# Modulated Viscosity-Dependent Parameters for MHD Blood Flow in Microvessels Containing Oxytactic Microorganisms and Nanoparticles

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

**CP-Model**. Noting that this simplified set of equations (

**CP-Model**) contains dimensionless parameters that function in viscosity. Henceforward, we must consider these parameters as variables, which will result in a new modulated set of governing equations, which we called the

**VP-Model**. It worth mentioning that this remarkable announcement leads to realistic results contrary to these sets of equations that treat the parameters that rely on viscosity as constants. Both non-linear coupled systems of equations of the

**CP-Model**and the

**VP-Model**were solved numerically via the computational software program Mathematica 11. The effects of various dimensionless parameters of interest are presented through plots. Physical interpretations of the results are thoroughly discussed, and the paramount outcomes are then summarized.

## 2. Physical Description

#### 2.1. The Carreau–Yasuda Model

#### 2.2. Model Formulation

#### 2.3. Governing Equations

#### 2.4. Transformations and Simplifications

#### 2.5. Variable Non-Dimensional Parameters

#### 2.6. Boundary Conditions

## 3. Results and Discussion

**CP-Model**and the system of Equations (44)–(49) labeled as the

**VP-Model**, along with the boundary conditions Equations (50) and (51), were solved numerically with the aid of the most sophisticated version of the computational software program Mathematica 11.3. The obtained solutions via the built-in command ParametricNDSolve were displayed through Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9 to analyze the problem’s physics. Parametric study for both the CP-Model and the VP-Model on temperature, nanoparticles volume fraction, microorganism density profiles, longitudinal velocity, and longitudinal pressure gradients is carried out. The selected values for all embedded parameters are in Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9 (see Table 1 and Table 2).

- (1)
- Show that the VP-Model is more reliable than the CP-Model.
- (2)
- Investigate the features of adding oxytactic microorganisms (as oxygen repellents) to the upstream of the blood flow.
- (3)
- Examine the magnetic field parameters’ role in the presence/absence of the Joule heating effect for the VP-Model case.
- (4)
- Discuss the theoretical significance of the current study’s ambient parameters supported with experimental agreements and theoretical studies whenever possible.

#### 3.1. CP-Model Versus VP-Model

#### Discussion on Some Previous Works

#### 3.2. Oxytactic Microorganism Parameters

#### 3.3. Magnetic Field Parameters and Joule Heating Effects

#### 3.4. Role of Nanoparticles Parameters

#### 3.5. Role of Carreau–Yasuda Fluid Parameters

#### 3.6. Role of Thermal Radiation, Buoyancy, Chemical Reaction and Flow Rate Effects

#### 3.7. Streamlines and Trapping

## 4. Conclusions

- (1)
- We have confirmed the VP-Model’s reliability over the CP-Model by referring to physical phenomena and past experimental results.
- (2)
- Disregarding the Joule heating effects in modeling MHD flow problems in the presence of heat and mass transfer will result in unrealistic outcomes.
- (3)
- Microorganism density is an increasing function of${N}_{tv},\alpha ,{\sigma}_{1},{\rho}_{ev},{\lambda}_{1}$and${M}_{v},$ whereas it decreases with increasing${\mathsf{\lambda}}_{2},{N}_{bv}$ and$m$.
- (4)
- Rising in oxygen concentrations causes the microorganism density to increase in the direction towards the hypoxic tumor regions; even a reduction in blood viscosity is regarded.
- (5)
- The impacts of${N}_{tv}$on the temperature and nanoparticle volume fraction profiles are quite the opposite. On the other hand, the influences of ${N}_{bv}$on the nanoparticle volume fraction and temperature profiles are similar.
- (6)
- In the presence of a generative chemical reaction, the temperature, nanoparticle concentration, and microorganism density profiles are boosted, but the opposite trend occurred for a destructive chemical reaction.
- (7)
- It is remarkable to observe that the impact of $\mathrm{We}$ on velocity in blood shear-thinning is quite the opposite of the case of blood shear-thickening.
- (8)
- Surprisingly, it is noticed that the pressure gradient starts showing an alternating increase/decrease behavior upon substituting positive constitutive integer values for the Carreau–Yasuda index parameter$a\left(a=1,2,3,\dots \right).$ The maximum pressure gradient occurs for the case of the Carreau fluid$(a=2$).
- (9)
- The number and size of the trapped bolus decreases with an increase in $\alpha $ and ${M}_{v}$, while increasing with an increase of $\beta .$

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**A schematic representation of blood in a microvessel containing microorganisms and nanoparticles.

**Figure 2.**$\left(a\right)$ Influence of $\beta $ on $\theta $. $\left(b\right)$ Influence of $\beta $ on $\theta .$ $\left(c\right)$ Influence of $\beta $ on $\phi .$ $\left(d\right)$ Influence of $\beta $ on $\phi .$

**Figure 3.**$\left(a\right)$ Influence of ${\rho}_{ev}$ on $\Omega $. $\left(b\right)$ Influence of ${\sigma}_{1}$on $\Omega .$ $\left(c\right)$ Influence of ${\lambda}_{1}$ on $\Omega .$ $\left(d\right)$ Influence of ${\lambda}_{1}$ on ${\Omega}^{\prime}.$ $\left(e\right)$ Influence of ${\lambda}_{2}$ on $\Omega $. $\left(f\right)$ Influence of ${\lambda}_{2}$ on ${\Omega}^{\prime}.$ $\left(g\right)$ Influence of ${R}_{bv}$ on $\theta .$ $\left(h\right)$ Influence of${R}_{bv}$ on $u.$

**Figure 4.**$\left(a\right)$ Influence of ${M}_{v}$ on $\theta $. $\left(b\right)$ Influence of ${M}_{v}$ on $\phi .$ $\left(c\right)$ Influence of $m$ on $\theta .$ $\left(d\right)$ Influence of $m$on $\phi .$ $\left(e\right)$ Influence of ${M}_{v}$ on $\frac{dp}{dx}$. $\left(f\right)$ Influence of $m$ on $\frac{dp}{dx}.$ $\left(g\right)$ Influence of ${M}_{v}$ on $\Omega .$ $\left(h\right)$ Influence of $m$ on $\Omega .$ $\left(i\right)$ Influence of ${M}_{0}$ on $\Omega .$

**Figure 5.**(

**a**) Influence of ${M}_{v}$ on $\theta $. (

**b**) Influence of $m$ on $\phi .$ (

**c**) Influence of ${M}_{v}$on $\theta .$ (

**d**) Influence of $m$ on $\phi $.

**Figure 6.**$\left(a\right)$ Influence of ${N}_{tv}$ on $\frac{dp}{dx}$. $\left(b\right)$ Influence of ${N}_{tv}$ on $\theta .$ $\left(c\right)$ Influence of ${N}_{tv}$ on $\Omega .$ $\left(d\right)$ Influence of ${N}_{tv}$on $\phi .$ $\left(e\right)$ Influence of ${N}_{bv}$ on $\frac{dp}{dx}$. $\left(f\right)$ Influence of${N}_{bv}$ on $\Omega .$ $\left(g\right)$ Influence of ${N}_{bv}$ on $\theta .$ (h) Influence of ${N}_{bv}$ on $\phi .$

**Figure 7.**$\left(a\right)$ Influence of $\beta $ on $\xi $. $\left(b\right)$ Influence of $\beta $ on $\xi .$ $\left(c\right)$ Influence of $\beta $ on $\Omega .$ $\left(d\right)$ Influence of $\beta $ on $\phi .$ $\left(e\right)$ Influence of $\mathsf{\alpha}$ on $\Omega $. $\left(f\right)$ Influence of $n$ on $\frac{dp}{dx}.$ $\left(g\right)$ Influence of $We$ on $u.$ $\left(h\right)$ Influence of $We$ on $u.$ $\left(i\right)$ Influence of $a$ on $\frac{dp}{dx}.$ $\left(j\right)$ Influence of $a$ on $u.$

**Figure 8.**$\left(a\right)$ Influence of${R}_{nv}$ on $\phi $. (

**b**) Influence of${R}_{nv}$ on $\theta .$ $\left(c\right)$ Influence of${\u03f5}_{v}$on $\phi .$ $\left(d\right)$ Influence of ${\u03f5}_{v}$on $\theta .$ $\left(e\right)$ Influence of${G}_{tv}$ on$\frac{dp}{dx}$. $\left(f\right)$ Influence of${G}_{cv}$ on $\frac{dp}{dx}.$ $\left(g\right)$ Influence of${\u03f5}_{v}$ on $\Omega .$ $\left(h\right)$ Influence of$q$ on $\frac{dp}{dx}.$

**Figure 9.**Streamline patterns for different values of$\left(a\right)\beta $. $\left(b\right)$ $\alpha .$ $\left(c\right)$ ${M}_{v}$.

Figure | Dimensionless Constant Parameters | |||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$c$ | $b$ | $d$ | $\varphi $ | ${M}_{0}$ | $\beta $ | ${\rho}_{e0}$ | ${N}_{b0}$ | ${N}_{t0}$ | ${G}_{c0}$ | ${G}_{t0}$ | ${\sigma}_{1}$ | ${R}_{n0}$ | ${R}_{b0}$ | ${E}_{c}$ | $m$ | $\alpha $ | $a$ | $q$ | ${P}_{r0}$ | $We$ | ${\lambda}_{1}$ | ${\lambda}_{2}$ | $n$ | ${\u03f5}_{0}$ | ||

Figure 2a | 0.3 | 0.5 | 1.1 | $\pi /2$ | 0.9 | - | 0.8 | 1.1 | 0.9 | 0.5 | 0.7 | 0.8 | 0.5 | 3 | 0.25 | 0.6 | 0.15 | 1 | $-0.1$ | 4 | 0.1 | 0.5 | 0.15 | 0.5 | 0.5 | |

Figure 2c | 0.3 | 0.5 | 1.1 | $\pi /2$ | 0.9 | - | 0.8 | 1.1 | 0.9 | 0.5 | 0.6 | 0.6 | 0.5 | 1 | 0.25 | 0.6 | 0.25 | 1 | $0$ | 4 | 0.15 | 0.3 | 0.1 | 0.5 | 0.2 | |

Figure 4i | 0.3 | 0.5 | 1.1 | $\pi /4$ | - | 0.35 | 0.8 | 1.2 | 0.5 | 0.5 | 0.5 | 0.7 | 0.5 | 2 | 0.25 | 0.5 | 0.25 | 1 | $-0.1$ | 3 | 0.15 | 0.5 | 0.15 | 0.5 | 0.8 | |

Figure 7a | 0.3 | 0.5 | 1.1 | $\pi /2$ | 0.7 | - | 0.85 | 0.8 | 0.8 | 0.6 | 0.6 | 0.8 | 0.5 | 1 | 0.25 | 0.7 | 0.25 | 1 | $0$ | 3 | 0.15 | 0.5 | 0.1 | 0.5 | 0.4 |

Figure | Dimensionless Variable Parameters | |||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$c$ | $b$ | $d$ | $\varphi $ | ${M}_{v}$ | $\beta $ | ${\rho}_{ev}$ | ${N}_{bv}$ | ${N}_{tv}$ | ${G}_{cv}$ | ${G}_{tv}$ | ${\sigma}_{1}$ | ${R}_{nv}$ | ${R}_{bv}$ | ${E}_{c}$ | $m$ | $\alpha $ | $a$ | $q$ | ${P}_{rv}$ | $We$ | ${\lambda}_{1}$ | ${\lambda}_{2}$ | $n$ | ${\u03f5}_{v}$ | ||

Figure 2b | 0.3 | 0.5 | 1.1 | $\frac{\pi}{2}$ | 0.9 | - | 0.8 | 1.1 | 0.9 | 0.5 | 0.7 | 0.8 | 0.5 | 3 | 0.25 | 0.6 | 0.15 | 1 | $-0.1$ | 4 | 0.1 | 0.5 | 0.15 | 0.5 | 0.5 | |

Figure 2d | 0.3 | 0.5 | 1.1 | $\frac{\pi}{2}$ | 0.9 | - | 0.8 | 1.1 | 0.9 | 0.5 | 0.6 | 0.6 | 0.5 | 1 | 0.25 | 0.6 | 0.25 | 1 | 0 | 1 | 0.1 | 0.3 | 0.1 | 0.5 | 0.2 | |

Figure 3a | 0.3 | 0.5 | 1.0 | $\frac{\pi}{2}$ | 0.8 | 0.15 | - | 1.2 | 0.8 | 0.9 | 0.7 | 0.7 | 0.5 | 2 | 0.25 | 0.6 | 0.2 | 1 | $-0.1$ | 3 | 0.1 | 0.5 | 0.15 | 0.5 | 0.4 | |

Figure 3b | 0.4 | 0.5 | 1.0 | $\frac{\pi}{6}$ | 0.8 | 0.2 | 0.7 | 1.2 | 0.7 | 0.5 | 0.6 | - | 0.5 | 1 | 0.25 | 0.6 | 0.15 | 1 | $-0.1$ | 3 | 0.1 | 0.5 | 0.15 | 0.5 | 0.8 | |

Figure 3c | 0.3 | 0.5 | 1.1 | $\frac{\pi}{3}$ | 0.9 | 0.2 | 0.7 | 1.3 | 0.8 | 0.6 | 0.6 | 0.8 | 0.5 | 1 | 0.25 | 0.7 | 0.15 | 1 | $-0.1$ | 3 | 0.15 | - | 0.15 | 0.5 | 0.4 | |

Figure 3d | 0.3 | 0.5 | 1.1 | $\frac{\pi}{3}$ | 0.9 | 0.2 | 0.7 | 1.3 | 0.8 | 0.6 | 0.6 | 0.8 | 0.5 | 1 | 0.25 | 0.7 | 0.15 | 1 | $-0.1$ | 3 | 0.15 | - | 0.15 | 0.5 | 0.4 | |

Figure 3e | 0.3 | 0.5 | 1.1 | $\frac{\pi}{2}$ | 0.9 | 0.2 | 0.7 | 1.3 | 0.8 | 0.6 | 0.6 | 0.8 | 0.5 | 1 | 0.25 | 0.7 | 0.15 | 1 | $-0.1$. | 3 | 0.15 | 0.5 | - | 0.5 | 0.4 | |

Figure 3f | 0.3 | 0.5 | 1.1 | $\frac{\pi}{2}$ | 0.9 | 0.2 | 07 | 1.3 | 0.8 | 0.6 | 0.6 | 0.8 | 0.5 | 1 | 0.25 | 0.7 | 0.15 | 1 | $-0.1$ | 3 | 0.15 | 0.5 | - | 0.5 | 0.4 | |

Figure 3g | 0.3 | 0.5 | 1.1 | $\frac{\pi}{3}$ | 0.8 | 0.4 | 0.7 | 1.2 | 0.9 | 0.7 | 0.7 | 0.5 | 0.5 | - | 0.25 | 0.6 | 0.15 | 1 | $-0.1$ | 4 | 0.15 | 0.5 | 0.15 | 0.5 | 0.8 | |

Figure 3h | 0.3 | 0.5 | 1.1 | $\frac{\pi}{2}$ | 0.8 | 0.3 | 0.9 | 1.2 | 0.6 | 0.7 | 0.8 | 0.6 | 0.5 | - | 0.25 | 0.7 | 0.15 | 1 | $-0.1$ | 3 | 0.15 | 0.5 | 0.15 | 0.5 | 0.4 | |

Figure 4a | 0.3 | 0.5 | 1.1 | $\frac{\pi}{2}$ | - | 0.25 | 0.9 | 1.1 | 0.7 | 0.6 | 0.6 | 0.8 | 0.5 | 1 | 0.25 | 0.7 | 0.2 | 1 | $-0.1$ | 3 | 0.15 | 0.5 | 0.15 | 0.5 | 0.5 | |

Figure 4b | 0.4 | 0.5 | 1.1 | $\frac{\pi}{2}$ | - | 0.25 | 0.9 | 1.1 | 0.7 | 0.6 | 0.6 | 0.8 | 0.5 | 1 | 0.25 | 0.7 | 0.2 | 1 | $-0.1$ | 3 | 0.15 | 0.5 | 0.15 | 0.5 | 0.2 | |

Figure 4c | 0.3 | 0.5 | 1.1 | $\frac{\pi}{2}$ | 0.9 | 0.25 | 0.9 | 1.1 | 0.7 | 0.6 | 0.6 | 0.8 | 0.5 | 1 | 0.25 | - | 0.2 | 1 | $-0.1$ | 3 | 0.15 | 0.5 | 0.15 | 0.5 | 0.2 | |

Figure 4d | 0.4 | 0.5 | 1.1 | $\frac{\pi}{2}$ | 0.9 | 0.25 | 0.9 | 1.1 | 0.7 | 0.6 | 0.6 | 0.8 | 0.5 | 1 | 0.25 | - | 0.2 | 1 | $-0.1$. | 3 | 0.15 | 0.5 | 0.15 | 0.5 | 0.2 | |

Figure 4e | 0.4 | 0.7 | 1.1 | $\frac{\pi}{6}$ | - | 0.3 | 0.7 | 1.2 | 0.7 | 0.7 | 0.8 | 0.7 | 0.5 | 1 | 0.25 | 0.9 | 0.25 | 1 | $-1.3$ | 3 | 0.15 | 0.5 | 0.12 | 0.5 | 0.5 | |

Figure 4f | 0.4 | 0.7 | 1.1 | $\frac{\pi}{6}$ | 0.6 | 0.3 | 0.7 | 1.2 | 0.8 | 0.7 | 0.8 | 0.7 | 0.5 | 3 | 0.25 | - | 0.25 | 1 | $-1.3$ | 3 | 0.15 | 0.5 | 0.15 | 0.5 | 0.5 | |

Figure 4g | 0.3 | 0.5 | 1.1 | $\frac{\pi}{4}$ | - | 0.35 | 0.8 | 1.2 | 0.5 | 0.5 | 0.5 | 0.7 | 0.5 | 2 | 0.25 | 0.5 | 0.25 | 1 | $-0.1$ | 4 | 0.15 | 0.5 | 0.15 | 0.5 | 0.8 | |

Figure 4h | 0.4 | 0.5 | 1.1 | $\frac{\pi}{4}$ | 0.9 | 0.35 | 0.8 | 1.2 | 0.5 | 0.6 | 0.5 | 0.7 | 0.5 | 2 | 0.25 | - | 0.25 | 1 | $\frac{\pi}{6}$ | 4 | 0.15 | 0.5 | 0.15 | 0.5 | 0.8 | |

Figure 5a | 0.3 | 0.5 | 1.1 | $\frac{\pi}{6}$ | - | 0.4 | 0.6 | 1.4 | 1.1 | 0.9 | 0.5 | 0.7 | 0.5 | 4 | 0.25 | 1 | 0.2 | 1 | $0.1$ | 5 | 0.15 | 0.5 | 0.15 | 0.5 | 0.2 | |

Figure 5b | 0.3 | 0.5 | 1.1 | $\frac{\pi}{6}$ | 1.1 | 0.45 | 0.6 | 1.4 | 0.9 | 0.8 | 1.4 | 0.7 | 0.5 | 4 | 0.25 | - | 0.2 | 1 | $0.1$ | 5 | 0.15 | 0.5 | 0.15 | 0.5 | 0.2 | |

Figure 5c | 0.3 | 0.5 | 1.1 | $\frac{\pi}{6}$ | - | 0.4 | 0.5 | 1.4 | 0.9 | 0.8 | 1.1 | 0.6 | 0.5 | 4 | 0.3 | 0.6 | 0.2 | 1 | $0.1$ | 5 | 0.15 | 0.5 | 0.15 | 0.5 | 0.2 | |

Figure 5d | 0.3 | 0.4 | 1.1 | $\frac{\pi}{2}$ | 1.3 | 0.3 | 0.6 | 0.9 | 0.8 | 0.8 | 1.4 | 0.6 | 0.5 | 1 | 0.3 | - | 0.2 | 1 | $0.1$ | 5 | 0.1 | 0.5 | 0.15 | 0.5 | 0.2 | |

Figure 6a | 0.3 | 0.5 | 1 | $\frac{\pi}{12}$ | 0.8 | 0.3 | 0.7 | 1.2 | - | 0.7 | 0.5 | 0.7 | 0.5 | 4 | 0.25 | 0.7 | 0.25 | 1 | $-1.3$ | 4 | 0.15 | 0.5 | 0.1 | 0.5 | 0.2 | |

Figure 6b | 0.3 | 0.4 | 1.1 | $\frac{\pi}{4}$ | 0.9 | 0.4 | 0.7 | 1.4 | - | 0.5 | 0.5 | 0.7 | 0.5 | 1 | 0.25 | 0.7 | 0.25 | 1 | $-0.1$ | 3 | 0.15 | 0.4 | 0.1 | 0.5 | 0.5 | |

Figure 6c | 0.4 | 0.5 | 1.1 | $\frac{\pi}{2}$ | 0.9 | 0.35 | 0.8 | 1.1 | - | 0.6 | 0.6 | 0.7 | 0.5 | 1 | 0.25 | 0.7 | 0.25 | 1 | $-0.1$ | 3 | 0.15 | 0.5 | 0.15 | 0.5 | 0.4 | |

Figure 6d | 0.4 | 0.5 | 1.1 | $\frac{\pi}{3}$ | 0.9 | 0.4 | 0.7 | 1.3 | - | 0.5 | 0.5 | 0.7 | 0.5 | 1 | 0.25 | 0.7 | 0.25 | 1 | $-0.1$ | 3 | 0.15 | 0.4 | 0.12 | 0.5 | 0.5 | |

Figure 6e | 0.3 | 0.5 | 1 | $\frac{\pi}{2}$ | 0.8 | 0.3 | 0.7 | - | 0.7 | 0.8 | 0.5 | 0.8 | 0.5 | 4 | 0.25 | 0.7 | 0.25 | 1 | $-1.3$ | 4 | 0.15 | 0.5 | 0.15 | 0.5 | 0.5 | |

Figure 6f | 0.3 | 0.5 | 0.9 | $\frac{\pi}{6}$ | 0.6 | 0.4 | 0.8 | - | 1 | 0.8 | 0.7 | 0.7 | 0.5 | 1 | 0.25 | 0.6 | 0.25 | 1 | $-0.1$ | 3 | 0.15 | 0.5 | 0.15 | 0.5 | 0.4 | |

Figure 6g | 0.3 | 0.5 | 1.1 | $\frac{\pi}{2}$ | 0.9 | 0.35 | 0.8 | - | 1 | 0.8 | 0.6 | 0.6 | 0.5 | 1 | 0.25 | 0.5 | 0.25 | 1 | $-0.1$ | 3 | 0.15 | 0.5 | 0.15 | 0.5 | 0.4 | |

Figure 6h | 0.3 | 0.5 | 0.9 | $\frac{\pi}{3}$ | 0.7 | 0.3 | 0.7 | - | 0.6 | 0.5 | 0.5 | 0.7 | 0.5 | 1 | 0.25 | 0.8 | 0.25 | 1 | $-0.1$ | 3 | 0.15 | 0.4 | 0.12 | 0.5 | 0.5 | |

Figure 7b | 0.3 | 0.4 | 1.1 | $\frac{\pi}{2}$ | 0.7 | - | 0.85 | 1.3 | 0.8 | 0.6 | 0.6 | 0.8 | 0.5 | 1 | 0.25 | 0.7 | 0.25 | 1 | 0 | 3 | 0.15 | 0.5 | 0.1 | 0.5 | 0.4 | |

Figure 7c | 0.3 | 0.5 | 1.1 | $\frac{\pi}{3}$ | 0.9 | - | 0.8 | 1.3 | 0.7 | 0.7 | 0.9 | 0.8 | 0.5 | 2 | 0.25 | 0.6 | 0.2 | 1 | $-0.1$ | 4 | 0.15 | - | - | 0.5 | 0.5 | |

Figure 7d | 0.3 | 0.5 | 1.1 | $\frac{\pi}{3}$ | 0.9 | - | 0.8 | 1.3 | 0.7 | 0.7 | 0.9 | 0.8 | 0.5 | 2 | 0.25 | 0.6 | 0.2 | 1 | $-0.1$ | 4 | 0.15 | - | - | 0.5 | 0.5 | |

Figure 7e | 0.4 | 0.6 | 1.1 | $\frac{\pi}{2}$ | 0.8 | 0.25 | 1 | 1.2 | 0.7 | 0.6 | 0.7 | 0.5 | 0.5 | 2 | 0.25 | 0.6 | - | 1 | $-0.1$ | 4 | 0.1 | 0.5 | 0.1 | 0.5 | 0.25 | |

Figure 7f | 0.3 | 0.5 | 0.9 | $\frac{\pi}{4}$ | 0.8 | 0.4 | 0.7 | 1.3 | 0.6 | 0.7 | 0.8 | 0.5 | 0.5 | 1 | 0.25 | 0.6 | 0.2 | 1 | $-1.1$ | 4 | 0.15 | 0.5 | 0.15 | - | 0.4 | |

Figure 7g | 0.3 | 0.5 | 0.9 | $\frac{\pi}{3}$ | 0.7 | 0.3 | 0.7 | 1.1 | 0.7 | 0.6 | 0.8 | 0.7 | 0.5 | 3 | 0.25 | 0.8 | 0.2 | 1 | $-0.5$ | 3 | - | 0.5 | 0.15 | - | 0.4 | |

Figure 7h | 0.3 | 0.5 | 0.9 | $\frac{\pi}{3}$ | 0.7 | 0.3 | 0.7 | 1.1 | 0.7 | 0.6 | 0.8 | 0.7 | 0.5 | 3 | 0.25 | 0.6 | 0.2 | 1 | $-0.5$ | 3 | - | 0.5 | 0.15 | - | 0.4 | |

Figure 7i | 0.3 | 0.5 | 1 | $\frac{\pi}{2}$ | 0.9 | 0.3 | 0.7 | 1.3 | 0.8 | 0.7 | 1.2 | 0.7 | 0.5 | 2 | 0.25 | 0.7 | 0.2 | - | $-1.2$ | 4 | 0.3 | 0.5 | 0.12 | 0.5 | 0.5 | |

Figure 7j | 0.3 | 0.5 | 0.9 | $\frac{\pi}{3}$ | 0.8 | 0.4 | 1 | 1.1 | 0.7 | 0.6 | 0.8 | 0.7 | 0.5 | 4 | 0.25 | 1 | 0.25 | - | $-1.3$ | 3 | 0.1 | 0.5 | 0.15 | 0.5 | 0.6 | |

Figure 8a | 0.3 | 0.5 | 1.1 | $\frac{\pi}{4}$ | 0.8 | 0.3 | 0.8 | 1.1 | 0.8 | 0.6 | 0.7 | 0.7 | - | 3 | 0.25 | 0.7 | 0.2 | 1 | $-0.1$ | 3 | 0.15 | 0.5 | 0.15 | 0.5 | 0.4 | |

Figure 8b | 0.3 | 0.5 | 0.9 | $\frac{\pi}{4}$ | 0.8 | 0.4 | 0.8 | 1.1 | 0.7 | 0.6 | 0.7 | 0.7 | - | 3 | 0.25 | 0.7 | 0.25 | 1 | $-0.1$ | 3 | 0.1 | 0.5 | 0.15 | 0.5 | 0.4 | |

Figure 8c | 0.3 | 0.5 | 1.1 | $\frac{\pi}{2}$ | 0.8 | 0.25 | 0.7 | 1.2 | 0.7 | 0.8 | 0.8 | 0.7 | 0.5 | 1 | 0.25 | 0.6 | 0.15 | 1 | $-0.1$ | 3 | 0.15 | 0.5 | 0.12 | 0.5 | - | |

Figure 8d | 0.3 | 0.5 | 1.1 | $\frac{\pi}{2}$ | 0.8 | 0.25 | 0.7 | 1.2 | 0.7 | 0.8 | 0.8 | 0.8 | 0.5 | 1 | 0.25 | 0.6 | 0.15 | 1 | $-0.1$ | 3 | 0.15 | 0.5 | 0.12 | 0.5 | - | |

Figure 8e | 0.3 | 0.5 | 1.1 | $\frac{\pi}{12}$ | 0.8 | 0.4 | 0.8 | 1.2 | 0.6 | 0.7 | - | 0.7 | 0.5 | 1 | 0.2 | 0.6 | 0.25 | 1 | $-1.3$ | 3 | 0.15 | 0.5 | 0.15 | 0.5 | 0.5 | |

Figure 8f | 0.3 | 0.5 | 1.1 | $\frac{\pi}{12}$ | 0.8 | 0.4 | 0.8 | 1.1 | 0.6 | - | 0.7 | 0.7 | 0.5 | 1 | 0.25 | 0.6 | 0.25 | 1 | $-1.3$ | 3 | 0.1 | 0.5 | 0.15 | 0.5 | 0.5 | |

Figure 8g | 0.3 | 0.5 | 1.1 | $\frac{\pi}{2}$ | 0.5 | 0.6 | 0.6 | 1 | 0.9 | 0.6 | 0.7 | 0.9 | 0.5 | 5 | 0.25 | 0.6 | 0.15 | 1 | $-0.2$ | 4 | 0.15 | 0.5 | 0.15 | 0.5 | - | |

Figure 8h | 0.3 | 0.5 | 1.1 | 0 | 0.8 | 0.4 | 0.8 | 1 | 0.6 | 0.5 | 0.7 | 0.7 | 0.5 | 1 | 0.25 | 0.6 | 0.25 | 1 | - | 4 | 0.1 | 0.5 | 0.15 | 0.5 | 0.5 | |

Figure 9a | 0.4 | 0.5 | 1.1 | $\frac{\pi}{4}$ | 0.8 | - | 0.8 | 1 | 0.5 | 0.8 | 0.8 | 0.7 | 0.5 | 1 | 0.25 | 0.7 | 0.15 | 1 | $-0.2$ | 4 | 0.15 | 0.5 | 0.15 | 0.5 | - | |

Figure 9b | 0.3 | 0.5 | 1.1 | $\frac{\pi}{4}$ | 0.8 | 0.3 | 0.8 | 1.2 | 0.5 | 0.8 | 0.8 | 0.7 | 0.5 | 1 | 0.25 | 0.7 | - | 1 | $-0.2$ | 4 | 0.15 | 0.5 | 0.15 | 0.5 | 0.5 | |

Figure 9c | 0.3 | 0.5 | 1.1 | $\frac{\pi}{4}$ | - | 0.3 | 0.8 | 1.2 | 0.5 | 0.8 | 0.8 | 0.7 | 0.5 | 1 | 0.25 | 0.7 | 0.15 | 1 | $-0.2$ | 4 | 0.15 | 0.5 | 0.15 | 0.5 | 0.5 |

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**MDPI and ACS Style**

Elogail, M.A.; Mekheimer, K.S.
Modulated Viscosity-Dependent Parameters for MHD Blood Flow in Microvessels Containing Oxytactic Microorganisms and Nanoparticles. *Symmetry* **2020**, *12*, 2114.
https://doi.org/10.3390/sym12122114

**AMA Style**

Elogail MA, Mekheimer KS.
Modulated Viscosity-Dependent Parameters for MHD Blood Flow in Microvessels Containing Oxytactic Microorganisms and Nanoparticles. *Symmetry*. 2020; 12(12):2114.
https://doi.org/10.3390/sym12122114

**Chicago/Turabian Style**

Elogail, M. A., and Kh. S. Mekheimer.
2020. "Modulated Viscosity-Dependent Parameters for MHD Blood Flow in Microvessels Containing Oxytactic Microorganisms and Nanoparticles" *Symmetry* 12, no. 12: 2114.
https://doi.org/10.3390/sym12122114