# An MHD Flow of Non-Newtonian Fluid Due to a Porous Stretching/Shrinking Sheet with Mass Transfer

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## Abstract

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## 1. Introduction

## 2. Physical Model

- The velocity in the axial direction is much larger than that in the transverse direction, i.e.,

- The velocity gradient in the transverse direction is much bigger than the velocity gradient in the axial direction.

#### 2.1. Analytical Solution of Momentum Problem

#### 2.2. Analytical Solution of Mass Transfer Problem

## 3. Results and Discussion

## 4. Conclusions

- As the parameter $\alpha $ increases, mass transpiration also increases in both the stretching and shrinking cases.
- Mass transpiration increases with increases in Casson fluid parameter $\Lambda $ and slip factor, and will decreases as $d$ increases.
- Transverse velocity will be higher for higher values of the slip factor in the stretching case and lower for higher values of the slip factor in the shrinking case.
- Axial velocity expands with $K$ or d for the stretching case and shrinks with $K$ or d for the shrinking case. The effect is reversed while varying $\Gamma $.
- Skin friction decreases with increases in d and it is greater in the shrinking sheet case than in the stretching sheet case.
- Skin friction increases with increases in K or $\Gamma $ in the stretching case and will decrease with increases in K or $\Gamma $ for the shrinking sheet case and become constant after a certain stage.
- The concentration profile will be higher for higher values of K or $\beta $; it will higher for higher values of $\Gamma $ in the stretching case and lower for higher values of $\Gamma $ in the shrinking case.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Symbol | Description | S.I. Units |

$a,b$ | stretching/shrinking rates | $\left({s}^{-1}\right)$ |

${B}_{0}$ | magnetic field strength | $\left(w{m}^{-2}\right)$ |

$C$ | concentration field | $\left(mol/{m}^{3}\right)$$c,d$ |

$c,d$ | Stretching/shrinking parameters along x and y axis | $(-)$ |

${D}_{B}$ | molecular diffusivity | $\left({m}^{2}{s}^{-1}\right)$ |

$D{a}^{-1}$ | Inverse Darcy number | $(-)$ |

${k}_{0}$ | material constant | $\left(W/mK\right)$ |

$K$ | viscoelastic parameter | $\left({m}^{-2}\right)$ |

${k}_{C}$ | chemical reaction parameter | $(-)$ |

${K}^{1}$ | permeability of porous medium | $\left({m}^{2}\right)$ |

$l$ | slip factor | $(-)$ |

$\Gamma $ | first order slip parameter | $(-)$ |

M | magnetic parameter | $(-)$ |

$Sc$ | Schmidt number | $(-)$ |

T | Temperature | $(K)$ |

${V}_{C}$ | Mass transpiration | $(-)$ |

$\left(u,v,w\right)$ | velocities along x, y and z direction respectively | $\left(m{s}^{-1}\right)$ |

$\left(x,y,z\right)$ | Cartesian coordinates | $(m)$ |

${w}_{0}$ | wall transpiration | $\left(m{s}^{-1}\right)$ |

Greek symbols | ||

$\beta $ | chemical reaction parameter | $(-)$ |

$\eta $ | Similarity variable | $(-)$ |

${\gamma}_{0}$ | porosity | $(-)$ |

$\gamma $ | porosity parameter | $(-)$ |

$\mu $ | dynamic viscosity o | $\left(kg{m}^{-1}{S}^{-1}\right)$ |

$\nu $ | Kinematic viscosity | $\left({m}^{2}{s}^{-1}\right)$ |

$\rho $ | density | $\left(kg{m}^{-3}\right)$ |

$\varphi $ | dimensionless concentration | $(-)$ |

$\sigma $ | Electric conductivity | $\left(S{m}^{-1}\right)$ |

$\Lambda $ | Brinkman ratio | $(-)$ |

Subscripts | ||

$hnf$ | Hybridnanofluid parameter | $(-)$ |

$w$ | Wall condition | $(-)$ |

$\infty $ | ambient condition | $(-)$ |

Abbreviations | ||

HNF | hybrid nanofluid | $(-)$ |

MHD | Magneto hydrodynamics | $(-)$ |

ODEs | Ordinary differential equations | $(-)$ |

PDEs | Partial differential equations | $(-)$ |

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**Figure 2.**The solution domain of ${V}_{C}$ versus K for different values of $\alpha $ in (

**a**) stretching case $\left(d=1\right)$ and (

**b**) shrinking case $\left(d=-1\right)$.

**Figure 3.**The solution domain of ${V}_{C}$ for different values of $\alpha $, (

**a**) versus $\Lambda $ and (

**b**) versus d in stretching case.

**Figure 4.**The transverse velocity $f\left(\eta \right)$ for different values of $\alpha $. Black curves denote stretching case $\left(d=1\right)$ and red curves denote shrinking case $\left(d=1\right)$.

**Figure 5.**The axial velocity ${f}_{\eta}\left(\eta \right)$ for different values of $K$ in (

**a**), $\Gamma $ in (

**b**), and $d$ in (

**c**). Black curves denote the stretching case and red curves denote the shrinking case.

**Figure 6.**The skin friction $-{f}_{\eta \eta}\left(0\right)$, (

**a**) verses $K$ for different values of $d$, (

**b**) verses $d$ for different values of $K$, and (

**c**) verses $\Gamma $ for different values of $d$; black curves denote stretching case and red curves denote shrinking case.

**Figure 7.**The concentration profile $\varphi \left(\eta \right)$ for different values of $K$ in (

**a**) stretching case and in (

**b**) shrinking case.

**Figure 8.**The concentration profile $\varphi \left(\eta \right)$ for different values of $\beta $ in (

**a**) stretching case and in (

**b**) shrinking case.

**Figure 9.**The concentration profile $\varphi \left(\eta \right)$ for different values of $\Gamma $ in (

**a**) stretching case and in (

**b**) shrinking case.

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**MDPI and ACS Style**

Mahabaleshwar, U.S.; Anusha, T.; Laroze, D.; Said, N.M.; Sharifpur, M.
An MHD Flow of Non-Newtonian Fluid Due to a Porous Stretching/Shrinking Sheet with Mass Transfer. *Sustainability* **2022**, *14*, 7020.
https://doi.org/10.3390/su14127020

**AMA Style**

Mahabaleshwar US, Anusha T, Laroze D, Said NM, Sharifpur M.
An MHD Flow of Non-Newtonian Fluid Due to a Porous Stretching/Shrinking Sheet with Mass Transfer. *Sustainability*. 2022; 14(12):7020.
https://doi.org/10.3390/su14127020

**Chicago/Turabian Style**

Mahabaleshwar, Ulavathi Shettar, Thippeswamy Anusha, David Laroze, Nejla Mahjoub Said, and Mohsen Sharifpur.
2022. "An MHD Flow of Non-Newtonian Fluid Due to a Porous Stretching/Shrinking Sheet with Mass Transfer" *Sustainability* 14, no. 12: 7020.
https://doi.org/10.3390/su14127020