# Cattaneo–Christov Double Diffusion (CCDD) on Sutterby Nanofluid with Irreversibility Analysis and Motile Microbes Due to a RIGA Plate

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## Abstract

**:**

## 1. Introduction

_{2}O

_{3}–Water/Ethylene Glycol over a Gailitis and Lielausis device with an effective Prandtl number.

## 2. Description of the Physical Model

## 3. Fluid Flow Problem

## 4. Entropy Generation Analysis

## 5. Homotopy Expression

## 6. The Zeroth Order Formulation

## 7. The mth Order Formulation

## 8. Convergence of Homotopy Solutions

## 9. Results and Discussion

## 10. Results and Discussion

## 11. Velocity Field

## 12. Temperature Field

## 13. Concentration Field

## 14. Microorganism Field

## 15. Entropy Generation

## 16. Bejan Number

## 17. Physical Entitles

## 18. Streamline and Isotherm Line

- The convergence of HAM solutions is ensured up to the 25th iteration.
- Deterioration and elevation behaviour in the momentum of fluid is manifested in relation to Deborah’s number and Reynold’s number when $\beta =-2.5$ and $\beta =2.5$.
- The velocity shows continuous improvement with increases in the Hartman number in cases of dilatant and pseudoplastic fluid.
- The growing estimate of the variable leads to increases in the dimensionless temperature field.
- Declining aptitude expressed in the concentration field against the Schmidt number and Prandtl number.
- A larger chemical reaction $Cr$ portrays a decline in the concentration, while the Biot number $Bi$ lead to the expansion in concentration.
- The microorganism field has deteriorated for the higher value of $Pe$ and microorganism difference parameter.
- The entropy generation number presented an increasing magnitude for large values of the Reynolds number and Brinkman number, for the case of pseudoplastic and dilatants fluid. Large values of the entropy generation number appear in the vicinity of the sheet due to high viscous effects.
- Enhancing the value of Deborah’s number and Reynold’s number results in Bejan’s profile decay in the case of dilatant fluid, while the opposite effect is observed in the case of shear-thinning.
- Skin fraction decelerated for the modified Hartman number $Z$ but accelerated against the Nusselt number for the heat source/sink parameter $\mathsf{\Upsilon}$ and the Sherwood number for the Thermophoresis parameter $Nt$.
- The density of motile microorganisms goes up with rising values of $Lb,Pe$ and $\varpi .$

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

$a$ | Constant |

$u,v$ | Velocity components |

$f$ | Similarity function for velocity |

${T}_{\infty}$ | Ambient temperature |

${T}_{w}$ | Wall constant temperature |

$T$ | Fluid Temperature |

$\chi $ | Microorganism concentration |

$\theta $ | Dimensionless temperature |

$\varphi $ | Dimensionless concentration |

${C}_{\infty}$ | Ambient concentration |

$C$ | Fluid concentration |

$\beta $ | Power index number |

${B}_{\circ}$ | $\mathrm{Magnetic}\mathrm{field}\mathrm{strength}\left({\mathrm{NA}}^{-1}{\mathrm{m}}^{-1}\right)$ |

$\delta $ | Deborah number |

${\mathrm{Re}}_{L}$ | Local Reynolds number |

$Z$ | Modified Hartmann number |

$M$ | Magnetic Parameter |

$Rd$ | Thermal Radiation |

$\mathsf{\Upsilon}$ | Heat source/sink parameter |

$Cr$ | Chemical reaction |

${\lambda}_{1}$ | Thermal Relaxation parameter |

${\lambda}_{2}$ | Concentration Relaxation parameter |

$Nt$ | Thermophoresis parameter |

$Nb$ | Brownian motion parameter |

$Bi$ | Biot number |

$\mathrm{Pr}$ | Prandtl number |

$Sc$ | Schmidt number |

$Pe$ | Peclet number |

$Lb$ | Lewis number |

$\varpi $ | Microorganism concentration difference parameter |

${D}_{B}$ | Mass diffusivity |

${D}_{T}$ | Thermophoresis diffusivity |

${J}_{\circ}$ | Current density |

${C}_{f}$ | Skin friction coefficient |

$N{u}_{x}$ | Nusselt number |

$S{h}_{x}$ | Sherwood number |

$W{h}_{x}$ | Microorganism density number |

$k$ | Thermal conductivity of the fluid $\left({\mathrm{m}}^{2}/\mathrm{s}\right)$ |

$Br$ | Brickman number |

${S}_{G}$ | Local volumetric entropy generation rate |

${N}_{G}$ | Entropy number |

$Be$ | Bejan number |

$\tau $ | Ratio of the effective heat capacity |

$\eta $ | Similarity variable |

$\rho $ | Fluid Density |

$\mu $ | Dynamic viscosity |

$\infty $ | Ambient condition |

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**Figure 2.**Plots of (

**a**) ${\hslash}_{f}$ the curve of ${f}^{\u2033}(0)$ (

**b**) ${\hslash}_{\theta}$ the curve of $\theta \prime (0)$ (

**c**) ${\hslash}_{\varphi}$ the curve of $\varphi \prime (0)$ (

**d**) ${\hslash}_{w}$ the curve of $W\prime (0)$.

**Figure 3.**(

**a**–

**d**). The impact of ${f}^{\prime}(\eta )$ numerous variables (

**a**) $\delta $ (

**b**) ${\mathrm{Re}}_{L}$ (

**c**) $\beta $ (

**d**) $Z$.

**Figure 4.**(

**a**–

**f**). The impact of $\theta \left(\eta \right)$ numerous variables (

**a**) ${\lambda}_{1}$ (

**b**) $Rd$ (

**c**) $Nt$ (

**d**) $Nb$ (

**e**) $Bi$ (

**f**) $Pr$.

**Figure 5.**(

**a**–

**d**) The impact of $\varphi (\eta )$ numerous variables (

**a**) $Nb$ (

**b**) $Nb$ (

**c**) $Cr$ (

**d**) $Sc$.

**Figure 7.**(

**a**–

**c**) The impact of ${N}_{G}$ numerous variables (

**a**) $\delta $ (

**b**) ${\mathrm{Re}}_{L}$ (

**c**) $Br$.

**Figure 8.**(

**a**–

**c**) The impact of $Be$ numerous variables (

**a**) $\delta $ (

**b**) ${\mathrm{Re}}_{L}$ (

**c**) $Br$.

**Figure 9.**(

**a**–

**d**) Variation in ${\mathrm{Re}}_{x}^{1/2}Cf,{\mathrm{Re}}_{x}^{-1/2},Nu{\mathrm{Re}}_{x}^{-1/2}\mathrm{Sh},{\mathrm{Re}}_{x}^{-1/2}{\mathrm{Wh}}_{x}$ numerous variables (

**a**)$Z,\delta $ (

**b**) $Nb,\mathsf{\Upsilon}$ (

**c**) $Nb,Nt$ (

**d**) $Lb,Pe$.

**Figure 10.**(

**a**–

**d**) 3D graph (

**a**) Skin friction for $\delta $ and $Z$ (

**b**) Nusselt number for $Rd$ and $\mathsf{\Upsilon}$. (

**c**) Sherwood number for $Nt$ and $Nb$ (

**d**) Motile density for $Pe$ and $Lb$.

**Table 1.**Convergence solutions of HAM for different order of approximations when $\beta =0.2,\delta =0.1,M=0.3,\mathrm{Pr}=3.0,Nb=Nt=0.1,Sc=1.0,\mathsf{\Upsilon}=0.1,Pe=0.3,Lb=0.2$.

Order of HAM | ||||
---|---|---|---|---|

Approximation | $-{\mathit{f}}^{\u2033}(0)$ | $-{\mathit{\theta}}^{\prime}(0)$ | $-{\mathit{\varphi}}^{\prime}(0)$ | $-{\mathit{W}}^{\prime}(0)$ |

1 | 0.874702 | 0.166821 | 1.21667 | 0.955833 |

5 | 0.768677 | 0.167973 | 1.41759 | 1.038657 |

10 | 0.765186 | 0.168223 | 1.42596 | 1.07814 |

15 | 0.765357 | 0.168144 | 1.42606 | 1.08236 |

20 | 0.765347 | 0.168158 | 1.42609 | 1.08264 |

25 | 0.765341 | 0.168158 | 1.42608 | 1.08261 |

30 | 0.765341 | 0.168157 | 1.42608 | 1.08258 |

35 | 0.765341 | 0.168157 | 1.42608 | 1.08258 |

40 | 0.765341 | 0.168157 | 1.42608 | 1.08258 |

Ali and Zaib [58] | Bvp4c | Current Result (HAM) | ||
---|---|---|---|---|

$Nt$ | $Nb$ | |||

0.1 | 0.1 | 0.092906 | 0.0923 | $0.093757137$ |

0.5 | 0.5 | 0.092126 | 0.0925 | $0.093975770$ |

**Table 3.**Numerical outcomes ${\mathrm{Re}}_{x}^{1/2}C{f}_{x}$ for numerous values of $Z,\delta ,{\mathrm{Re}}_{L},A$.

$\mathit{Z}$ | $\mathit{\delta}$ | ${\mathbf{Re}}_{\mathit{L}}$ | $\mathit{A}$ | ${\mathbf{Re}}_{\mathit{x}}^{1/2}\mathit{C}{\mathit{f}}_{\mathit{x}}$ | |
---|---|---|---|---|---|

$\mathit{\beta}=-2.5$ | $\mathit{\beta}=2.5$ | ||||

0.1 | 0.4 | 0.3 | 0.2 | 0.92454 | 0.96057 |

0.3 | 0.82986 | 0.86558 | |||

0.5 | 0.73214 | 0.79555 | |||

0.7 | 0.4 | 0.8 | 0.63177 | 0.74173 | |

0.1 | 0.6 | 0.3 | 0.94253 | 0.98097 | |

0.3 | 0.84824 | 0.89991 | |||

0.5 | 0.79896 | 0.82019 | |||

0.7 | 0.6 | 0.72767 | 0.77578 | ||

0.1 | 0.8 | 0.3 | 1.6 | 0.96645 | 1.00572 |

0.3 | 0.88142 | 0.94353 | |||

0.5 | 0.84492 | 0.88265 | |||

0.7 | 0.8 | 0.80704 | 0.82265 | ||

0.1 | 1.0 | 0.3 | 2.4 | 0.98264 | 1.01726 |

0.3 | 1.0 | 0.90723 | 0.96739 | ||

0.5 | 0.85289 | 0.91848 | |||

0.7 | 1.0 | 0.82750 | 0.87050 |

**Table 4.**Numerical outcomes of ${\mathrm{Re}}_{x}^{-1/2}N{u}_{x}$ numerous values of $Rd,\mathsf{\Upsilon},Bi,\mathrm{Pr},Nb,Nt$.

$\mathit{R}\mathit{d}$ | $\mathsf{\Upsilon}$ | $\mathit{B}\mathit{i}$ | $\mathbf{Pr}$ | $\mathit{N}\mathit{b}$ | $\mathit{N}\mathit{t}$ | ${\mathbf{Re}}_{\mathit{x}}^{-1/2}\mathit{N}{\mathit{u}}_{\mathit{x}}$ | |
---|---|---|---|---|---|---|---|

${\mathit{\lambda}}_{1}=-2.5$ | ${\mathit{\lambda}}_{1}=2.5$ | ||||||

0.1 | 0.1 | 0.1 | 1.0 | 0.1 | 0.1 | 0.10375 | 0.09868 |

0.2 | 1.3 | 0.2 | 0.11564 | 0.10996 | |||

0.3 | 0.1 | 0.2 | 1.6 | 0.3 | 0.23375 | 0.23171 | |

0.4 | 1.9 | 0.1 | 0.4 | 0.27543 | 0.25036 | ||

0.5 | 0.2 | 0.3 | 2.0 | 0.2 | 0.1 | 0.37615 | 0.32808 |

0.6 | 2.2 | 0.2 | 0.40300 | 0.35108 | |||

0.7 | 0.4 | 2.6 | 0.2 | 0.3 | 0.52608 | 0.44059 | |

0.8 | 2.8 | 0.4 | 0.55644 | 0.46506 | |||

0.9 | 0.3 | 0.5 | 3.0 | 0.3 | 0.1 | 0.66801 | 0.53560 |

1.0 | 3.1 | 0.2 | 0.69913 | 0.55870 | |||

1.1 | 0.6 | 3.4 | 0.3 | 0.3 | 0.80631 | 0.61842 | |

1.2 | 4.0 | 0.3 | 0.4 | 0.83636 | 0.63832 | ||

1.3 | 0.4 | 0.7 | 4.2 | 0.4 | 0.1 | 0.92220 | 0.67117 |

1.4 | 4.5 | 0.2 | 0.94881 | 0.68554 | |||

1.5 | 0.8 | 4.9 | 0.3 | 1.02733 | 0.70633 | ||

1.8 | 5.5 | 0.4 | 0.4 | 1.09963 | 0.73584 |

**Table 5.**Numerical outcomes f ${\mathrm{Re}}_{x}^{-1/2}S{h}_{x}$ for numerous values of $Cr,\mathrm{Pr},Sc$.

$\mathit{C}\mathit{r}$ | $\mathbf{Pr}$ | $\mathit{S}\mathit{c}$ | $\mathit{N}\mathit{b}$ | $\mathit{N}\mathit{t}$ | ${\mathbf{Re}}_{\mathit{x}}^{-1/2}\mathit{S}{\mathit{h}}_{\mathit{x}}$ | |
---|---|---|---|---|---|---|

${\mathit{\lambda}}_{2}=-2.5$ | ${\mathit{\lambda}}_{2}=2.5$ | |||||

0.3 | 2.5 | 1.0 | 0.1 | 0.1 | 0.88698 | 1.29324 |

0.6 | 2.7 | 1.1 | 0.2 | 1.05551 | 1.46176 | |

0.9 | 2.9 | 1.2 | 0.3 | 1.19741 | 1.60365 | |

1.2 | 3.0 | 1.3 | 0.4 | 1.32527 | 1.73152 | |

1.5 | 3.3 | 0.2 | 0.1 | 1.45829 | 1.86454 | |

1.8 | 3.5 | 0.2 | 1.57097 | 1.97722 | ||

2.0 | 3.7 | 1.4 | 0.3 | 1.64331 | 2.04955 | |

2.3 | 4.0 | 0.4 | 1.74712 | 2.15337 | ||

2.6 | 4.1 | 1.8 | 0.3 | 0.1 | 1.85226 | 2.25851 |

2.7 | 0.2 | 1.88221 | 2.28846 | |||

2.9 | 0.3 | 1.94866 | 2.35491 | |||

3.0 | 4.2 | 1.9 | 0.4 | 1.97869 | 2.38494 | |

3.2 | 1.9 | 0.4 | 0.1 | 2.05108 | 2.45733 | |

3.3 | 4.3 | 0.4 | 0.2 | 2.08112 | 2.48737 | |

3.4 | 0.3 | 2.11116 | 2.51741 | |||

3.5 | 4.5 | 1.9 | 0.4 | 2.14119 | 2.54744 |

**Table 6.**Numerical outcomes ${\mathrm{Re}}_{x}^{-1/2}W{h}_{x}$ for numerous values of $Pe,Lb,\varpi $.

$\mathit{P}\mathit{e}$ | $\mathit{L}\mathit{b}$ | $\mathit{\varpi}$ | $\mathit{N}\mathit{b}$ | $\mathit{N}\mathit{t}$ | ${\mathbf{Re}}_{\mathit{x}}^{-1/2}\mathit{W}{\mathit{h}}_{\mathit{x}}$ |
---|---|---|---|---|---|

1.0 | 0.7 | 0.3 | 0.1 | 0.1 | 1.28167 |

1.2 | 0.8 | 0.5 | 0.2 | 1.37667 | |

0.7 | 0.3 | 1.45567 | |||

0.9 | 0.9 | 0.4 | 1.60451 | ||

1.3 | 0.3 | 0.2 | 0.1 | 1.46258 | |

0.5 | 0.2 | 1.54708 | |||

1.4 | 0.9 | 0.7 | 0.3 | 1.65217 | |

1.0 | 0.9 | 0.4 | 1.79951 | ||

1.5 | 0.3 | 0.3 | 0.1 | 1.57958 | |

1.6 | 0.5 | 0.2 | 1.73667 | ||

1.7 | 0.7 | 0.3 | 1.85154 | ||

1.8 | 0.9 | 0.4 | 2.08983 | ||

1.5 | 0.3 | 0.4 | 0.1 | 1.79321 | |

0.5 | 0.2 | 1.91432 | |||

1.9 | 1.6 | 0.7 | 0.3 | 2.04217 | |

2.0 | 1.7 | 0.9 | 0.4 | 2.31083 |

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## Share and Cite

**MDPI and ACS Style**

Ahmed, M.F.; Zaib, A.; Ali, F.; Bafakeeh, O.T.; Khan, N.B.; Mohamed Tag-ElDin, E.S.; Oreijah, M.; Guedri, K.; Galal, A.M.
Cattaneo–Christov Double Diffusion (CCDD) on Sutterby Nanofluid with Irreversibility Analysis and Motile Microbes Due to a RIGA Plate. *Micromachines* **2022**, *13*, 1497.
https://doi.org/10.3390/mi13091497

**AMA Style**

Ahmed MF, Zaib A, Ali F, Bafakeeh OT, Khan NB, Mohamed Tag-ElDin ES, Oreijah M, Guedri K, Galal AM.
Cattaneo–Christov Double Diffusion (CCDD) on Sutterby Nanofluid with Irreversibility Analysis and Motile Microbes Due to a RIGA Plate. *Micromachines*. 2022; 13(9):1497.
https://doi.org/10.3390/mi13091497

**Chicago/Turabian Style**

Ahmed, Muhammad Faizan, A. Zaib, Farhan Ali, Omar T Bafakeeh, Niaz B. Khan, El Sayed Mohamed Tag-ElDin, Mowffaq Oreijah, Kamel Guedri, and Ahmed M. Galal.
2022. "Cattaneo–Christov Double Diffusion (CCDD) on Sutterby Nanofluid with Irreversibility Analysis and Motile Microbes Due to a RIGA Plate" *Micromachines* 13, no. 9: 1497.
https://doi.org/10.3390/mi13091497