# Stability and Convergence Analysis of a Biomagnetic Fluid Flow Over a Stretching Sheet in the Presence of a Magnetic Field

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model of the Flow

## 3. Mathematical Formulation

## 4. Numerical Method

## 5. Stability and Convergence Analysis

## 6. Results and Discussion

#### 6.1. Justification of the Grid Space

#### 6.2. Steady-State Solution

#### 6.3. Estimation of Parameters

_{0}and M

_{0}are the magnetic induction number and magnetization number, respectively. For pure blood, it has been found that the saturation magnetization of $60\text{}{\mathrm{Am}}^{-1}$ is attained for a magnetic field strength above 6 T [8,11]. It is possible to attain order of magnitudes greater magnetization artificially, i.e., via the addition of magnetic nanoparticles, using much lower magnetic field strengths, such as 1 T or even less. The electrical conductivity of blood could also be artificially increased, and thus, the range of values of ${M}_{F}$ and ${M}_{M}$ could vary significantly depending on the specific application. For the present case, we considered some reference values of the magnetic field induction B

_{0}, and we considered ${M}_{F}$ and ${M}_{M}$ as determined by Equation (34). It is apparent that someone could arrive at totally different values depending on a specialized application, such as reinforcing polarization or electrical conductivity, by adding magnetic nanoparticles or electrolytes, respectively. Some corresponding values of ferromagnetic $\left({M}_{F}\right)$ and magnetohydrodynamic $\left({M}_{M}\right)$ numbers with a reference magnetic induction are given in Table 1.

## 7. Conclusions

- The fluid primary velocity and temperature distribution decreased in the area of the boundary layer with the simultaneous increase of the MHD and FHD parameters, whereas the corresponding secondary velocities were increased.
- The increment of the magnetic field resulted in a reduction of the skin friction coefficient on the wall, whereas the heat transfer on the wall was increased.
- With a greater elapsing of time $\left(\tau \right)$, the velocity and temperature profiles were found to be enhanced.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

$(u,v)$ | Velocity components in the $x,y$ direction $(\mathrm{m}\xb7{\mathrm{s}}^{-1})$ |

$(x,y)$ | Cartesian coordinates $(\mathrm{m})$ |

$p$ | Fluid pressure $(\mathrm{N}\xb7{\mathrm{m}}^{-}{}^{2})$ |

$\overrightarrow{M}$ | Magnetization $(\mathrm{A}\xb7{\mathrm{m}}^{-1})$ |

$\overrightarrow{H}$ | Magnetic field intensity $(\mathrm{A}\xb7{\mathrm{m}}^{-1})$ |

$\overrightarrow{B}$ | Magnetic induction $(\text{}Te\mathrm{s}la\text{},T)$ |

$T$ | Fluid temperature inside the boundary layer $(\mathrm{K})$ |

${T}_{c}$ | Fluid temperature far away from the sheet $(\mathrm{K})$ |

${T}_{w}$ | Temperature of the sheet $(\mathrm{K})$ |

$(U,V)$ | Dimensionless velocity components in the $x\text{}and\text{}y$ directions |

$P$ | Dimensionless pressure |

$\overline{T}$ | Dimensionless temperature |

$t$ | Time $(\mathrm{s})$ |

$\rho $ | Density of fluid $(\text{}\mathrm{k}\mathrm{g}\xb7{\mathrm{m}}^{-3})$ |

$\nu $ | Kinematic viscosity $({\mathrm{m}}^{2}\xb7{\mathrm{s}}^{-1})$ |

$\mu $ | Dynamic viscosity $(\mathrm{k}\mathrm{g}\xb7{\mathrm{m}}^{-1}{\mathrm{s}}^{-1})$ |

${\mu}_{0}$ | Magnetic permeability $(\mathrm{N}\xb7{\mathrm{A}}^{-2})$ |

$\sigma $ | Electrical conductivity $(\mathrm{s}\xb7{\mathrm{m}}^{-1})$ |

${c}_{p}$ | Specific heat constant pressure $\text{}(\mathrm{g}\xb7\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{K}}^{-1})$ |

$\mathrm{k}$ | Thermal conductivity $(\text{}\mathrm{g}\xb7{\mathrm{m}}^{-1}{\mathrm{s}}^{-1}{\mathrm{K}}^{-1})$ |

${P}_{r}$ | Prandtl number (dimensionless) |

${E}_{c}$ | Eckert number (dimensionless) |

$\epsilon $ | Dimensionless Curie temperature |

${M}_{F}$ | Ferromagnetic interaction parameter (dimensionless) |

${M}_{M}$ | Magnetohydrodynamic parameter (dimensionless) |

$\tau $ | Dimensionless time |

## Appendix A

_{3}| ≤ 1, the highest negative value is C

_{3}= −1, therefore one stability condition is

_{r}≥ 0.733, M

_{F}≤ 1.73 × 10

^{8}and ${M}_{M}\le 2.1\times {10}^{4}.$

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$\mathbf{Reference}\mathbf{Magnetic}\mathbf{Field}\mathbf{Flux}(\overrightarrow{\mathit{B}})$ | Magnetohydrodynamic Number (M_{M}) | Ferromagnetic Number (M_{F}) |
---|---|---|

2 T | 0.000062 | 767.8 |

4 T | 0.00025 | 1535.5 |

6 T | 0.00056 | 2303.5 |

8 T | 0.001 | 3071.2 |

9 T | 0.0013 | 3455.1 |

10 T | 0.0016 | 3839.2 |

**Table 2.**Comparison of the Nusselt number for different values of ${P}_{r}$ with ${M}_{F}=0,{M}_{M}=0,H={H}_{x}={H}_{y}=P={E}_{c}=0$.

${\mathit{P}}_{\mathit{r}}$ | Present Result | Khan et al. [29] |
---|---|---|

0.2 | 0.1689 | 0.1694 |

0.7 | 0.4524 | 0.4544 |

2 | 0.909 | 0.9109 |

7 | 1.8930 | 1.8960 |

20 | 3.3532 | 3.3541 |

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**MDPI and ACS Style**

Murtaza, M.G.; Tzirtzilakis, E.E.; Ferdows, M.
Stability and Convergence Analysis of a Biomagnetic Fluid Flow Over a Stretching Sheet in the Presence of a Magnetic Field. *Symmetry* **2020**, *12*, 253.
https://doi.org/10.3390/sym12020253

**AMA Style**

Murtaza MG, Tzirtzilakis EE, Ferdows M.
Stability and Convergence Analysis of a Biomagnetic Fluid Flow Over a Stretching Sheet in the Presence of a Magnetic Field. *Symmetry*. 2020; 12(2):253.
https://doi.org/10.3390/sym12020253

**Chicago/Turabian Style**

Murtaza, Md. Ghulam, Efstratios Emmanouil Tzirtzilakis, and Mohammad Ferdows.
2020. "Stability and Convergence Analysis of a Biomagnetic Fluid Flow Over a Stretching Sheet in the Presence of a Magnetic Field" *Symmetry* 12, no. 2: 253.
https://doi.org/10.3390/sym12020253