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
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, , and .
- 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 |
Material parameter | |
Radiation parameter | |
Unsteadiness parameter | |
Buoyancy parameter | |
Brownian motion parameter | |
Thermo-phoresis parameter | |
Prandtl number | |
Lewis number | |
R | Thermal radiation parameter |
Dufour parameter | |
Soret parameter | |
Schmidt number | |
Q | Chemical reaction |
Suction/Injection parameter | |
N | Microrotation vector |
Electrical conductivity | |
Thermal diffusivity | |
Spin gradient viscosity | |
g | Gravity |
Mean absorption coefficient | |
Stefan-Boltzmann constant | |
T | Temperature |
Sheet temperature | |
Ambient temperature | |
Reference temperature | |
Solutal concentration | |
Ambient solutal concentration | |
Velocity of sheet | |
Reference solutal concentration | |
Nanoparticle volume fraction | |
Ambient nanoparticle concentration | |
Reference nanoparticle concentration | |
Thermal diffusivity | |
Molecular diffusivity | |
Brownian diffusivity | |
Soret diffusivity | |
Dufour diffusivity | |
Dynamic viscosity | |
k | Vortex viscosity |
Fluid density | |
Velocity components | |
Cartesian coordinates | |
Reduced skin friction co-efficient | |
Local Nusselt number | |
Reduced Sherwood number |
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---|---|---|---|---|---|---|---|---|
[29] | [11] | [47] | [45] | [46] | (a) | (b) | ||
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 % |
---|---|---|---|---|---|
(a) | (b) | ||||
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 | [50] | [51] | Our Results | [50] | [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 |
M | |||||||||
---|---|---|---|---|---|---|---|---|---|
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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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
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 StyleAbdal, 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