# Thermo-Diffusion and Multislip Effects on MHD Mixed Convection Unsteady Flow of Micropolar Nanofluid over a Shrinking/Stretching Sheet with Radiation in the Presence of Heat Source

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation

## 3. Implementation of Method

## 4. Results and Discussion

## 5. Conclusions

- The fluid velocity, temperature solutal and nano-particle profile are seen to increase with an increment in unsteadiness parameter.
- The fluid velocity and micro-rotation declines while temperature shows opposite behavior with the enhancement in magnetic parameter, suction, hydro-dynamic, and thermal slips.
- The skin-friction coefficient decline with the increment of slip parameters, magnetic and unsteadiness parameter but shows the opposite effect for increasing values of hydrodynamic slip and thermal buoyancy.
- The reduced Nusselt number decreases with the enhancement in suction, radiation, thermophoresis parameter, thermal, and solutal slips.
- The Sherwood number increases with an increase in magnetic parameter, suction parameter, and hydro-dynamic slip.
- The fluid velocity and micro-rotation increase with the increment in K, ${f}_{w}$, and ${S}_{f}$.
- Temperature and solutal concentration increase with the increment in thermophoresis parameter, Schmidt number, Brownian motion parameter, Soret parameter, and thermal slip while the nano-particle concentration declines as the values of thermophoresis parameter, lewis number, and nano-particle slip increase.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

M | Magnetic parameter |

$\kappa $ | Material parameter |

$Rd$ | Radiation parameter |

$\delta $ | Unsteadiness parameter |

$\lambda $ | Buoyancy parameter |

$Nb$ | Brownian motion parameter |

$Nt$ | Thermo-phoresis parameter |

$Pr$ | Prandtl number |

$Ln$ | Lewis number |

R | Thermal radiation parameter |

$Nd$ | Dufour parameter |

$Sr$ | Soret parameter |

$Sc$ | Schmidt number |

Q | Chemical reaction |

${f}_{w}$ | Suction/Injection parameter |

N | Microrotation vector |

$\sigma $ | Electrical conductivity |

$\alpha $ | Thermal diffusivity |

$\gamma $ | Spin gradient viscosity |

g | Gravity |

${k}^{*}$ | Mean absorption coefficient |

${\sigma}^{*}$ | Stefan-Boltzmann constant |

T | Temperature |

${T}_{w}$ | Sheet temperature |

${T}_{\infty}$ | Ambient temperature |

${T}_{0}$ | Reference temperature |

${C}_{w}$ | Solutal concentration |

${C}_{\infty}$ | Ambient solutal concentration |

$U(x,t)$ | Velocity of sheet |

${C}_{0}$ | Reference solutal concentration |

${\chi}_{w}$ | Nanoparticle volume fraction |

${\chi}_{\infty}$ | Ambient nanoparticle concentration |

${\chi}_{0}$ | Reference nanoparticle concentration |

${D}_{T}$ | Thermal diffusivity |

${D}_{s}$ | Molecular diffusivity |

${D}_{B}$ | Brownian diffusivity |

${D}_{CT}$ | Soret diffusivity |

${D}_{Tc}$ | Dufour diffusivity |

$\mu $ | Dynamic viscosity |

k | Vortex viscosity |

$\rho $ | Fluid density |

$u,v$ | Velocity components |

$(u,v)$ | Cartesian coordinates |

${C}_{fr}$ | Reduced skin friction co-efficient |

${N}_{nr}$ | Local Nusselt number |

$S{h}_{r}$ | Reduced Sherwood number |

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**Figure 3.**Influence of K and ${f}_{w}$ on ${f}^{\prime}$. (

**a**) No hydrodynamic slip. (

**b**) With hydrodynamic slip.

**Figure 10.**Influence of M and $\delta $ on $\varphi $. (

**a**) No hydrodynamic slip. (

**b**) With hydrodynamic slip.

$\mathit{Pr}$ | Liaqat et al. | Fazle et al. | Ishak et al. | Dulal et al. | Haile et al. | Ishak et al. [47] | Our Results | Error in % |
---|---|---|---|---|---|---|---|---|

[29] | [11] | [47] | [45] | [46] | (a) | (b) | $|(\frac{\mathit{b}-\mathit{a}}{\mathit{a}})|\times 100$ | |

0.72 | 0.8086 | 0.8088 | - | - | - | 0.8086313498 | 0.8086339299 | 0.0004 |

1.00 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0004 | 1.000000000 | 1.0000080213 | 0.0008 |

3.00 | 1.9236 | 1.9237 | 1.9237 | 1.9236 | 1.9234 | 1.923682594 | 1.9236777221 | 0.0004 |

10.0 | 3.7206 | 3.7207 | 3.7207 | 3.7207 | 3.7205 | 3.720673901 | 3.7206681683 | 0.0002 |

100 | 12.2946 | - | 12.2941 | 12.2962 | 12.2962 | 12.294083260 | 12.294051659 | 0.0002 |

M | Gireesha et al. [48] | Mudassar et al. [49] | Bagh et al. [5] | Our Results | Error in % |
---|---|---|---|---|---|

$\mathit{\beta}=0$ | (a) | (b) | $|(\frac{\mathit{b}-\mathit{a}}{\mathit{a}})|\times 100$ | ||

0.0 | 1.000 | 1.000000 | 1.0000080 | 1.0000130 | 0.00130 |

0.2 | 1.095 | 1.095445 | 1.0954458 | 1.0954463 | 0.00013 |

0.5 | 1.224 | 1.224745 | 1.2247446 | 1.2247454 | 0.00003 |

1.0 | 1.414 | 1.414214 | 1.4142132 | 1.4142180 | 0.00002 |

1.2 | 1.483 | 1.483240 | 1.4832393 | 1.4832402 | 0.00001 |

1.5 | 1.581 | 1.581139 | 1.5811384 | 1.5811396 | 0.00003 |

2.0 | 1.732 | 1.732051 | 1.7320504 | 1.7320516 | 0.00003 |

M | K | $-{\mathit{f}}^{\u2033}(0)$ [50] | $-{\mathit{f}}^{\u2033}(0)$ [51] | Our Results | $-{\mathit{g}}^{\prime}(0)$ [50] | $-{\mathit{g}}^{\prime}(0)$ [51] | Our Results |
---|---|---|---|---|---|---|---|

0.0 | 0.2 | 0.9098 | 0.90976 | 0.909798 | 0.0950 | 0.09500 | 0.094895 |

0.5 | 1.1148 | 1011437 | 1.114378 | 0.1051 | 0.10509 | 0.105088 | |

1.0 | 1.2871 | 1.28711 | 1.287148 | 0.1121 | 0.11212 | 0.112048 | |

0.0 | 1.4142 | 1.41423 | 1.414228 | 0.0000 | 0.00000 | 0.000000 | |

0.5 | 1.1408 | 1.14073 | 1.140772 | 0.2112 | 0.21116 | 0.211165 | |

2.0 | 0.7697 | 0.76958 | 0.769755 | 0.3586 | 0.35855 | 0.358646 |

**Table 4.**Numerical values for different physical constraints M, ${\lambda}_{1}$, ${\lambda}_{2}$, ${\lambda}_{3}$, $\delta $, ${L}_{n}$, $-{f}^{\u2033}(0)$, $-{g}^{\prime}(0)$, $-{\theta}^{\prime}(0)$, $-{\varphi}^{\prime}(0)$.

M | ${\mathit{\lambda}}_{1}$ | ${\mathit{\lambda}}_{2}$ | ${\mathit{\lambda}}_{3}$ | $\mathit{\delta}$ | ${\mathit{L}}_{\mathit{n}}$ | $-{\mathit{f}}^{\u2033}(0)$ | $-{\mathit{g}}^{\prime}(0)$ | $-{\mathit{\theta}}^{\prime}(0)$ | $-{\mathit{\varphi}}^{\prime}(0)$ |
---|---|---|---|---|---|---|---|---|---|

0.5 | 0.945865 | 0.183972 | 0.708908 | 1.705238 | |||||

1.0 | 0.1 | 0.1 | 0.1 | 0.2 | 5.0 | 1.033896 | 0.190324 | 0.694612 | 1.691997 |

3.0 | 1.271767 | 0.199565 | 0.659158 | 1.655360 | |||||

0.5 | 0.1 | 0.945865 | 0.183972 | 0.708908 | 1.705238 | ||||

0.5 | 0.1 | 0.1 | 0.2 | 5.0 | 0.848947 | 0.174640 | 0.725500 | 1.719654 | |

1.0 | 0.737892 | 0.165374 | 0.742820 | 1.734753 | |||||

0.5 | 0.1 | 0.1 | 0.945865 | 0.183972 | 0.708908 | 1.705238 | |||

0.5 | 0.1 | 0.2 | 5.0 | 0.908433 | 0.180855 | 0.715054 | 1.710706 | ||

1.0 | 0.863521 | 0.177375 | 0.722139 | 1.717023 | |||||

0.5 | 0.1 | 0.1 | 0.1 | 0.2 | 5.0 | 0.945865 | 0.183972 | 0.708908 | 1.705238 |

0.5 | 0.913618 | 0.182366 | 0.713483 | 1.709592 | |||||

1.0 | 0.874410 | 0.180529 | 0.718900 | 1.714746 | |||||

0.5 | 0.1 | 0.1 | 0.1 | 0.2 | 5.0 | 0.945865 | 0.183972 | 0.708908 | 1.705238 |

0.8 | 1.030296 | 0.145774 | 0.886881 | 1.852248 | |||||

1.0 | 1.054940 | 0.137761 | 0.931947 | 1.889387 | |||||

0.5 | 0.1 | 0.1 | 0.1 | 0.2 | 5.0 | 0.945865 | 0.183972 | 0.708908 | 1.705238 |

10.0 | 0.950153 | 0.184260 | 0.710358 | 2.175390 | |||||

13.0 | 0.951190 | 0.184309 | 0.711291 | 2.338004 |

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## Share and Cite

**MDPI and ACS Style**

Abdal, S.; Ali, B.; Younas, S.; Ali, L.; Mariam, A.
Thermo-Diffusion and Multislip Effects on MHD Mixed Convection Unsteady Flow of Micropolar Nanofluid over a Shrinking/Stretching Sheet with Radiation in the Presence of Heat Source. *Symmetry* **2020**, *12*, 49.
https://doi.org/10.3390/sym12010049

**AMA Style**

Abdal S, Ali B, Younas S, Ali L, Mariam A.
Thermo-Diffusion and Multislip Effects on MHD Mixed Convection Unsteady Flow of Micropolar Nanofluid over a Shrinking/Stretching Sheet with Radiation in the Presence of Heat Source. *Symmetry*. 2020; 12(1):49.
https://doi.org/10.3390/sym12010049

**Chicago/Turabian Style**

Abdal, Sohaib, Bagh Ali, Saba Younas, Liaqat Ali, and Amna Mariam.
2020. "Thermo-Diffusion and Multislip Effects on MHD Mixed Convection Unsteady Flow of Micropolar Nanofluid over a Shrinking/Stretching Sheet with Radiation in the Presence of Heat Source" *Symmetry* 12, no. 1: 49.
https://doi.org/10.3390/sym12010049