# Numerical Solution of Biomagnetic Power-Law Fluid Flow and Heat Transfer in a Channel

^{1}

^{2}

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^{‡}

## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

## 3. Numerical Formulation and Solution Procedure

#### 3.1. Non-Dimensionalisation and Transformation to Stream Function-Vorticity

#### 3.2. Constrained Interpolated Profile (CIP) Based Finite Difference Method

#### 3.3. Boundary Conditions

#### 3.4. Dimensionless Parameters

#### 3.5. Grid Independence and Method Validation

^{®}. An extensive grid independence test is performed to check for the grid independent solution. The test is based on the u-velocity profile along the various x location of the channel, where the focus is emphasised near the location on the magnetic source. Five different size of grids configuration have been performed on the case flow problem $n=1$, $M{n}_{F}=1312$ and $N=0.056$. The magnetic source is located at $(a,b)=(2.5,-0.05)$. As shown in Figure 2, when the grid size is refined from $400\times 40$ to $500\times 100$, there is a significant change on velocity profile, particularly near the magnetic source and in the interval of the vortex size. The numerical simulation is sensitive on the vertical grid, while no significant change occurred when refining the horizontal grid. For the temperature profile, the local heat transfer on the lower wall of the channel has been used as a reference to test the grid independence. Only the lower wall is shown since most of the significant changes occurred near the magnetic source. The same grid independence test result is observed as seen in Figure 3, where the grids size have reached grid independence for temperature profile. These grids sizes have also been tested with smaller time step and no significant changes is accounted. It is observed that the grid size configuration of $400\times 80$ is sufficient to capture the velocity profile change near the magnetic location.

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

FHD | Ferrohydrodynamics |

MHD | Magnetohydrodynamics |

EMF | electromagnetic field |

MRI | Magnetic Resonance Imaging |

T | Tesla |

RF | radiofrequency field |

CIP | Constrained Interpolation Profile |

FDM | Finite Difference method |

FEM | Finite Element method |

FDLBM | Finite Difference Lattice Boltzmann method |

DRBEM | Dual Reciprocity Boundary Element method |

LBM | Lattice Boltzmann method |

CVFEM | Control Volume Finite Element method |

SOR | Successive Overrelaxation |

CIP-FDM | Constrained Interpolation Profile Finite Difference Method |

nomenclatures | |

${\overline{\mu}}_{a}$ | viscosity |

m | fluid consistency |

${I}_{2}$ | second invariant of the rate of strain tensor |

n | Power-law index |

$\overrightarrow{v}$ | velocity vector |

$\rho $ | density |

p | pressure |

$\overrightarrow{\tau}$ | stress tensor |

$\overrightarrow{J}$ | current density |

$\overline{M}$ | magnetisation |

$\overline{B}$ | magnetic field |

${\overline{\mu}}_{0}$ | magnetic permeability |

$\overline{H}$ | magnetic field intensity |

${c}_{p}$ | specific heat |

$\kappa $ | thermal conductivity |

$\overline{\sigma}$ | electrical conductivity |

${\overline{T}}_{w}$ | temperature of the wall |

${\overline{T}}_{f}$ | temperature of the fluid |

$x,y$ | Cartesian coordinates |

T | temperature |

u | x-velocity |

v | y-velocity |

t | time |

$(a,b)$ | coordinate of magnetic source |

$\psi $ | stream function |

w | vorticity |

$Re$ | Reynolds number |

$M{n}_{F}$ | Magnetic number |

N | Stuart number |

$Pr$ | Prandtl number |

$Ec$ | Eckert number |

$\u03f5$ | temperature number |

${C}_{f}$ | local skin friction coefficient, wall shear stress |

$Nu$ | Nusselt number, heat transfer coefficient |

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**Figure 4.**Stream function and vorticity contour for $Re=10$, $M{n}_{F}$ = 131,250 obtained by using present method and comparable with [33].

**Figure 5.**Comparison between present results and the results by Bell and Surana [60], Neofytou [61] for non-Newtonian power-law at various values of power-law index n, and Newtonian fluid flow by Ghia et al. [80]. (

**a**) u-velocity profiles along the vertical centreline of the cavity. (

**b**) v-velocity profiles along the horizontal centreline of the cavity for Re = 100.

**Figure 6.**Streamline plot for power-law index: (

**a**) n = 0.6 (shear thinning). (

**b**) n = 1.0 (Newtonian). (

**c**) n = 1.4 (shear thickening).

**Figure 7.**Temperature plot for power-law index: (

**a**) n = 0.6 (shear thinning). (

**b**) n = 1.0 (Newtonian). (

**c**) n = 1.4 (shear thickening).

**Figure 8.**Variations of wall shear stress and heat transfer parameter for n = 0.6 with different intensity of magnetic field number.

**Figure 9.**Variations of wall shear stress and heat transfer parameter for n = 1 with different intensity of magnetic field.

**Figure 10.**Variations of wall shear stress and heat transfer parameter for $n=1.4$ with different intensity of magnetic field.

**Figure 11.**Figure 11. Variations of wall shear stress parameter C

_{f}and heat transfer parameter Nu for Mn

_{F}= 1312, N = 0.056 for different power-law index n.

**Table 1.**Value of reference magnetic field ${\overline{B}}_{0}$ and the corresponding values of $M{n}_{F}$ and N for $Re=100$.

${\overline{\mathit{B}}}_{0}$ (Tesla) | ${\mathit{Mn}}_{\mathit{F}}$ | N |
---|---|---|

0 | 0 | 0 |

1 | 164 | $0.0008$ |

4 | 656 | $0.014$ |

8 | 1316 | $0.056$ |

n | ${\overline{\mathit{B}}}_{0}=1$ T | ${\overline{\mathit{B}}}_{0}=4$ T | ${\overline{\mathit{B}}}_{0}=8$ T |
---|---|---|---|

$n=0.6$ | $2<x<5.66$ | $2<x<7.61$ | $2<x<8.42$ |

$n=1.0$ | $2<x<4.69$ | $2<x<5.59$ | $2<x<5.94$ |

$n=1.4$ | $2<x<4.04$ | $2<x<4.61$ | $2<x<4.87$ |

${\overline{\mathit{B}}}_{0}=1$ T | ${\overline{\mathit{B}}}_{0}=4$ T | ${\overline{\mathit{B}}}_{0}=8$ T | |
---|---|---|---|

$n=0.6$ | ${u}_{min}=-0.1792$ | ${u}_{min}=-0.2831$ | ${u}_{min}=-0.4106$ |

$n=1.0$ | ${u}_{min}=-0.2258$ | ${u}_{min}=-0.3783$ | ${u}_{min}=-0.4848$ |

$n=1.4$ | ${u}_{min}=-0.2357$ | ${u}_{min}=-0.4068$ | ${u}_{min}=-0.5093$ |

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Halifi, A.S.; Shafie, S.; Amin, N.S.
Numerical Solution of Biomagnetic Power-Law Fluid Flow and Heat Transfer in a Channel. *Symmetry* **2020**, *12*, 1959.
https://doi.org/10.3390/sym12121959

**AMA Style**

Halifi AS, Shafie S, Amin NS.
Numerical Solution of Biomagnetic Power-Law Fluid Flow and Heat Transfer in a Channel. *Symmetry*. 2020; 12(12):1959.
https://doi.org/10.3390/sym12121959

**Chicago/Turabian Style**

Halifi, Adrian S., Sharidan Shafie, and Norsarahaida S. Amin.
2020. "Numerical Solution of Biomagnetic Power-Law Fluid Flow and Heat Transfer in a Channel" *Symmetry* 12, no. 12: 1959.
https://doi.org/10.3390/sym12121959