# Thermal Case Study of Cilia Actuated Transport of Radiated Blood-Based Ternary Nanofluid under the Action of Tilted Magnetic Field

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}, silicon dioxide SiO

_{2}, and aluminum oxide Al

_{2}O

_{3}in the base fluid (blood). The base fluid is modeled as hyperbolic tangent fluid due to its shear thinning properties. Moreover, the simultaneous effects of inclined magnetic field and thermal radiation are also considered. To the best of our understanding such flow setting has never been studied before now. The numerical solutions for flow and thermodynamical quantities were computed via the shooting method. A comparison between the thermal features of mono/hybrid/ternary nanofluids and traditional blood is presented. The anticipated study may offer a deep insight in bioengineering pump design to be utilized in drug delivery and thermal pipe heat sinks.

## 2. Mathematical Modelling and Governing Equations

#### 2.1. Problem Formulation

_{2}-SiO

_{2}-Al

_{2}O

_{3}/blood that is acquired by the dispersion of 1% volume fraction of each tri-hybrid nanosized metallic particle of Titanium dioxide (TiO

_{2}), silicon dioxide or silica (SiO

_{2}), and Alumina (Al

_{2}O

_{3}) in conventional blood (base fluid) characterized by hyperbolic tangent fluid. Furthermore, the internal surface of the pump is imbedded with a layer of fabricated cilia that blow in organization to start a set of travelling waves moving with constant speed c along the pump wall, named as metachronal waves, as depicted in Figure 1. The present flow is modeled in a rectangular coordinate system in such a way that the wave propagation along the $\overline{X}$-axis and the $\overline{Y}$-axis is orthogonal to the motion.

_{0}is the location of nanofluid particle.

_{∞}is the infinite shear rate viscosity; the effective viscosity for the ternary fluid is measured by μ

_{thnf}at zero shear rate; and Γ is the time constant. Since the tangent hyperbolic fluid shows shear-thinning properties, it is supposed that μ

_{∞}is zero and Γ$\dot{\gamma}$ is smaller than 1; thus, Equation (5) leads to

#### 2.2. Governing Equations in Laboratory Frame

**B**= [B

_{0}sin(χ), B

_{0}cos(χ)] are given as [16,17,18]

^{−2}] as

^{4}about the temperature difference (∆T) in the occurrence of insignificant temperature gradient in the symmetric channel is written as ${\overline{T}}^{4}\cong 4\overline{T}$ (∆T)

^{3}− 3(∆T)

^{4}. Consequently, Equation (14) leads to

#### 2.3. Thermophysical Features of Trihybrid Nanofluid (TiO_{2}-SiO_{2}-Al_{2}O_{3}/Blood)

_{2}), Silox (SiO

_{2}), and Alumina (Al

_{2}O

_{3}) mixed thoroughly in a base liquid, assumed to be pure blood. Titanium dioxide is an alkaline and highly corrosion-resistant coolant agent. It is harmless and is suitable for medical uses. Furthermore, silica and alumina are considered owing to their non-reactive nature and thermodynamical stability, respectively. At the reference temperature 25 °C, the numerical values of thermophysical properties for the trihybrid nanofluid are shown in Table 1.

_{thnf}, and k

_{thnf}) can be expressed, respectively, as [40]

_{p}, σ, k, µ, and ϕ are used for the density, the effective heat capacity, the electrical conductivity, the thermal conductivity, the viscosity, and volume fraction of metallic nanogranules

#### 2.4. Governing Equations in Wave Frame

_{n}are the Hartmann number, the Weissenberg number, the Prandtl number, the Eckert number, and the thermal radiation parameter, which are, respectively, defined as

#### 2.5. Entropy Analysis

_{G}

_{0}deliver the total entropy number N

_{S}as

_{0}) is maintained as one.

## 3. Solution of the Problem

## 4. Computational Results and Discussion

_{S}), and the Bejan number (Be) for the trihybrid nanofluid. Different graphs for various values of the applicable parameters of interest are provided through Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10. In the entire evaluation, the net volumetric proportion of metallic nanogranules, ϕ (=ϕ

_{1}+ ϕ

_{2}+ ϕ

_{3}) scattered in the traditional base fluid is assumed as 3%. Therefore, the blood-based trihybrid nanofluid is deemed to be obtained by combining 1% fraction of each kind of metallic nanogranule (such as titanium dioxide, silicon dioxide, and alumina) in the hyperbolic tangent nanofluid. Since the current parametric study was conducted for pertinent parameters of interest, some of the parameters are assigned with fixed values, such as Ec = n = 0.2, α = 0.3, β = 0.006, and χ = π/3

#### 4.1. Flow and Pumping Characteristics

**Figure 2.**Alterations in the axial velocity distribution for distinct values of (

**a**)Weissenberg number (We) and (

**b**) the Hartmann number (Ha) at fixed values of ε = 0.2 and x = 0.6.

**Figure 3.**Alterations in the pressure gradient profile for distinct values of (

**a**) Weissenberg number (We), (

**b**) Hartmann number (Ha), and (

**c**) cilia length parameter (ε).

**Figure 4.**Alterations in pressure rise per metachronal wavelength for distinct values of (

**a**) Weissenberg number (We), (

**b**) Hartmann number (Ha), and (

**c**) cilia length parameter (ε).

#### 4.2. Thermal Characteristics of the Ternary Fluid

_{n}), and the Prandtl number (Pr). Figure 5a shows that the lofty values of the Hartmann number (Ha) engender an enrichment in ternary nanofluid temperature because of significant convection. This rise in temperature is more pronounced across the pump center than its peripheral area. Figure 5b demonstrates that the elevated values of the thermal radiation number (R

_{n}) retain the capability to substantially drop the medium temperature. This assertion is elaborated since the converse relation between thermal radiation and heat conduction stimulates a cooling process in the medium by dropping the liquid temperature down. Consequently, the radiated nanofluid might be acknowledged as a more expedient cooling agent than the traditional fluid. Figure 5c investigates the consequence of the Prandtl number (Pr) on fluid temperature. In thermal flows, the Prandtl number monitors the comparative of consequence of momentum diffusive rate to the thermal diffusive rate. Small values of Pr indicate that the heat diffuses swiftly compared to velocity dispersion. The higher values of Pr indicate the rapid dispersion of momentum rather than heat. Therefore, a considerable increment in liquid temperature is seen for substantial values of Pr.

**Figure 5.**Alterations in temperature profile for various values of (

**a**) the Hartmann number (Ha), (

**b**) the thermal radiation number (R

_{n}), and (

**c**) the Prandtl number (Pr).

_{s}, Figure 6a–c is exhibited. Figure 6a determines that the elevated values of the Hartman number (Ha) lead to an enlargement in entropy formation near the pump boundary. Around the pump center, the alterations in entropy generation number are minimal. This fact can be confirmed from Table 3, which exhibits the comparative glimpse of heat transfer rate in ternary/binary/mono nanofluid and traditional blood for different values of Ha. From Table 3, it is noticed that around the pump surface, the heat transfer rate enhances for large values of Ha. At the same value of Ha, the heat transfer rate is maximum for the trihybrid nanofluid and minimum for blood model. For instance, at Ha = 0.5, the heat transfer rate is lifted to 6.11% for mono nanofluid, 11.07% for hybrid nanofluid, and 18.23% for trihybrid nanofluid compared to pure blood. Figure 6b communicates that the loft values of the thermal radiation number (R

_{n}) curtail entropy generation around the peripheral zone of the micropump, whereas the effects of R

_{n}near the pump center are negligible. Thus, it can be reported that the radiated ternary nanofluid may be considered as an important factor for the better the functioning of thermal system. Figure 6c conveys that the overall entropy production inside the micropump amplifies for large values of Pr.

**Figure 6.**Variations in entropy generation number distribution for different values of (

**a**) the Hartmann number (Ha), (

**b**) the thermal radiation number (R

_{n}), and (

**c**) the Prandtl number (Pr).

_{n}) assist the leading effect of overall irreversibility of the system over the heat transfer irreversibility. Figure 7c illuminates that the Bejan number seems to be a boosting function of the Prandtl number (Pr). Near the pump center, the Bejan number moves to zero, which indicates the absence of heat transfer irreversibility effects, but close to the pump wall, the heat transfer effects are appreciable.

**Figure 7.**Alterations in the Bejan number distribution for distinct values of (

**a**) the Hartmann number (Ha), (

**b**) the thermal radiation number (R

_{n}), and (

**c**) the Prandtl number (Pr).

#### 4.3. Streamline Pattern and Trapping Phenomenon

**Figure 8.**Streamline pattern for (

**a**) We = 0.01 and (

**b**) We = 0.08 at fixed values of Ha = 1 and ε = 0.4.

**Figure 9.**Streamline pattern for (

**a**) ε = 0.4 and (

**b**) ε = 0.4 at fixed values of Ha = 1 and We = 0.01.

**Figure 10.**Streamline pattern for (

**a**) Ha = 1 and (

**b**) Ha = 1.5 at fixed values of We = 0.01 and ε = 0.4.

## 5. Conclusions

- Around the pump surface, the exalted values of the Hartman number and the Weissenberg number induce an acceleration in fluid velocity.
- In the pump deep part, the ternary fluid pressure gradient enhances for the small values of the magnetic parameter and extended cilia.
- In the pumping zone, the pressure rise per metachronal wavelength in an escalating function of cilia length and small Hartmann number, whereas the same parameters induce a deceleration in pressure rise in the co-pumping zone.
- The temperature of the medium dampens down by contemplating the thermally radiated ternary fluid in a weak magnetic environment.
- Appropriately small values of the Hartmann number and substantially elevated values of the radiation number effectively contribute to entropy reduction in the micropump.
- Weak magnetic field and strong radiation effects assist with the domination of the fluid friction irreversibility over the heat transfer irreversibility.
- The small values of the magnetic parameter and prolonged cilia aids the fluid flow rate augmentation.
- The heat transfer rate is raised by about 6% in mono nanofluid, 11%–12% in bi-hybrid nanofluid, and 12%–18% in ternary fluid compared to conventional blood model for different values of the magnetic parameter.
- The present study may provide a profound perception in bioengineered medical instruments and thermal pipe heat sink. It can further assist with commencing a new course for the consideration of various other ternary nanofluids as desirable coolants under resistive forces, that is, electric and induced magnetic fields, porous media, and gravity effects. Such other possible future extensions may be the consideration of temperature and velocity jumps at the boundary, which could further explore the mechanism of thermo-microfluidics devices.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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Physical Entities | Base Fluid | Solid Nanogranules | ||
---|---|---|---|---|

(Blood) | TiO_{2} | SiO_{2} | Al_{2}O_{3} | |

ρ (kgm^{−3}) | 1063 | 4250 | 2200 | 3970 |

σ (1/Ωm) | 0.8 | 2.4 × 10^{6} | 3.5 × 10^{6} | 36.9 × 10^{6} |

C_{p} (JK^{−1}kg^{−1}) | 3594 | 686.2 | 754 | 765 |

k (Wm^{−1}K^{−1}) | 0.492 | 8.9538 | 1.4013 | 40 |

**Table 2.**The comparison of numerical values of axial velocity distribution (u(y)) near the pump center when We = 0.001, ε = 0.25, and n = 0.2.

Ha | Perturbation Solution | Numerical Solution |
---|---|---|

0.5 | 2.31025 | 2.31118 |

1 | 2.25681 | 2.28942 |

1.5 | 2.21310 | 2.25518 |

2 | 2.10603 | 2.21105 |

2.5 | 2.04358 | 2.16001 |

**Table 3.**The heat transfer rate for blood, mono nanofluid, hybrid nanofluid, and tri-hybrid nanofluid at fixed values of Ec = 0.1, Pr = 1, and R

_{n}= 1.

Ha | Blood | Mono Nanofluid | % Increase | Bi-Hybrid Nanofluid | % Increase | Ternary Nanofluid | % Increase |
---|---|---|---|---|---|---|---|

0.5 | 0.87458 | 0.92805 | 6.11 | 0.97141 | 11.07 | 1.03398 | 18.23 |

1 | 1.03909 | 1.10407 | 6.25 | 1.15685 | 11.33 | 1.21239 | 16.68 |

1.5 | 1.31190 | 1.39597 | 6.41 | 1.46436 | 11.62 | 1.50839 | 14.98 |

2 | 1.69127 | 1.80186 | 6.54 | 1.89194 | 11.86 | 1.92020 | 13.54 |

2.5 | 2.17514 | 2.31953 | 6.64 | 2.43729 | 12.05 | 2.44572 | 12.45 |

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**MDPI and ACS Style**

Saleem, N.; Ashraf, T.; Daqqa, I.; Munawar, S.; Idrees, N.; Afzal, F.; Afzal, D.
Thermal Case Study of Cilia Actuated Transport of Radiated Blood-Based Ternary Nanofluid under the Action of Tilted Magnetic Field. *Coatings* **2022**, *12*, 873.
https://doi.org/10.3390/coatings12060873

**AMA Style**

Saleem N, Ashraf T, Daqqa I, Munawar S, Idrees N, Afzal F, Afzal D.
Thermal Case Study of Cilia Actuated Transport of Radiated Blood-Based Ternary Nanofluid under the Action of Tilted Magnetic Field. *Coatings*. 2022; 12(6):873.
https://doi.org/10.3390/coatings12060873

**Chicago/Turabian Style**

Saleem, Najma, Tahreem Ashraf, Ibtisam Daqqa, Sufian Munawar, Nazeran Idrees, Farkhanda Afzal, and Deeba Afzal.
2022. "Thermal Case Study of Cilia Actuated Transport of Radiated Blood-Based Ternary Nanofluid under the Action of Tilted Magnetic Field" *Coatings* 12, no. 6: 873.
https://doi.org/10.3390/coatings12060873