# Hall Current and Soret Effects on Unsteady MHD Rotating Flow of Second-Grade Fluid through Porous Media under the Influences of Thermal Radiation and Chemical Reactions

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## Abstract

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## 1. Introduction

## 2. Formulation of the Problem

_{0}(1 + A ε e

^{nt})

_{0}is the balance of the suction velocity with a non-zero positive constant. Suppose that the magnetic field has enormous strength. In that case, the exhaustive Ohms commandment to incorporate the Hall currents is unique. With the Hall currents assumed, the generalized Ohm’s law [32] can be expressed as follows:

## 3. Solution of the Problem

#### 3.1. Skin Friction

#### 3.2. Nusselt Number

#### 3.3. Sherwood Number

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Nomenclature | |

A | real positive constant |

B_{0} | applied magnetic field (A/m) |

x, y | dimensional co-ordinates (m) |

u, v | velocity components in x and y directions (m/s) |

C | non-dimensional fluid concentration (kg/m^{3}) |

D_{1} | coefficient of thermal diffusivity (m^{2} s^{−1}) |

g | acceleration due to gravity (m s^{−2}) |

Gr | thermal Grashof number |

Gm | mass Grashof number |

C_{w} | the uniform concentration of the fluid at the plate (kg m^{−3}) |

Kc | chemical reaction parameter (w/mk) |

m | Hall parameter |

C_{∞} | the concentration of the fluid far away from the plate (kg m^{−3}) |

B | magnetic field vector (A/m) |

q_{w} | local surface heat flux (W m^{−2}) |

Nu | local Nusselt number |

E | electric field vector (c) |

V | velocity vector (m/s) |

k_{1} | thermal conductivity (W/m K) |

K | permeability parameter |

q_{m} | local surface mass flux (kg s^{−1} m^{−2}) |

P_{e} | electron pressure (Pascal) |

Pr | Prandtl number |

Sc | Schmidt number |

J_{x}, J_{y} | current densities in x and y directions |

k | permeability of porous medium (m^{2}) |

H | heat source parameter |

J | current density vector (A/m^{2}) |

C_{p} | specific heat at a constant pressure (J/kg·K) |

D | coefficient of mass diffusivity (m^{2}/s) |

Sh | local Sherwood number |

w | slip velocity (m s^{−1}) |

w_{0} | scale of suction velocity |

So | Soret number |

t | time (s) |

u_{0} | plate velocity (m s^{−1}) |

q_{r} | radiative heat flux |

F | radiation parameter (cm^{−2}) |

T_{w} | the uniform temperature of the fluid on the plate (K) |

T_{∞} | the temperature of the fluid far away from the plate (K) |

R | rotation parameter |

S | second grade fluid |

M | Hartmann number |

N | constant |

Greek symbols | |

β | coefficient of thermal expansion of the fluid |

Ω | angular velocity (s^{−1}) |

τ_{w} | local wall shear stress (pascal) |

τ_{e} | electron collision time (s) |

ω_{e} | cyclotron frequency (e/mB) |

ϕ | non-dimensional concentration (mol/m^{3}) |

v | kinematic viscosity (m^{2}/s) |

τ | local skin friction coefficient |

β * | coefficient of mass expansion of the solid |

Θ | non-dimensional temperature (K) |

ρ | fluid density (Kg/m^{3}) |

σ | electrical conductivity (S/m) |

Subscripts and Superscripts | |

∞ | free stream conditions |

i | ions |

w | conditions on the wall |

e | electrons |

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**Figure 2.**The velocity patterns for u and v plotted vs. M, with k = 0.50, R = 1.0, Gr = 5.0, S = 1.0, Gm = 3.0, m = 0.20, Pr = 0.710, H = 1.0, Sc = 0.220, Kc = 1.0, n = 0.50, t = 0.50, So = 0.50, F = 1.0.

**Figure 3.**The velocity patterns for u and v plotted vs. Gr, with k = 0.50, R = 1.0, M = 2.0, S = 1.0, Gm = 3.0, m = 0.20, Pr = 0.710, H = 1.0, Sc = 0.220, Kc = 1.0, n = 0.50, t = 0.50, So = 0.50, F = 1.0.

**Figure 4.**The velocity patterns for u and v plotted vs. Gm, with k = 0.50, R = 1.0, Gr = 5.0, S = 1.0, M = 2.0, m = 0.20, Pr = 0.710, H = 1.0, Sc = 0.220, Kc = 1.0, n = 0.50, t = 0.50, So = 0.50, F = 1.0.

**Figure 5.**The velocity patterns for u and v plotted vs. k, with M = 2.0, R = 1.0, Gr = 5.0, S = 1.0, Gm = 3.0, m = 0.20, Pr = 0.710, H = 1.0, Sc = 0.220, Kc = 1.0, n = 0.50, t = 0.50, So = 0.50, F = 1.0.

**Figure 6.**The velocity patterns for u and v plotted vs. R, with k = 0.50, M = 2.0, Gr = 5.0, S = 1.0, Gm = 3.0, m = 0.20, Pr = 0.710, H = 1.0, Sc = 0.220, Kc = 1.0, n = 0.50, t = 0.50, So = 0.50, F = 1.0.

**Figure 7.**The velocity patterns for u and v plotted vs. m, with k = 0.50, R = 1.0, Gr = 5.0, S = 1.0, Gm = 3.0, M = 2.0, Pr = 0.710, H = 1.0, Sc = 0.220, Kc = 1.0, n = 0.50, t = 0.50, So = 0.50, F = 1.0.

**Figure 8.**The velocity patterns for u and v plotted vs. F, with k = 0.50, R = 1.0, Gr = 5.0, S = 1.0, Gm = 3.0, m = 0.20, Pr = 0.710, H = 1.0, Sc = 0.220, Kc = 1.0, n = 0.50, t = 0.50, So = 0.50, M = 2.0.

**Figure 9.**The velocity patterns for u and v plotted vs. So, with k = 0.50, R = 1.0, Gr = 5.0, S = 1.0, Gm = 3.0, m = 0.20, Pr = 0.710, H = 1.0, Sc = 0.220, Kc = 1.0, n = 0.50, t = 0.50, M = 2.0, F = 1.0.

M | k | R | Pr | Gr | Gm | Sc | Kc | H | So | m | τ |
---|---|---|---|---|---|---|---|---|---|---|---|

2.0 | 0.50 | 1.0 | 0.710 | 3.0 | 5.0 | 0.22 | 1.0 | 1.0 | 1.0 | 0.20 | 0.6248 |

3.0 | 0.7245 | ||||||||||

4.0 | 0.7645 | ||||||||||

1.5 | 1.9475 | ||||||||||

2.0 | 2.0657 | ||||||||||

2.0 | 1.6578 | ||||||||||

3.0 | 1.7458 | ||||||||||

3.0 | 0.8542 | ||||||||||

7.0 | 0.8154 | ||||||||||

6 | 1.9875 | ||||||||||

9 | 2.2547 | ||||||||||

10 | 1.8752 | ||||||||||

15 | 2.0124 | ||||||||||

0.4 | 2.5479 | ||||||||||

0.6 | 2.9875 | ||||||||||

2.0 | 1.9875 | ||||||||||

3.0 | 2.0214 | ||||||||||

2.0 | 1.9647 | ||||||||||

3.0 | 2.6578 | ||||||||||

2.0 | 19875 | ||||||||||

3.0 | 2.2448 | ||||||||||

0.40 | 2.1254 | ||||||||||

0.60 | 2.1004 |

Kc | H | F | Pr | Nu |
---|---|---|---|---|

1.0 | 1.0 | 1.0 | 0.710 | 0.445215 |

2.5 | 0.645214 | |||

3.5 | 0.712032 | |||

2.5 | 1.578521 | |||

3.5 | 1.945214 | |||

2.5 | 0.978521 | |||

3.5 | 0.978521 | |||

3.5 | 3.478542 | |||

7.5 | 7.578512 |

Kc | Sc | So | T | F | Sh |
---|---|---|---|---|---|

1.0 | 0.220 | 0.50 | 0.50 | 1.0 | 0.578521 |

2.5 | 0.645876 | ||||

3.5 | 0.778521 | ||||

0.35 | 0.712014 | ||||

0.45 | 0.978520 | ||||

2.5 | 2.801234 | ||||

3.5 | 1.312478 | ||||

1.5 | 0.498741 | ||||

2.5 | 0.445214 | ||||

2.5 | 1.278521 | ||||

3.5 | 1.045210 |

M | K | Gr | Gm | Previous Results Deepthi et al. [10] | Present Values |
---|---|---|---|---|---|

2.0 | 0.50 | 5.0 | 3.0 | 0.70785200 | 0.7458752 |

3.0 | 0.44587710 | 0.4785214 | |||

4.0 | 0.33254700 | 0.3785214 | |||

1.00 | 0.75266440 | 0.7785214 | |||

1.50 | 0.85657100 | 0.8021448 | |||

10.0 | 0.97521200 | 0.9321477 | |||

15.0 | 1.00214700 | 1.0785214 | |||

6.0 | 0.76578280 | 0.7785214 | |||

9.0 | 0.81032570 | 0.8654785 |

_{0}= 0.50, m = 1.0, Sc = 0.220, Kc = H = 1.0, and So = 0.

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## Share and Cite

**MDPI and ACS Style**

Bafakeeh, O.T.; Raghunath, K.; Ali, F.; Khalid, M.; Tag-ElDin, E.S.M.; Oreijah, M.; Guedri, K.; Khedher, N.B.; Khan, M.I.
Hall Current and Soret Effects on Unsteady MHD Rotating Flow of Second-Grade Fluid through Porous Media under the Influences of Thermal Radiation and Chemical Reactions. *Catalysts* **2022**, *12*, 1233.
https://doi.org/10.3390/catal12101233

**AMA Style**

Bafakeeh OT, Raghunath K, Ali F, Khalid M, Tag-ElDin ESM, Oreijah M, Guedri K, Khedher NB, Khan MI.
Hall Current and Soret Effects on Unsteady MHD Rotating Flow of Second-Grade Fluid through Porous Media under the Influences of Thermal Radiation and Chemical Reactions. *Catalysts*. 2022; 12(10):1233.
https://doi.org/10.3390/catal12101233

**Chicago/Turabian Style**

Bafakeeh, Omar T, Kodi Raghunath, Farhan Ali, Muhammad Khalid, El Sayed Mohamed Tag-ElDin, Mowffaq Oreijah, Kamel Guedri, Nidhal Ben Khedher, and Muhammad Ijaz Khan.
2022. "Hall Current and Soret Effects on Unsteady MHD Rotating Flow of Second-Grade Fluid through Porous Media under the Influences of Thermal Radiation and Chemical Reactions" *Catalysts* 12, no. 10: 1233.
https://doi.org/10.3390/catal12101233