# Entropy Generation and Consequences of MHD in Darcy–Forchheimer Nanofluid Flow Bounded by Non-Linearly Stretching Surface

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## Abstract

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## 1. Introduction

## 2. Mathematical Modeling

#### Entropy Generation Modeling

## 3. Solution Methodology

## 4. Analysis

## 5. Conclusions

- Entropy optimization, heat and mass transport in Darcy–Forchheimer and MHD type nanofluid flow surrounded by a non-linear stretching surface is analyzed.
- Rate of Bejan number shows mixed behavior for elevated values of temperature difference parameter. An enhancement is noted for larger values.
- Skin friction enhances for all the parameters involved in momentum equation.
- Heat transfer rate declines while mass transfer intensifies for elevated numbers of thermal radiation parameters.
- Resistive force due to inertia and enhanced friction are the source of enhancement in heat convection.
- The concentration of nanoparticles reduces near the surface, whereas an enhancement is noted in the case of thermophoresis due to stronger thermophoretic force.
- An enhanced variation in stream lines is noted at distance far away from the origin. Near to the origin, this variation is very narrow.
- We observed that a more porous medium offers more retardational force (friction), which continuously diminishes the velocity of the fluid.
- Contour graphs given at $M=0.1$ and $M=0.5$ show an enhanced variation at distance sufficiently away from origin. Near to the origin, the variation is very narrow.
- Stream function graphs at $M=0.1$ and $M=0.5$ show a very narrow variation in the stream lines. At $M=0.1$ curves are not spread much as compared to case $M=0.5$. Stronger magnetic effect boosts the opposing Lorentz forces which occur in the way of fluid motion and stream lines get affected.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$\mathrm{RK}45$ | Runge-Kutta 45 Method |

MHD | Magnetohydrodynamics |

PDE | Partial Differential Equation |

ODE | Ordinary Differential Equation |

${u}_{1},{u}_{2}$ | Cartesian velocity coordinates/m·s${}^{-1}$ |

$x,y$ | Cartesian distance coordinates/m |

${u}_{w}=m{x}^{n}$ | Velocity (stretching)/m·s${}^{-1}$ |

m | Stretching rate/s${}^{-1}$ |

$\mu $ | Dynamic viscosity/Pa·s |

${B}_{0}$ | Magnetic impact/intensity/A·m${}^{-1}$ |

$\nu $ | Kinematic viscosity/m${}^{2}\xb7$s${}^{-1}$ |

${\rho}_{f}$ | Density/kg·m${}^{-3}$ |

${D}_{Br}$ | Brownian diffusion |

${D}_{Th}$ | thermophoresis |

T | Temperature distributions /K |

C | Concentration distributions/kg·m${}^{-3}$ |

$\sigma $ | Electric conductivity of the base fluid/($\Omega $ m)${}^{-1}$ |

${\left(\rho c\right)}_{np}$ | Nanoparticles’ heat capacity/J·m${}^{-3}\xb7$k${}^{-1}$ |

${\left(\rho c\right)}_{fl}$ | Fluid’s heat capacity/J·m${}^{-3}\xb7$k${}^{-1}$ |

${C}_{b}$ | Drag force coefficient |

K | Permeability |

$\tau $ | Ratio of heat capacity of fluid and nanoparticles |

${q}_{r}$ | Radiative heat flux |

${\sigma}^{\prime}$ | Stephen boltzmann constant |

$k\prime $ | Mean absorption constant |

$\alpha $ | Thermal diffusivity/m${}^{2}\xb7{\mathrm{s}}^{-1}$ |

k | Thermal conductivity/$\mathrm{W}\xb7{\mathrm{m}}^{-1}\xb7{\mathrm{K}}^{-1}$ |

Dimensionless Parameters | |

M | Magnetic parameter |

$Pr$ | Prandtl number |

$Sc$ | Schmidt number |

${N}_{b}$ | Brownian diffusion |

${N}_{t}$ | thermophoresis |

$S{h}_{x}$ | Sherwood factor |

$N{u}_{x}$ | Nusselt factor |

${S}_{G}$ | Entropy generation rate in two dimensions |

${R}_{D}$ | Difference ratio |

${N}_{G}$ | Entropy generation rate |

${\beta}_{1}$ | Temperature difference |

$B{r}_{1}$ | Brinkman number |

${L}_{1}$ | Diffusive parameter |

${F}_{r}$ | Forchheimer number |

$\lambda $ | Porosity |

${R}_{d}$ | Radiation parameter |

$Ec$ | Eckert number |

${\mu}_{0}$ | Viscosity at initial position |

$\eta $ | Variable |

$\varphi $ | Concentration distribution (dimensionless) |

$\theta $ | Temperature distribution (dimensionless) |

${f}^{\prime}$ | Velocity (dimensionless) |

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**Table 1.**Numerical outcomes of skin-friction given at $n=1.2$ while values of other parameters are varied one by one.

M | ${\mathit{F}}_{\mathit{r}}$ | $\mathit{\lambda}$ | $-{\mathit{Re}}_{\mathit{x}}{\mathit{C}}_{\mathit{x}}$ |
---|---|---|---|

$0.0$ | $0.3$ | $0.6$ | $1.6772$ |

$0.5$ | $1.75074$ | ||

$1.0$ | $2.325338$ | ||

$0.5$ | $0.0$ | $0.6$ | $1.63528$ |

$0.3$ | $1.75074$ | ||

$0.6$ | $1.85947$ | ||

$0.2$ | $0.5$ | $0.0$ | $1.56812$ |

$0.6$ | $1.75074$ | ||

$1.2$ | $2.419236$ |

**Table 2.**Numerical outcomes of heat transfer (Nusselt) rate and mass transfer (Sherwood) rate given at $n=1.2$ while values of other parameters are varied one by one.

M | ${\mathit{F}}_{\mathit{r}}$ | $\mathit{\lambda}$ | ${\mathit{R}}_{\mathit{d}}$ | Pr | ${\mathit{N}}_{\mathit{t}}$ | ${\mathit{N}}_{\mathit{b}}$ | $\mathit{Ec}$ | $\mathit{Sc}$ | Nusselt | Sherwood |
---|---|---|---|---|---|---|---|---|---|---|

$0.0$ | $0.3$ | $0.6$ | $0.5$ | $1.0$ | $0.1$ | $0.2$ | $0.7$ | $1.0$ | 0.330523 | 0.5403 |

$0.5$ | 0.294335 | 0.539539 | ||||||||

$1.0$ | 0.146822 | 0.196834 | ||||||||

$0.5$ | $0.0$ | $0.6$ | $0.5$ | $1.0$ | $0.1$ | $0.2$ | $0.7$ | $1.0$ | 0.298447 | 0.550958 |

$0.3$ | 0.294335 | 0.539539 | ||||||||

$0.6$ | 0.29052 | 0.529184 | ||||||||

$0.5$ | $0.3$ | $0.0$ | $0.5$ | $1.0$ | $0.1$ | $0.2$ | $0.7$ | $1.0$ | 0.386878 | 0.542202 |

$0.6$ | 0.294335 | 0.539539 | ||||||||

$1.2$ | 0.157756 | 0.222705 | ||||||||

$0.5$ | $0.3$ | $0.6$ | $0.0$ | $1.0$ | $0.1$ | $0.2$ | $0.7$ | $1.0$ | 0.397385 | 0.501379 |

$0.5$ | 0.294335 | 0.539539 | ||||||||

$1.0$ | 0.24219 | 0.561276 | ||||||||

$0.5$ | $0.3$ | $0.6$ | $0.5$ | $1.0$ | $0.1$ | $0.2$ | $0.7$ | $1.0$ | 0.294335 | 0.539539 |

$1.5$ | 0.374306 | 0.509569 | ||||||||

$2.0$ | 0.43873 | 0.487011 | ||||||||

$0.5$ | $0.3$ | $0.6$ | $0.5$ | $1.0$ | $0.1$ | $0.2$ | $0.7$ | $1.0$ | 0.294335 | 0.539539 |

$0.4$ | 0.271793 | 0.309337 | ||||||||

$0.8$ | 0.244013 | 0.081471 | ||||||||

$0.5$ | $0.3$ | $0.6$ | $0.5$ | $1.0$ | $0.1$ | $0.2$ | $0.7$ | $1.0$ | 0.294335 | 0.539539 |

$0.4$ | 0.26720 | 0.589734 | ||||||||

$0.6$ | 0.241696 | 0.606221 | ||||||||

$0.5$ | $0.3$ | $0.6$ | $0.5$ | $1.0$ | $0.1$ | $0.2$ | $0.0$ | $1.0$ | 0.39101 | 0.495559 |

$0.5$ | 0.321959 | 0.526972 | ||||||||

$1.0$ | 0.252895 | 0.55839 | ||||||||

$0.5$ | $0.3$ | $0.6$ | $0.5$ | $1.0$ | $0.1$ | $0.2$ | $0.7$ | $1.0$ | 0.294335 | 0.539539 |

$1.5$ | 0.290366 | 0.76239 | ||||||||

$2.0$ | 0.288085 | 0.952778 |

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## Share and Cite

**MDPI and ACS Style**

Rasool, G.; Shafiq, A.; Khan, I.; Baleanu, D.; Sooppy Nisar, K.; Shahzadi, G.
Entropy Generation and Consequences of MHD in Darcy–Forchheimer Nanofluid Flow Bounded by Non-Linearly Stretching Surface. *Symmetry* **2020**, *12*, 652.
https://doi.org/10.3390/sym12040652

**AMA Style**

Rasool G, Shafiq A, Khan I, Baleanu D, Sooppy Nisar K, Shahzadi G.
Entropy Generation and Consequences of MHD in Darcy–Forchheimer Nanofluid Flow Bounded by Non-Linearly Stretching Surface. *Symmetry*. 2020; 12(4):652.
https://doi.org/10.3390/sym12040652

**Chicago/Turabian Style**

Rasool, Ghulam, Anum Shafiq, Ilyas Khan, Dumitru Baleanu, Kottakkaran Sooppy Nisar, and Gullnaz Shahzadi.
2020. "Entropy Generation and Consequences of MHD in Darcy–Forchheimer Nanofluid Flow Bounded by Non-Linearly Stretching Surface" *Symmetry* 12, no. 4: 652.
https://doi.org/10.3390/sym12040652