# Entropy Analysis for Hydromagnetic Darcy–Forchheimer Flow Subject to Soret and Dufour Effects

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## Abstract

**:**

## 1. Introduction

## 2. Formulation

## 3. Engineering Contents of Interest

#### 3.1. Nusselt Number

#### 3.2. Sherwood Number

## 4. Entropy

## 5. Solution Methodology

## 6. Graphical Results and Review

#### 6.1. Velocity

#### 6.2. Temperature

#### 6.3. Concentration

#### 6.4. Entropy Generation Rate

## 7. Closing Points

- The theermal field and velocity for the magnetic field had opposing trends.
- A decrease in velocity was noted for the Forchheimer number and suction variable.
- The velocity versus the porosity parameter was decreased.
- Similar behavior for the concentration and temperature against suction was noticed.
- The temperatures for the Eckert and Prandtl numbers were dissimilar.
- Radiation for the entropy and temperature had a similar role.
- The concentration decayed via larger approximation of the Soret number and reaction parameter.
- A decay in concentration against the Schmidt number held.
- Entropy generation enhancement against the Brinkman number and diffusion variable was noticed.
- The entropy rate was boosted with variation in the diffusion variable.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

$u,v$ | Velocity components (ms${}^{-1}$) | $x,y$ | Cartesian coordinates (m) |

t | Time (s) | ${v}_{0}>0$ | Suction velocity (ms${}^{-1}$) |

$\rho $ | Density (kgm${}^{-3}$) | $\sigma $ | Electrical conductivity (Sm${}^{-1}$) |

T | Temperature (K) | ${c}_{p}$ | Specific heat (Jkg${}^{-1}$K${}^{-1}$) |

${k}_{p}$ | Porous medium permeability (m${}^{2}$) | ${C}_{b}$ | Drag coefficient |

${T}_{w}$ | Wall temperature (K) | $\alpha $ | Thermal diffusivity (m${}^{2}$ s${}^{-1}$) |

k | Thermal conductivity (Wm${}^{-1}$K${}^{-1}$) | ${T}_{\infty}$ | Ambient temperature (K) |

${\sigma}^{\ast}$ | Stefan–Boltzman constant (Wm${}^{-2}$K${}^{-4}$) | ${K}_{T}$ | Thermal diffusion ratio |

${C}_{s}$ | Concentration susceptibility | ${k}^{\ast}$ | Mean absorption coefficient (cm${}^{-1}$) |

C | Concentration | ${k}_{r}$ | Reaction rate (s) |

${C}_{w}$ | Wall concentration | ${D}_{B}$ | Mass diffusivity (m${}^{2}$ s${}^{-1}$) |

${L}_{1}$ | Reference length (m) | ${C}_{\infty}$ | Ambient concentration |

${u}_{w}$ | Stretching velocity (ms${}^{-1}$) | a | Stretching rate constant (s${}^{-1}$) |

$N{u}_{x}$ | Nusselt number | ${q}_{w}$ | Heat flux (Wm${}^{2}$) |

$S{h}_{x}$ | Sherwood number | ${j}_{w}$ | Mass flux |

R | Molar gas constant (kgm${}^{2}$ s${}^{-2}$K${}^{-1}$mol${}^{-1}$) | M | Magnetic variable |

$\lambda $ | Porosity variable | $Fr$ | Forchheimer number |

S | Suction parameter | Pr | Prandtl number |

$Rd$ | Radiation variable | $Du$ | Dufour number |

$Ec$ | Eckert number | $\gamma $ | Reaction variable |

$Sr$ | Soret number | Re | Reynold number |

$Sc$ | Schmidt number | ${N}_{G}$ | Entropy rate |

${\alpha}_{1}$ | Temperature ratio variable | $Br$ | Brinkman number |

${\alpha}_{2}$ | Concentration ratio variable | L | Diffusion variable |

${T}_{m}$ | Mean fluid temperature (K) | ${B}_{0}$ | Magnetic field strength |

## References

- Darcy, H. Les Fontaines Publiques de la Ville dr Dijion; Dalmont, V., Ed.; Typ. Hennuyer: Paris, France, 1856; pp. 647–658. [Google Scholar]
- Forchheimer, P. Wasserbewegung durch boden. Z. Vereins Dtsch. Ingenieure
**1901**, 45, 1782–1788. [Google Scholar] - Muskat, M. The Flow of Homogeneous Fluids through Porous Media; JW Edwards, Inc.: Ann Arbor, MI, USA, 1946. [Google Scholar]
- Hayat, T.; Muhammad, T.; Al-Mezal, S.; Liao, S.J. Darcy-Forchheimer flow with variable thermal conductivity and Cattaneo-Christov heat flux. Int. J. Numer. Methods Heat Fluid Flow
**2016**, 26, 2355–2369. [Google Scholar] [CrossRef] - Pal, D.; Mondal, H. Hydromagnetic convective diffusion of species in Darcy-Forchheimer porous medium with non-uniform heat source/sink and variable viscosity. Int. Commun. Heat Mass Transf.
**2012**, 39, 913–917. [Google Scholar] [CrossRef] - Mallawi, F.; Ullah, M.Z. Conductivity and energy change in Carreau nanofluid flow along with magnetic dipole and Darcy-Forchheimer relation. Alex. Eng. J.
**2021**, 60, 3565–3575. [Google Scholar] [CrossRef] - Alshomrani, A.S.; Ullah, M.Z. Effects of homogeneous-heterogeneous reactions and convective condition in Darcy-Forchheimer flow of carbon nanotubes. J. Heat Transf.
**2019**, 141, 012405. [Google Scholar] [CrossRef] - Seth, G.S.; Mandal, P.K. Hydromagnetic rotating flow of Casson fluid in Darcy-Forchheimer porous medium. MATEC Web Conf.
**2018**, 192, 02059. [Google Scholar] [CrossRef] - Khan, S.A.; Hayat, T.; Alsaedi, A. Irreversibility analysis in Darcy-Forchheimer flow of viscous fluid with Dufour and Soret effects via finite difference method. Case Stud. Therm. Eng.
**2021**, 26, 101065. [Google Scholar] [CrossRef] - Azam, M.; Xu, T.; Khan, M. Numerical simulation for variable thermal properties and heat source/sink in flow of Cross nanofluid over a moving cylinder. Int. Commun. Heat Mass Transf.
**2020**, 118, 104832. [Google Scholar] [CrossRef] - Wu, Y.; Kou, J.; Sun, S. Matrix acidization in fractured porous media with the continuum fracture model and thermal Darcy-Brinkman-Forchheimer framework. J. Pet. Sci. Eng.
**2022**, 211, 110210. [Google Scholar] [CrossRef] - Haider, F.; Hayat, T.; Alsaedi, A. Flow of hybrid nanofluid through Darcy-Forchheimer porous space with variable characteristics. Alex. Eng. J.
**2021**, 60, 3047–3056. [Google Scholar] [CrossRef] - Tayyab, M.; Siddique, I.; Jarad, F.; Ashraf, M.A.; Ali, B. Numerical solution of 3D rotating nanofluid flow subject to Darcy-Forchheimer law, bio-convection and activation energy. S. Afr. J. Chem. Eng.
**2022**, 40, 48–56. [Google Scholar] [CrossRef] - Nawaz, M.; Sadiq, M.A. Unsteady heat transfer enhancement in Williamson fluid in Darcy-Forchheimer porous medium under non-Fourier condition of heat flux. Case Stud. Therm. Eng.
**2021**, 28, 101647. [Google Scholar] [CrossRef] - Ali, L.; Wang, Y.; Ali, B.; Liu, X.; Din, A.; Mdallal, Q.A. The function of nanoparticle’s diameter and Darcy-Forchheimer flow over a cylinder with effect of magnetic field and thermal radiation. Case Stud. Therm. Eng.
**2021**, 28, 101392. [Google Scholar] [CrossRef] - Bejawada, S.G.; Reddy, Y.D.; Jamshed, W.; Eid, M.R.; Safdar, R.; Nisar, K.S.; Isa, S.S.P.M.; Alam, M.M.; Parvin, S. 2D mixed convection non-Darcy model with radiation effect in a nanofluid over an inclined wavy surface. Alex. Eng. J.
**2022**, 61, 9965–9976. [Google Scholar] [CrossRef] - Eid, M.R.; Mahny, K.L.; Al-Hossainy, A.F. Homogeneous-heterogeneous catalysis on electromagnetic radiative Prandtl fluid flow: Darcy-Forchheimer substance scheme. Surf. Interfaces
**2021**, 24, 101119. [Google Scholar] [CrossRef] - Rastogi, R.P.; Madan, G.L. Dufour Effect in Liquids. J. Chem. Phys.
**1965**, 43, 4179–4180. [Google Scholar] [CrossRef] - Rastogi, R.P.; Nigam, R.K. Cross-phenomenological coefficients. Part 6—Dufour effect in gases. Trans. Faraday Soc.
**1966**, 62, 3325–3330. [Google Scholar] [CrossRef] - Rastogi, R.P.; Yadava, B.L.S. Dufour effect in liquid mixtures. J. Chem. Phys.
**1969**, 51, 2826–2830. [Google Scholar] [CrossRef] - Moorthy, M.B.K.; Senthilvadivu, K. Soret and Dufour effects on natural convection flow past a vertical surface in a porous medium with variable viscosity. J. Math. Phys.
**2012**, 2012, 634806. [Google Scholar] [CrossRef] - El-Arabawy, H.A.M. Soret and dufour effects on natural convection flow past a vertical surface in a porous medium with variable surface temperature. J. Math. Stat.
**2009**, 5, 190–198. [Google Scholar] [CrossRef] - Reddy, G.J.; Raju, R.S.; Manideep, C.; Rao, J.A. Thermal diffusion and diffusion thermo effects on unsteady MHD fluid flow past a moving vertical plate embedded in porous medium in the presence of Hall current and rotating system. Trans. A. Razmadze Math. Inst.
**2016**, 170, 243–265. [Google Scholar] [CrossRef] [Green Version] - Dursunkaya, Z.; Worek, W.M. Diffusion-thermo and thermal-diffusion effects in transient and steady natural convection from vertical surface. Int. J. Heat Mass Transf.
**1992**, 35, 2060–2067. [Google Scholar] [CrossRef] - Khan, S.A.; Hayat, T.; Khan, M.I.; Alsaedi, A. Salient features of Dufour and Soret effect in radiative MHD flow of viscous fluid by a rotating cone with entropy generation. Int. J. Hydrogen Energy
**2020**, 45, 4552–14564. [Google Scholar] [CrossRef] - Bekezhanova, V.B.; Goncharova, O.N. Influence of the Dufour and Soret effects on the characteristics of evaporating liquid flows. Int. J. Heat Mass Transf.
**2020**, 154, 119696. [Google Scholar] [CrossRef] - Jiang, N.; Studer, E.; Podvin, B. Physical modeling of simultaneous heat and mass transfer: Species interdiffusion, Soret effect and Dufour effect. Int. J. Heat Mass Transf.
**2020**, 156, 119758. [Google Scholar] [CrossRef] - Bejan, A. Second law analysis in heat transfer. Energy Int. J.
**1980**, 5, 721–732. [Google Scholar] [CrossRef] - Bejan, A. Entropy Generation Minimization; CRC Press: New York, NY, USA, 1996. [Google Scholar]
- Buonomo, B.; Pasqua, A.; Manca, O.; Nappo, S.; Nardini, S. Entropy generation analysis of laminar forced convection with nanofluids at pore length scale in porous structures with Kelvin cells. Int. Commun. Heat Mass Transf.
**2022**, 132, 105883. [Google Scholar] [CrossRef] - Khan, S.A.; Hayat, T.; Alsaedi, A.; Ahmad, B. Melting heat transportation in radiative flow of nanomaterials with irreversibility analysis. Renew. Sustain. Energy Rev.
**2021**, 140, 110739. [Google Scholar] [CrossRef] - Tayebi, T.; Öztop, H.F.; Chamkha, A.J. Natural convection and entropy production in hybrid nanofluid filled-annular elliptical cavity with internal heat generation or absorption. Therm. Sci. Eng. Prog.
**2020**, 19, 100605. [Google Scholar] [CrossRef] - Abbas, Z.; Naveed, M.; Hussain, M.; Salamat, N. Analysis of entropy generation for MHD flow of viscous fluid embedded in a vertical porous channel with thermal radiation. Alex. Eng. J.
**2020**, 59, 3395–3405. [Google Scholar] [CrossRef] - Rahmanian, S.; Koushkaki, H.R.; Shahsavar, A. Numerical assessment on the hydrothermal behaviour and entropy generation characteristics of boehmite alumina nanofluid flow through a concentrating photovoltaic/thermal system considering various shapes for nanoparticle. Sustain. Energy Technol. Assess.
**2022**, 52, 102143. [Google Scholar] [CrossRef] - Nayak, M.K.; Mabood, F.; Dogonchi, A.S.; Khan, W.A. Electromagnetic flow of SWCNT/MWCNT suspensions with optimized entropy generation and cubic auto catalysis chemical reaction. Int. Commun. Heat Mass Transf.
**2020**, 2020, 104996. [Google Scholar] [CrossRef] - Kumawat, C.; Sharma, B.K.; Al-Mdallal, Q.M.; Gorji, M.R. Entropy generation for MHD two phase blood flow through a curved permeable artery having variable viscosity with heat and mass transfer. Int. Commun. Heat Mass Transf.
**2022**, 133, 105954. [Google Scholar] [CrossRef] - Liu, Y.; Jian, Y.; Tan, W. Entropy generation of electromagnetohydrodynamic (EMHD) flow in a curved rectangular microchannel. Int. J. Heat Mass Transf.
**2018**, 127, 901–913. [Google Scholar] [CrossRef] - Alotaibi, H.; Eid, M.R. Thermal analysis of 3D electromagnetic radiative nanofluid flow with suction/blowing: Darcy–Forchheimer scheme. Micromachines
**2021**, 12, 1395. [Google Scholar] [CrossRef] - Eid, M.R.; Mabood, F. Entropy analysis of a hydromagnetic micropolar dusty carbon NTs-kerosene nanofluid with heat generation: Darcy–Forchheimer scheme. J. Therm. Anal. Calorim.
**2021**, 143, 2419–2436. [Google Scholar] [CrossRef] - Swain, I.; Pattanayak, H.; Das, M.; Singh, T. Finite difference solution of free convective heat transfer of non-Newtonian power law fluids from a vertical plate. Glob. J. Pure Appl. Math.
**2015**, 11, 339–348. [Google Scholar] - Adekanye, O.; Washington, T. Nonstandard finite difference scheme for a Tacoma narrows bridge model. Appl. Math. Model.
**2018**, 62, 223–236. [Google Scholar] [CrossRef] - Hayat, T.; Ullah, H.; Ahmad, B.; Alhodaly, M.S. Heat transfer analysis in convective flow of Jeffrey nanofluid by vertical stretchable cylinder. Int. Commun. Heat Mass Transf.
**2021**, 120, 104965. [Google Scholar] [CrossRef] - Khan, Z.H.; Makinde, O.D.; Ahmad, R.; Khan, W.A. Numerical study of unsteady MHD flow and entropy generation in a rotating permeable channel with slip and Hall effects. Commun. Theor. Phys.
**2018**, 70, 641–650. [Google Scholar] [CrossRef] - Bidin, B.; Nazar, R. Numerical solution of the boundary layer flow over an exponentially stretching sheet with thermal radiation. Eur. J. Sci. Res.
**2009**, 33, 710–717. [Google Scholar]

**Table 1.**Comparison of Nusselt numbers with [44].

Pr | Bidin and Nazar [44] | Recent Outcomes |
---|---|---|

1.0 | 0.9547 | 0.954710 |

2.0 | 1.4714 | 1.471409 |

3.0 | 1.8961 | 1.896115 |

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**MDPI and ACS Style**

Khan, S.A.; Hayat, T.
Entropy Analysis for Hydromagnetic Darcy–Forchheimer Flow Subject to Soret and Dufour Effects. *Math. Comput. Appl.* **2022**, *27*, 80.
https://doi.org/10.3390/mca27050080

**AMA Style**

Khan SA, Hayat T.
Entropy Analysis for Hydromagnetic Darcy–Forchheimer Flow Subject to Soret and Dufour Effects. *Mathematical and Computational Applications*. 2022; 27(5):80.
https://doi.org/10.3390/mca27050080

**Chicago/Turabian Style**

Khan, Sohail A., and Tasawar Hayat.
2022. "Entropy Analysis for Hydromagnetic Darcy–Forchheimer Flow Subject to Soret and Dufour Effects" *Mathematical and Computational Applications* 27, no. 5: 80.
https://doi.org/10.3390/mca27050080