# Heat and Mass Transfer Analysis on Magneto Micropolar Fluid Flow with Heat Absorption in Induced Magnetic Field

## Abstract

**:**

## 1. Introduction

## 2. Flow Model of the Physical Problem

- All the physical properties of fluid are considered to be constant but the influence of density variation with temperature is assumed only in the body force term, in accordance with the well-known Boussinesq’s approximation.
- Since the plate is of semi-infinite extent and the fluid motion is unsteady so all the flow variables will depend upon the distance variable along the plate x, distance variable normal to the plate y and the time variable $\tau .$
- The microrotation vector of the form $\mathbf{G}=\left(0,\hspace{0.17em}\hspace{0.17em}0,\hspace{0.17em}\hspace{0.17em}\overline{\Gamma}\right)$ is considered here.
- The viscous dissipation and joule heating terms in the energy equation have been assumed for high speed flow as well as a constant heat sink ${h}_{s}$ is used for heat absorption [35].
- The level of concentration of foreign mass has been taken very high for observing the thermal diffusion effect on flow. The mass diffusion effect has also been studied here.
- The magnetic Reynolds number is taken to be large enough so that the induced magnetic field vector of the form $\mathbf{H}=\left({H}_{x},\hspace{0.17em}{H}_{y},\hspace{0.17em}0\right)$ is applicable. The divergence equation of Maxwell’s equation $\nabla .\mathbf{H}=0$ for the magnetic field gives ${H}_{y}=$ constant $={H}_{0}$ (say).

## 3. Mathematical Formulation

## 4. Numerical Computation

## 5. Discussion of the Results

## 6. Conclusions

- All of the flow variables except microrotational velocity and induced magnetic field increase significantly until the steady-state value with time.
- Only the Grashof number enhances the fluid velocity near at the plate. It is concluded that the Grashof number plays an important role on fluid velocity in case of cooling problem.
- The microrotational velocity of fluid particles is positively influenced by Darcy number but negatively influenced by spin gradient viscosity.
- The induced magnetic field strength is stronger for the lowest magnetic force or diffusivity numbers.
- The fluid temperature is found to be high in case of strong mass diffusion. Particularly, the fluid temperature is grater for air than water.
- The species concentration is increasingly affected by the both heat sink and thermal diffusion. It is also confirmed that the concentration level of fluid is greater for lighter particles than heavier particles.

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$\overline{C}$ | species concentration |

$C$ | dimensionless species concentration |

${c}_{s}$ | concentration susceptibility |

${c}_{p}$ | specific heat at constant pressure |

${C}_{\infty}$ | species concentration of uniform flow |

${D}_{a}$ | Darcy number |

${D}_{m}$ | coefficient of mass diffusivity |

${D}_{f}$ | Dufour number |

${E}_{c}$ | Eckert number |

$g$ | local acceleration due to gravity |

$\mathit{G}$ | microrotation vector |

${G}_{r}$ | Grashof number |

${G}_{m}$ | modified Grashof number |

${h}_{s}$ | constant heat sink |

$\mathbf{H}$ | induced magnetic field vector |

$H$ | dimensionless induced magnetic field component |

${H}_{0}$ | induced magnetic field strength |

${H}_{w}$ | induced magnetic field at the wall |

${H}_{x},\hspace{0.17em}\hspace{0.17em}{H}_{y}$ | induced magnetic field components |

$j$ | microinertia per unit mass |

$K$ | permeability of the medium |

$m$ | constant mass flux per unit area |

$M$ | magnetic force number |

${P}_{m}$ | magnetic diffusivity number |

${P}_{r}$ | Prandtl number |

$Q$ | constant heat flux per unit area |

$S$ | microrotational constant |

${S}_{c}$ | Schmidt number |

${S}_{o}$ | Soret number |

$t$ | dimensionless time |

$\overline{T}$ | fluid temperature |

$T$ | dimensionless fluid temperature |

${T}_{m}$ | mean fluid temperature |

${T}_{\infty}$ | fluid temperature of uniform flow |

$u,\hspace{0.17em}\hspace{0.17em}v$ | velocity components |

$U,\hspace{0.17em}\hspace{0.17em}V$ | dimensionless velocity components |

${U}_{0}$ | dimensionless constant velocity |

$x$ | spatial coordinate along to the plate |

$X$ | dimensionless spatial coordinate along to the plate |

$y$ | spatial coordinate normal to the plate |

$Y$ | dimensionless spatial coordinate normal to the plate |

Greek Symbols | |

$\nabla $ | divergence vector |

$\Delta $ | microrotational number |

$\Delta t$ | dimensionless time-step |

$\Delta X$ | dimensionless mesh sizes along $X$ direction |

$\Delta Y$ | dimensionless mesh sizes along $Y$ direction |

$\Lambda $ | spin gradient viscosity |

$\alpha $ | heat absorption parameter |

$\beta $ | thermal expansion coefficient |

${\beta}^{*}$ | concentration expansion coefficient |

$\chi $ | vortex viscosity |

$\gamma $ | spin-gradient viscosity |

$\kappa $ | thermal conductivity |

${\kappa}_{T}$ | thermal diffusion ratio |

$\lambda $ | vortex viscosity |

${\mu}_{e}$ | magnetic permeability |

$\rho $ | density of the fluid |

$\sigma $ | electrical conductivity |

$\tau $ | time |

$\Gamma $ | dimensionless microrotational component |

$\overline{\Gamma}$ | microrotational component |

$\upsilon $ | kinematic viscosity |

Subscripts | |

$W$ | at the wall of the plate |

$\infty $ | free stream conditions |

$i,\hspace{0.17em}\hspace{0.17em}j$ | grid points along $X$ and $Y$ axis respectively |

Superscript | |

$n$ | number of time-steps |

′ | at the end of a time-step |

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**Figure 4.**Velocity profiles for different values of (

**a**) Grashof number (

**b**) Darcy number (

**c**) magnetic force number.

**Figure 5.**Velocity profiles for different values of (

**a**) heat absorption parameter (

**b**) Prandtl number (

**c**) Schmidt number.

**Figure 6.**Microrotational velocity profiles for different values of (

**a**) Grashof number (

**b**) Darcy number (

**c**) spin gradient viscosity.

**Figure 7.**Microrotational velocity profiles for different values of (

**a**) Soret number (

**b**) Prandtl number (

**c**) Schmidt number.

**Figure 8.**Induced Magnetic field profiles for different values of (

**a**) magnetic diffusivity number (

**b**) Darcy number (

**c**) magnetic force number.

**Figure 9.**Temperature profiles for different values of (

**a**) Eckert number (

**b**) Prandtl number (

**c**) Dufour number.

**Figure 10.**Concentration profiles for different values of (

**a**) heat absorption parameter (

**b**) Soret number (

**c**) Schmidt number.

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Haque, M.M.
Heat and Mass Transfer Analysis on Magneto Micropolar Fluid Flow with Heat Absorption in Induced Magnetic Field. *Fluids* **2021**, *6*, 126.
https://doi.org/10.3390/fluids6030126

**AMA Style**

Haque MM.
Heat and Mass Transfer Analysis on Magneto Micropolar Fluid Flow with Heat Absorption in Induced Magnetic Field. *Fluids*. 2021; 6(3):126.
https://doi.org/10.3390/fluids6030126

**Chicago/Turabian Style**

Haque, Md. Mohidul.
2021. "Heat and Mass Transfer Analysis on Magneto Micropolar Fluid Flow with Heat Absorption in Induced Magnetic Field" *Fluids* 6, no. 3: 126.
https://doi.org/10.3390/fluids6030126