# Analytical Modeling of MHD Flow over a Permeable Rotating Disk in the Presence of Soret and Dufour Effects: Entropy Analysis

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## Abstract

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## 1. Introduction

## 2. Mathematical Formulation

_{0}which is considered unchanged by taking the small magnetic Reynolds number is imposed in the direction normal to the surface of the disk. The induced magnetic field due to motion of the electrically-conducting fluid is negligible. Uniform suction is also applied at the surface of the disk. The flow description and geometrical coordinates are shown in Figure 1. The governing equations, respectively, of continuity, momentum, energy and species diffusion in laminar incompressible flow are given by:

_{p}is the specific heat at constant pressure, D is the molecular diffusion coefficient, K

_{T}is the thermal diffusion ratio, C

_{s}is the concentration susceptibility, and T

_{m}is the mean fluid temperature. The appropriate boundary conditions subject to uniform suction w

_{0}through the disk are:

_{0}

^{2}/Ω·ρ is the magnetic interaction parameter, Pr = ν·ρ·c

_{p}/k is the Prandtl number, Sc = ν/D is the Schmidt number, Sr = D·(T

_{∞}− T

_{w})·K

_{T}/ν·T

_{m}(C

_{∞}− C

_{w}) is the Soret number, Du = D·(C

_{∞}− C

_{w})·K

_{T}/C

_{s}·c

_{p}·ν·(T

_{∞}− T

_{w}) is the Dufour number, and F, G, H, θ, and φ are the dimensionless functions of modified dimensionless vertical coordinate η. The transformed boundary conditions are:

_{0}/(ν·Ω)

^{1/2}is the suction/injection parameter and Ws < 0 shows uniform suction at the disk surface.

## 3. Entropy Generation Analysis

## 4. HAM Solution

## 5. Optimal Convergence Control Parameters

## 6. Results and Discussion

_{s}. The values of the physical flow parameters are mentioned in each of the graphs and tables. Table 1 and Table 2 illustrate a comparison between our results and those reported by Turkyilmazoglu [49] and Kelson and Desseaux [50] for ${F}^{\prime}(0)$ and ${G}^{\prime}(0)$ as well as different values of magnetic interaction parameter and suction parameter. Excellent agreement is observed. The diluting chemical species of most common interest have Schmidt number between 0.1 and 10.0. Thus, we chose Schmidt number 0.22, 0.64, 0.78, and 1, which represent the Schmidt number of helium, ammonia, carbon monoxide, and carbon dioxide, respectively.

_{G, av}(Figure 9). It is clear that Schmidt number follows the same trend as Prandtl number. From Figure 10 maximum values of the averaged entropy generation number occur when the values of both Soret and Dufour numbers are maximized simultaneously. Finally, all entropy generation related figures reveal that as the magnetic interaction parameter increases, the averaged entropy generation number also increases.

## 7. Conclusions

- (a)
- HAM is shown to demonstrate excellent potential, convergence and accuracy for simulating flow over rotating disk problems.
- (b)
- As the magnetic field becomes stronger, the velocity profiles in radial, tangential and axial directions decrease and the thermal boundary layer and concentration field increase.
- (c)
- When suction is applied at the disk surface, the radial, tangential and axial velocity profiles decrease. The usual decay of temperature and concentration profiles occurs for larger values of the suction parameter.
- (d)
- The thermal boundary-layer thickness decreases with increasing Prandtl number. Furthermore, as the Schmidt number increases, the concentration boundary layer thickness decreases.
- (e)
- The thermal boundary layer increases by increasing Dufour number or simultaneously decreasing Soret number. As the Dufour number increases or Soret number decreases, the rate of mass transfer (concentration boundary layer thickness) decreases at the disk.
- (f)
- The averaged entropy generation number increases by increasing the magnetic interaction parameter, suction parameter, Prandtl number, and Schmidt number. In addition, the maximum values of averaged entropy generation number occur when the values of both Soret and Dufour numbers are maximized simultaneously.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

B | external uniform magnetic field |

B_{0} | constant magnetic flux density |

C | fluid concentration |

c_{p} | specific heat at constant pressure |

C_{s} | concentration susceptibility |

D | molecular diffusion coefficient |

E | electric field |

F | self-similar radial velocity |

G | self-similar tangential velocity |

H | self-similar axial velocity |

J | current density field |

k | thermal conductivity |

K_{T} | thermal diffusion ratio |

L | characteristic length |

P | pressure |

Q | electric charge density |

r | radial direction in cylindrical polar coordinates |

R_{g} | ideal gas constant |

${\dot{S}}_{gen}^{\u2034}$ | volumetric rate of local entropy generation |

${\dot{S}}_{0}^{\u2034}$ | characteristic entropy generation rate |

T | fluid temperature |

u | velocity component in the radial directio |

v | velocity component in the tangential direction |

w | velocity component in the axial direction |

w_{0} | uniform suction |

z | normal direction in cylindrical polar coordinates |

Dimensionless parameters | |

Br | rotational Brinkman number |

Du | Dufour number |

N_{G} | entropy generation number |

M | magnetic interaction parameter |

Pr | Prandtl number |

R | dimensionless radial coordinate |

Re | rotational Reynolds number |

Sc | Schmidt number |

Sr | Soret number |

W_{s} | suction parameter |

Greek symbols | |

α | dimensionless temperature difference |

β | dimensionless concentration difference |

λ | diffusive constant parameter |

η | a scaled boundary-layer coordinate |

θ | self-similar temperature |

μ | dynamic viscosity |

ν | kinematic viscosity |

ρ | density |

σ | electrical conductivity |

φ | self-similar concentration |

ϕ | tangential direction in cylindrical polar coordinates |

Φ | viscous dissipation function |

Ω | angular velocity of the disk |

$\forall $ | volume |

Subscripts | |

av | average condition |

m | mean condition |

w | condition of the wall |

$\infty $ | condition of the free steam |

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**Figure 3.**The residual error of (

**a**) Equation (64) and (

**b**) Equation (65) obtained by 20th order approximation of the HAM solution

**Figure 4.**Effect of magnetic interaction parameter on: (

**a**) axial velocity; (

**b**) radial velocity; (

**c**) tangential velocity; (

**d**) temperature distribution; and (

**e**) concentration profiles.

**Figure 5.**Effect of suction parameter on: (

**a**) axial velocity; (

**b**) radial velocity; (

**c**) tangential velocity; (

**d**) temperature distribution; and (

**e**) concentration profiles.

**Figure 6.**(

**a**) Effect of Prandtl number on the temperature distribution; and (

**b**) effect of Schmidt number on the concentration profile.

**Figure 7.**Effects of Soret and Dufour numbers on: (

**a**) temperature distribution; and (

**b**) concentration profiles.

**Figure 8.**Change of ${N}_{G,av}$ with respect to magnetic interaction parameter for different values of suction parameter when $Du=0.2,\text{}Sr=0.25,\text{}Pr=0.71,\text{}Sc=1$, and $Re=Br=5$.

**Figure 9.**Change of ${N}_{G,av}$ with respect to magnetic interaction parameter for different values of Prandtl number when $Du=0.2,\text{}Sr=0.25,\text{}{W}_{s}=-1,\text{}Sc=1$, and $Re=Br=5$.

**Figure 10.**Change of ${N}_{G,av}$ with respect to magnetic interaction parameter for different values of Schmidt number when $Du=0.2,\text{}Sr=0.25,\text{}Pr=0.71,\text{}{W}_{s}=-1$ and $Re=Br=5$.

**Figure 11.**Change of ${N}_{G,av}$ with respect to magnetic interaction parameter for different values of Soret and Dufour numbers when $Pr=0.71,\text{}Sc=0.78,\text{}{W}_{s}=-1$ and $Re=Br=5$.

M | W_{s} | Ref. [49] | Ref. [50] | Present |
---|---|---|---|---|

0 | 0 | - | 0.510233 | 0.510186 |

−1 | - | 0.389569 | 0.389559 | |

−2 | - | 0.242421 | 0.242416 | |

1 | 0 | 0.309258 | - | 0.309237 |

−1 | 0.251044 | - | 0.251039 | |

−2 | 0.188719 | - | 0.188718 | |

4 | 0 | 0.165703 | - | 0.165701 |

−1 | 0.149016 | - | 0.149015 | |

−2 | 0.129438 | - | 0.129438 |

**Table 2.**Numerical values of the tangential skin friction coefficient $-\text{\hspace{0.17em}}{G}^{\prime}(0)$.

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**MDPI and ACS Style**

Freidoonimehr, N.; Rashidi, M.M.; Abelman, S.; Lorenzini, G. Analytical Modeling of MHD Flow over a Permeable Rotating Disk in the Presence of Soret and Dufour Effects: Entropy Analysis. *Entropy* **2016**, *18*, 131.
https://doi.org/10.3390/e18050131

**AMA Style**

Freidoonimehr N, Rashidi MM, Abelman S, Lorenzini G. Analytical Modeling of MHD Flow over a Permeable Rotating Disk in the Presence of Soret and Dufour Effects: Entropy Analysis. *Entropy*. 2016; 18(5):131.
https://doi.org/10.3390/e18050131

**Chicago/Turabian Style**

Freidoonimehr, Navid, Mohammad Mehdi Rashidi, Shirley Abelman, and Giulio Lorenzini. 2016. "Analytical Modeling of MHD Flow over a Permeable Rotating Disk in the Presence of Soret and Dufour Effects: Entropy Analysis" *Entropy* 18, no. 5: 131.
https://doi.org/10.3390/e18050131