability of convection in a non-Newtonian vertical fluid, the Electroosmosis forces EOF-driven boundary layer flow, oxytactic microorganisms and nanoparticles, and the bio-thermal convection of Prandtl nanofluid have been made by [14,15,16,17,18].

**Novelty**

- In the whole literature, there are several studies which are related to a Cross fluid model attached with horizontal and vertical cylinders, but a Cross fluid model has not yet been used with an inclined cylinder. Therefore, this attempt is performed to carry out the results with Cross fluid along an inclined cylinder geometry.
- The spectral relaxation technique has not been utilized yet for the case of numerical computation of Cross nanofluid flow subjected to an elastic cylinder.

## 2. Flow Analysis

_{0}is constant. The symbols Tw, Cw, and Nw indicate wall temperature as well as concentration and density phenomenon of the microorganisms. The free stream effect on temperature (T), concentration (C) field, and microorganisms (N) are ${T}_{\infty},{C}_{\infty}$, and ${N}_{\infty}$, respectively.

## 3. Rheology of Cross Fluid

## 4. Modeled Equations

_{c}depict the thermal-based conductivity and reaction rate type constant, W

_{c}depicts swimming capacity of the cells, and ρ

_{p}indicates nanoparticle density. Abbreviations such as $D$, $\overline{D}$, and $\tilde{D}$ indicate diffusion phenomena in terms of Brownian, thermophoretics and microorganisms. Additionally, conditions at the boundary are premeditated by [52,53]

## 5. Method of Solution

## 6. Numerical Procedure: Spectral Relaxation Technique

#### 6.1. Advantages of Spectral Relaxation Technique

- Less expensive
- Easier to implement than finite element methods
- They shine best when high accuracy is sought
- They are useful in simple domains with smooth solutions.

#### 6.2. Applied Spectral Relaxation Technique on Current Problem

## 7. Validness of Study

## 8. Data Description, Collection, Interpretation, and Discussion

#### 8.1. Data Collection

#### 8.2. Data Description

#### 8.3. Data Handling

#### 8.4. Interpretation of Numerically Outcomes

## 9. Results and Discussion

_{f}but diminishes for parameters such as n, γ, M, and A. The rate at which heat is delivered amplifies for all distinguished parameters, and the microorganisms’ density phenomenon diminishes for We, Rb, n, γ, and M and amplifies in the case of A. It is observed that that the surface drags and heat deliverance rate amplifies as a result of an amplification in We. Relaxation of the fluid escalates by the virtue of a magnification in We, which decreases and amplifies the drag phenomenon and heat deliverance rate but decreases microorganisms’ density. It is well established that a positive variation in bio convective Rayleigh number Rb amplifies buoyancy forces, and this change produces a decrement in the velocity field, which amplifies drag phenomenon. Microorganism’s density diminishes in the case of $0\le Rb\le 1\text{}$and increases in the case of $1\le Rb\le \infty .\text{}$The heat deliverance rate amplifies owing a positive variation in Rb. It is noticed that the fluid behavior is shear thickening in the case of n > 1, and shear thinning for n < 1. Viscous forces dominate the inertial forces, which lessens the fluid motion, density of microorganisms, and increases the fluid temperature. It is quite clear that the fluid is allowed to relax and regain its original shape, similar to an elastic rubber that magnifies the fluid viscosity. An amplification in $\mathsf{\Gamma}$ provides a resistance to the fluid flow and, moreover, produces a decrement in drag phenomenon and microorganisms’ density but amplifies temperature at the other end. The Lorentz force is produced as a result of a positive variation in M, which acts like a barrier in the way of fluid flow which reduces drag phenomenon, microorganism’s density, and increases temperature. It is observed that fluid behavior is unsteady as a result of an incremental change in A because the viscosity of fluid increases, which lessens the surface drag phenomenon, microorganism’s density, and amplifies the heat deliverance rate.

#### 9.1. Velocity Profile

#### 9.2. Temperature Profile

_{e}and thermal conductivity $\epsilon $ on the temperature field $\theta \left(\eta \right)$ are displayed in Figure 9 and Figure 10. It is observed from Figure 4 that an amplification in We amplifies the fluid viscosity. In the light of Figure 3, a positive variation in viscosity amplifies the fluid temperature. The thermal boundary layer thickness (TBL) is amplifies, and fluid moves away from the wall in an upward direction, owing to a magnification in the We ranges from 1 to 4. As a result, the temperature field escalates as shown in Figure 9. The Weissenberg number relies on the relaxation time, or the time required by the fluid to relax and regain its original position. Fluid becomes more viscous as a result of magnification in We. Shear thickening behavior is observed in the case of a magnification in We, which escalates the temperature phenomenon. The ability of any material to conduct heat is called thermal conductivity. In liquids, the thermal conductivity phenomenon takes place as a result of an intermolecular collision, which furthermore amplifies the temperature of the amplified fluid. Molecules collide more frequently as a result of a magnification in $\epsilon $ and share more K.E with each other, and the heat transfer exchange phenomenon of this situation intensifies more favorably in the case of nanofluids. The fluid flows away from the wall and TBL escalates in the case of a positive variation in $\epsilon $ from 1 to 5.5, as displayed in Figure 10. Figure 11 and Figure 12 highlighted the influence of Prandtl number Pr and inclined angle $\omega $ on $\theta \left(\eta \right)$. The Prandtl number is the ratio of momentum to thermal diffusivity. Heat diffuses more by the virtue of an increment in Pr, which lessens the temperature inside the fluid. The fluid approaches towards the vertical wall and TBL abates. The abatement in $\theta \left(\eta \right)$ is more dominant in the case of no inclined angle, in comparison to inclined angle $\omega $, as displayed in Figure 11. From Figure 12 it is observed that a magnification in $\omega $ amplifies$\text{}\theta \left(\eta \right)$. The viscosity of fluid diminishes as a result of an augmentation in $\omega $, which furthermore lessens the velocity and ultimately escalates the temperature phenomenon. That is why an amplification in $\omega $ is the major factor responsible for an increment in TBL. The impact of curvature parameter γ is shown in Figure 13. Enhancing the curvature parameter bending of fluid particles increases the resistance within the wall, causing the rise in temperature.

#### 9.3. Concentration Profile

_{r}on the concentration field$\text{}\varphi \left(\eta \right)$. It is observed that a positive change in We diminishes the concentration phenomenon. The concentration boundary layer thickness decreases and moves forward, owing to the magnification of We, as shown in Figure 14. Reaction rate K

_{r}is directly proportional to concentration. Concentration of the fluid diminishes owing to a variation in K

_{r}, which depreciates the concentration boundary layer thickness as fluid moves towards the wall in a downward direction, as shown in Figure 15. The effect of thermal conductivity $\epsilon $ on the mass fraction field is highlighted in Figure 16. The temperature amplifies owing to a magnification in $\epsilon $, which diminishes the viscosity phenomenon and concentration of the fluid as well. The velocity of fluid decreases owing to a magnification in $\omega $. Velocity is linked with the viscosity phenomenon. A positive variation in $\omega $ diminishes the viscosity and concentration field $\varphi \left(\eta \right)$ as shown in Figure 17.

#### 9.4. Statistical Graphs

_{x}and Nn

_{x}diminish, owing to an amplification in viscosity. Amplification in Rb escalates surface drag phenomenon much better in contrast to heat transfer and motile density of microorganisms, as displayed in Figure 19. Figure 20 displayed the effect of power law index n on various physical quantities. The power law index is related to the viscosity phenomenon and is opposite to the temperature effect, and shear thickening behavior is reported by the virtue of a magnification in n, which amplifies the surface drag and density of motile microorganisms and diminishes the heat transfer phenomenon. Fluid behavior is shear thinning for n < 1, Newtonian for n = 1, and ashear thickening in the case of n > 1. Blood is a perfect example of this phenomenon. Blood behavior is shear thinning and shear thickening according to the situation. That is why magnification diminishes velocity and magnifies temperature. Figure 21, Figure 22 and Figure 23 are designed to highlight the influence of γ, M, and A on various physical quantities. It is observed that an enhancement in γ amplifies the fluid velocity and lessens the drag phenomenon of the surface and microorganism’s density but magnifies the heat transfer rate. It is well established that an amplification in curvature parameter amplifies the fluid viscosity phenomenon, which diminishes the velocity of fluid flow and amplifies temperature because viscosity is inversely linked with temperature. Paint is the example of this phenomenon. Fluid possessing electric current when moving through a magnetic creates a resistive-type force, known as the Lorentz force. The Lorentz force is basically a resistive force that diminishes the drag surface phenomenon and density of motile microorganisms. This resistive force, on the other hand, amplifies the fluid temperature and heat transfer rate of the fluid depicted in Figure 22. The influence of unsteady parameter A on the surface drag coefficient, heat transfer, and motile microorganisms are highlighted in Figure 23. It is quite evident that an augmentation in an unsteady phenomenon brings about a magnification in fluid viscosity. This augmentation in viscosity is a major factor responsible for a decrement in fluid velocity. Velocity of fluid is inversely related to drag phenomenon of a surface. That is why an amplification in A diminishes the surface drag friction. Density is also related to viscosity. The fluid is denser when the viscosity amplifies, which is why an abatement in fluid viscosity lessens the density of motile microorganisms.

## 10. Conclusions

- Shear thinning behavior is noticed as a result of a magnification in a Weissenberg number, which brings a decrement in fluid viscosity and amplifies velocity of the fluid flow.
- The fluid behavior is shear thickening owing to magnification in power law index n, which amplifies the fluid viscosity and velocity phenomenon.
- It is well-established that the molecules collide more randomly and exchange more KE with each other by virtue of a magnification in the thermal conductivity parameter $\epsilon ,$ which provides an amplification in the temperature field.
- It is observed that the microorganism swimming speed amplifies as a result of an augmentation in w, which brings about an increment in the microorganisms flow field.
- Transport of heat is rapid on a cylinder as compared to a flat surface. Molecules collide more randomly and enhance K.E in the case of cylindrical surface in contrast to elastic surface.
- The drag friction phenomenon is inversely related to the fluid flow motion. Amplification in We amplifies and encourages the viscosity phenomenon, which diminishes the fluid velocity and escalates the drag friction phenomenon.
- Viscosity is inversely linked with temperature. Amplification in Weissenberg number We and power law index n amplifies the fluid viscosity, which diminishes the fluid velocity and amplifies temperature.
- A positive variation in M encourages the resistive force, called Lorentz force, which provides hurdle to the fluid flow and diminishes the velocity phenomenon, but a magnification M provides a substantial heat to the fluid flow subjected to a cylinder.
- From the obtained results, it is revealed that more heat is produced in the case of nanofluid in contrast to simple base fluid.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Parameters | Units | Parameters | Units |

Gravity (g) | $\frac{m}{{s}^{2}}$ | Activation energy (E) | Kg∙J∙mol^{−1} |

material constant $(\tilde{\text{}\gamma )}$ | N∙m^{−2} | Power law index (n) | N∙m^{−2} |

Magnetic field (${B}_{0}$) | Kg∙s^{−2}∙A^{_1} | cell swimming speed (Wc) | m∙s^{−1} |

Density ($\rho $) | Kg∙m^{-−3} | Temperature (T) | K |

Angles ($\omega ,\text{}\mathsf{\Omega}$) | Radian | Concentration (C) | Mol∙m^{−3} |

Thermal conductivity (${k}_{f}$) | W∙m^{−1}∙k^{−1} | microorganisms concentration (N) | Mol∙m^{−3} |

Thermophoresis diffusion (D) | m^{2}∙s^{−1} | curvature parameter (γ) | m^{−1} |

Brownian diffusion $\overline{(D})$ | m^{2}∙s^{−1} | An unsteadiness parameter (A) | m∙s^{−1} |

Heat capacity (${C}_{P}$) | J∙Kg^{−1}∙K^{−1} | Buoyancy force (Nr) | Kg∙m^{2} |

Reaction rate$\text{}({k}_{r})$ | mol∙s^{−1} | Electrical conductivity ($\sigma $) | Sem∙m^{−1} |

Temperature difference (δ) | K | - | - |

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Γ | Khan and Alshomrani [59] | Ragni et al. [60] | Present |
---|---|---|---|

0.10 | 1.0000000 | 1.0000000 | 1.0000000 |

0.25 | 1.0943730 | 1.0943780 | 1.0943770 |

0.50 | 1.1887270 | 1.1887150 | 1.1887291 |

0.75 | 1.2818190 | 1.2818330 | 1.2818263 |

1.00 | 1.4533730 | 1.4593080 | 1.4533642 |

**Table 2.**Influence of various sundry parameters on surface drag frictiuon, heat transfer as microorganism’s density.

We | R_{b} | N | Γ | M | A | Surface Drag Friction Cf | Heat Transfer Nu | Microorganisms’ Density Nn |
---|---|---|---|---|---|---|---|---|

1.0 | - | - | - | - | - | 1.235892 | 1.02357 | 0.985636 |

10.0 | - | - | - | - | - | 1.259863 | 1.033987 | 0.965874 |

25.0 | - | - | - | - | - | 1.278524 | 1.042548 | 0.947865 |

50.0 | - | - | - | - | - | 1.298552 | 1.053952 | 0.925479 |

- | 0.10 | - | - | - | - | 1.309874 | 1.07899 | 0.995687 |

- | 0.15 | - | - | - | - | 1.324587 | 1.098745 | 0.987541 |

- | 0.25 | - | - | - | - | 1.345469 | 1.102155 | 0.978745 |

- | 0.35 | - | - | - | - | 1.369874 | 1.112548 | 0.964587 |

- | - | 0.50 | - | - | - | 1.121548 | 1.087855 | 1.124579 |

- | - | 0.80 | - | - | - | 1.102545 | 1.103215 | 1.112459 |

- | - | 1.10 | - | - | - | 1.087854 | 1.125449 | 1.101255 |

- | - | 1.35 | - | - | - | 1.069874 | 1.145687 | 1.096548 |

- | - | - | 1.0 | - | - | 1.045698 | 1.101255 | 1.095785 |

- | - | - | 2.0 | - | - | 1.032578 | 1.111854 | 1.092549 |

- | - | - | 3.0 | - | - | 1.026548 | 1.125487 | 1.091458 |

- | - | - | 4.0 | - | - | 1.01236 | 1.135459 | 1.089652 |

- | - | - | - | 0.10 | - | 1.894521 | 1.884579 | 1.884578 |

- | - | - | - | 0.50 | - | 1.890023 | 1.889578 | 1.874578 |

- | - | - | - | 1.00 | - | 1.880457 | 1.890022 | 1.86458 |

- | - | - | - | 1.12 | - | 1.874587 | 1.898453 | 1.846578 |

- | - | - | - | - | 0.10 | 1.564874 | 1.601254 | 1.615458 |

- | - | - | - | - | 0.15 | 1.554248 | 1.614588 | 1.624578 |

- | - | - | - | - | 0.25 | 1.542154 | 1.624574 | 1.634588 |

- | - | - | - | - | 0.35 | 1.532645 | 1.631246 | 1.641249 |

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**MDPI and ACS Style**

Rasool, G.; Shah, S.Z.H.; Sajid, T.; Jamshed, W.; Cieza Altamirano, G.; Keswani, B.; Artidoro Sandoval Núñez, R.; Sánchez-Chero, M.
Spectral Relaxation Methodology for Chemical and Bioconvection Processes for Cross Nanofluid Flowing around an Oblique Cylinder with a Slanted Magnetic Field Effect. *Coatings* **2022**, *12*, 1560.
https://doi.org/10.3390/coatings12101560

**AMA Style**

Rasool G, Shah SZH, Sajid T, Jamshed W, Cieza Altamirano G, Keswani B, Artidoro Sandoval Núñez R, Sánchez-Chero M.
Spectral Relaxation Methodology for Chemical and Bioconvection Processes for Cross Nanofluid Flowing around an Oblique Cylinder with a Slanted Magnetic Field Effect. *Coatings*. 2022; 12(10):1560.
https://doi.org/10.3390/coatings12101560

**Chicago/Turabian Style**

Rasool, Ghulam, Syed Zahir Hussain Shah, Tanveer Sajid, Wasim Jamshed, Gilder Cieza Altamirano, Bright Keswani, Rafaél Artidoro Sandoval Núñez, and Manuel Sánchez-Chero.
2022. "Spectral Relaxation Methodology for Chemical and Bioconvection Processes for Cross Nanofluid Flowing around an Oblique Cylinder with a Slanted Magnetic Field Effect" *Coatings* 12, no. 10: 1560.
https://doi.org/10.3390/coatings12101560