# Thermal Aspects of Casson Nanoliquid with Gyrotactic Microorganisms, Temperature-Dependent Viscosity, and Variable Thermal Conductivity: Bio-Technology and Thermal Applications

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## Abstract

**:**

## 1. Introduction

_{2}O

_{3}/water nanoparticles. Mahanthesh et al. [9] numerically executed the Hall effects and shape factor of nanoparticles configured by a rotating frame. Hayat et al. [10] scrutinized the Darcy flow of Jeffrey nanofluid with nonlinear radiation prospective. Maleki et al. [11] showed the MAR and artificial neural network applications while examining the thermal consequences of ZnO nano-materials. The Joule heating aspects in dissipative flow of nanofluid with chemical reaction effects was worked out by Shahzad et al. [12]. Ibrahim [13] presented the slip flow of tangent hyperbolic nanoparticles by utilizing convective condition approach. Wang et al. [14] used difference approximation to study the MgO nanoparticles thermal performance. The oscillatory surface flow of micropolar nanofluid with station point phenomenon has been focused by Nadeem et al. [15]. Khan and Shehzad [16] analyzedflow of third grade nanofluid with appliances of analytical approach. Ibrahim and co-researchers [17] employed the modified heat and mass flux relations for tangent hyperbolic nanofluid flow over a moving configuration. Eid and Mabdood [18] investigated the entropy generation analysis for thermally developed flow of micropolar dusty carbon nano-materials under the influence of heat generation features. In another investigation, Eid et al. [19] focused on chemically reactive flow of Carreau nanofluid induced by nonlinear stretched configuration. Al-Hossainy et al. [20] examined the rheological features of Casson nanofluid immersed in porous medium where numerical solution was computed by using famous spectral quasi-linearization numerical scheme with excellent accuracy. The utilization of gold nanoparticles in blood flow of Sisko fluid under the influence of nonlinear thermal radiation and suction/injection aspects was directed by Eid et al. [21]. Boumaiza and co-investigators [22] examined the Falkner-Skan flow of nanoparticles (copper, alumina, and magnetite) in presence of external magnetic force. Eid et al. [23] analyzed the shape factors of nanoparticles in blood flow with additional impact of nonlinear thermal radiation and heat source/sink. An interesting numerical approach namely finite element scheme was followed to simulate the solution. Rehman et al. [24] discussed the heat transfer phenomenon in rotatory flow of magnetized nanoparticles induced by rigid disk. Ragupathi et al. [25] focused on thermal performances of Fe

_{3}O

_{4}and Al

_{2}O

_{3}nanoparticles flow over a Riga surface. The flow of ferrofluid in a cavity was numerically tackled by Li et al. [26]. Saranya et al. [27] studied the thermal features of ferrofluid subject to the aligned magnetic force. Besthapu et al. [28] utilized the slip effects in stagnation point flow of nanofluid over a convectively heated surface. Abdelmalek et al. [29] investigated the thermally developed flow of viscous nanofluid in grooved computational channel. Rasool et al. [30] analyzed the aspects I entropy generation in flow of Williamson nanofluid over a nonlinear stretched surface. Reddy and Chamkha [31] analyzed the heat transfer enhancement in flow of Al

_{2}O

_{3}–water and TiO

_{2}–water nanoparticles in presence of diffusion-thermo features. RamReddyet al. [32] examined the Soret effects in mixed convection flow of nanofluid with the help of convective boundary conditions. The Falkner-Skan flow of Williamson nanofluid in presence of variable Prandtl number was focused by Basha et al. [33].

- Develop an unsteady mathematical model for flow of Casson nanofluid induced by a periodically oscillating stretched surface.
- The bioconvection aspects of nanoparticles are studied in presence of gyrotactic microorganisms.
- In current analysis, the viscosity of fluid is assumed to be temperature dependent.
- The novel features like mixed convection, activation energy, and nonlinear thermal radiation are also utilized to examine the heat and mass transfer phenomenon.
- The physical consequences for each flow parameter are illustrated graphically.

## 2. Mathematical Modeling

- The magnetic force features are taken into account by imposing it in vertical directions. Following to the assumption of very large magnetic diffusivity, the effects of induced magnetic field and Hall current are neglected.
- The viscosity of fluid is assumed to be temperature dependent by using famous Reynolds exponential concept.
- The activation energy features are utilizedin the concentration by using Arrhenius relations.
- The nanofluid temperature, concentration, and gyrotactic microorganisms are symbolizedby T, C, and N, respectively.
- Let T
_{w}be the surface temperature, C_{w}is surface concentration, while N_{w}is for surface motile density.

## 3. Homotopy Analysis Method

## 4. Convergence Analysis

## 5. Solution Verification

## 6. Discussion

## 7. Final Remarks

- The temperature-dependent viscosity, thermophoresis parameter, and Casson fluid parameter effectively improve the nanofluid temperature.
- The radiation parameter and heating source constant increases the temperature profile.
- Presence of activation energy and Casson fluid parameter improves the concentration field of nano-materials.
- The increment in Peclet number bioconvection and Lewis number declined microorganisms field while this physical quantity get maximum variation with Casson fluid parameter and bioconvection Rayleigh number.
- The wall shear force oscillates with time which increases for viscosity parameter and Casson fluid parameter.
- The results from the present flow model havevarious fundamental applications in solar energy systems, heat transfer enhancement, cooling and heating processes, environmental applications, thermal engineering, bio-sensors, enzymes, energy consumptions, bio-fuels applications and bio-technology.
- The obtained results can be further extended for different non-Newtonian fluid models by performing the stability analysis and utilizing distinct features like entropy generation, Joule heating, variable thermal conductivity, porous medium etc.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

$\left(u,v\right)$ | Velocity component |

$C$ | Concentration |

${T}_{w}$ | Surface temperature |

${N}_{w}$ | Surface motile density |

${\mu}^{\ast}$ | Temperature dependent viscosity |

${\beta}^{\ast}$ | Coefficient of volume suspension |

${\alpha}_{\otimes}$ | Thermal diffusivity |

$\nu $ | Knematic viscosity |

${\Lambda}^{\otimes}$ | Effective heat nanoparticles and effective base liquid heat capacity |

${E}_{a}$ | Activation energy |

${\rho}_{p}$ | Nanoparticles density |

${\rho}_{m}$ | Motile microorganism density |

$n$ | Rate constant |

${b}^{\oplus}$ | Chemotaxis constant |

$\kappa $ | Boltzmann constant |

$\theta $ | Dimensionless temperature profile |

$S$ | oscillating frequency to stretching rate ratio |

$\mathsf{\Omega}$ | Hartmann number |

$Rb$ | Bioconvection Rayleigh number |

$\mathrm{Pr}$ | Prandtl number |

$\sigma $ | Reaction constant |

$Nt$ | Thermophoresis parameter |

$Pe$ | Peclet number |

$k$ | Thermal conductivity |

${q}_{s}$ | Mass flux |

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

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

$T$ | Temperature |

${\rho}_{f}$ | Fluid density |

${K}_{r}$ | Reaction rate |

${D}_{B}$ | Diffusion constant |

$\kappa $ | Boltzmann constant |

${w}^{\oplus}$ | Swimming cells speed |

$\varphi $ | Concentration profile |

$\chi $ | Motile microorganism |

$\lambda $ | Mixed convection parameter |

$Nb$ | Brownian motion parameter |

$Nr$ | Buoyancy ratio constant |

$\varpi $ | microorganisms concentration difference |

$Le$ | Lewis number |

$E$ | activation energy constant |

$Lb$ | Bioconvection Lewis number |

${q}_{h}$ | Heat flux at wall |

${q}_{n}$ | Motile microorganism flux |

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

$N{n}_{x}$ | Local motile density number |

$N$ | Gyrotactic microorganisms |

${C}_{w}$ | surface concentration |

$t$ | Time |

$\mathsf{\Gamma}$ | Casson fluid parameter |

${\sigma}^{\otimes}$ | Electrical conductivity |

${D}_{m}$ | Microorganisms diffusion constant |

${B}_{0}$ | Magnetic field strength |

$g$ | Gravity |

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**Figure 3.**Variation in $\theta $ for (

**a**) $\mathsf{\Omega}$ and $\mathsf{\Gamma},$ (

**b**) $Nt$ and $Nb,$ (

**c**) $Nr$ and $Rb,$ (

**d**) $\mathrm{Pr}$ and $\delta ,$ (

**e**) $Rd$ and ${\theta}_{w}$.

**Figure 4.**Variation in $\varphi $ for (

**a**) $Nt$ and $\mathsf{\Gamma},$ (

**b**) $S$ and $\mathsf{\Omega},$ (

**c**) $Nr$ and $E$.

**Figure 6.**Variation in ${\mathrm{Re}}_{x}^{1/2}{C}_{f}$ for (

**a**) $\mathsf{\Gamma}$ and (

**b**) $\delta $.

**Table 1.**Comparison of ${f}_{\eta \eta}\left(0,\tau \right)$ with [48] when $S=1,\mathsf{\Gamma}\to \infty ,$ $\mathsf{\Omega}=0,$ $\delta =0,$ $\lambda =0,$ $Nr=0$ and $Rb=0$.

$\mathit{\tau}$ | Abbas et al. [48] | Present Results |
---|---|---|

$\tau =1.5\pi $ | 11.678656 | 11.678657 |

$\tau =5.5\pi $ | 11.678707 | 11.678708 |

$\tau =9.5\pi $ | 11.678656 | 11.678656 |

**Table 2.**Illustration of $-{\theta}_{\eta}\left(0,\tau \right),$ $-{\phi}_{\eta}\left(0,\tau \right)$ and $-{\chi}_{\eta}\left(0,\tau \right)$ for different flow parameters when $\tau =\pi /2$.

$\mathit{\delta}$ | $\mathit{N}\mathit{r}$ | $\mathit{R}\mathit{b}$ | $\mathit{\Gamma}$ | $\mathit{\Omega}$ | $\mathit{\lambda}$ | $-{\mathit{\theta}}_{\mathit{\eta}}\left(0,\mathit{\tau}\right)$ | $-{\mathit{\phi}}_{\mathit{\eta}}\left(0,\mathit{\tau}\right)$ | $-{\mathit{\chi}}_{\mathit{\eta}}\left(0,\mathit{\tau}\right)$ |
---|---|---|---|---|---|---|---|---|

0.2 0.4 0.6 | 0.3 | 0.1 | 0.3 | 0.5 | 0.5 | 0.45639 0.43208 0.41438 | 0.42355 0.41327 0.40768 | 0.57866 0.55554 0.53154 |

0.1 | 0.2 0.4 0.6 | 0.50523 0.47768 0.45455 | 0.47764 0.44542 0.41457 | 0.55657 0.52098 0.50896 | ||||

0.2 0.4 0.6 | 0.51214 0.48365 0.45456 | 0.44458 0.42898 0.40624 | 0.54811 0.53632 0.51614 | |||||

0.2 0.4 0.6 | 0.50256 0.47248 0.42695 | 0.46384 0.43657 0.41468 | 0.57213 0.54112 0.525333 | |||||

0.2 0.4 0.6 | 0.52659 0.47647 0.44892 | 0.44128 0.426589 0.39321 | 0.55526 0.53456 0.505254 | |||||

0.2 0.4 0.6 | 0.51653 0.54895 0.57035 | 0.47486 0.51559 0.53236 | 0.55546 0.565478 0.59598 |

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**MDPI and ACS Style**

Al-Khaled, K.; Khan, S.U.
Thermal Aspects of Casson Nanoliquid with Gyrotactic Microorganisms, Temperature-Dependent Viscosity, and Variable Thermal Conductivity: Bio-Technology and Thermal Applications. *Inventions* **2020**, *5*, 39.
https://doi.org/10.3390/inventions5030039

**AMA Style**

Al-Khaled K, Khan SU.
Thermal Aspects of Casson Nanoliquid with Gyrotactic Microorganisms, Temperature-Dependent Viscosity, and Variable Thermal Conductivity: Bio-Technology and Thermal Applications. *Inventions*. 2020; 5(3):39.
https://doi.org/10.3390/inventions5030039

**Chicago/Turabian Style**

Al-Khaled, Kamel, and Sami Ullah Khan.
2020. "Thermal Aspects of Casson Nanoliquid with Gyrotactic Microorganisms, Temperature-Dependent Viscosity, and Variable Thermal Conductivity: Bio-Technology and Thermal Applications" *Inventions* 5, no. 3: 39.
https://doi.org/10.3390/inventions5030039