Thermally Charged MHD Bi-Phase Flow Coatings with Non-Newtonian Nanofluid and Hafnium Particles along Slippery Walls
Abstract
1. Introduction
2. Mathematical Analysis
2.1. Governing Equations
2.2. Boundary Conditions
3. Numerical Results and Discussion
3.1. Procedure
3.2. Validation
3.3. Discussion
4. Conclusions
- The velocity of each phase increases due to an increase in the slip parameter.
- The magnetic field does not support the flow and ends up causing a force of resistance.
- The molecules additives of base fluid reduce the force of friction and hence velocities of both phases are galvanized.
- The temperature of the flow escalates for higher values of Brinkman number.
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Nomenclature
velocity of base fluid | |
distance between plates | |
pressure | |
time | |
Hartmann number | |
reference velocity | |
magnetic field | |
number density of the particles | |
velocity of particle | |
body force | |
body force | |
thermal conductivity | |
specific heat | |
dimensionless constant | |
brinkman number | |
drag force coefficient | |
Greek Symbols | |
γ | couple stress parameter |
ρf | density of base fluid |
η1 | constant associated to couple stress fluid |
Θ0 | temperature of lower wall |
μs | viscosity |
electric conductivity | |
ρ | density of the suspension |
β | slip length |
Θ | dimensional temperature |
Θl | temperature of upper wall |
β1 | slip parameter |
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−1.0 | 1.0000 | 1.0178 | 0 | 0.0178 |
−0.9 | 1.0572 | 1.0748 | 0.0572 | 0.0748 |
−0.8 | 1.1124 | 1.1297 | 0.1124 | 0.1297 |
−0.7 | 1.1638 | 1.1808 | 0.1638 | 0.1808 |
−0.6 | 1.2102 | 1.2271 | 0.2102 | 0.2271 |
−0.5 | 1.2508 | 1.2674 | 0.2508 | 0.2674 |
−0.4 | 1.2847 | 1.3012 | 0.2847 | 0.3012 |
−0.3 | 1.3116 | 1.3279 | 0.3116 | 0.3279 |
−0.2 | 1.3310 | 1.3473 | 0.3310 | 0.3473 |
−0.1 | 1.3427 | 1.3589 | 0.3427 | 0.3589 |
0.0 | 1.3466 | 1.3629 | 0.3466 | 0.3629 |
0.1 | 1.3427 | 1.3589 | 0.3427 | 0.3589 |
0.2 | 1.3310 | 1.3473 | 0.3310 | 0.3473 |
0.3 | 1.3116 | 1.3279 | 0.3116 | 0.3279 |
0.4 | 1.2847 | 1.3012 | 0.2847 | 0.3012 |
0.5 | 1.2508 | 1.2674 | 0.2508 | 0.2674 |
0.6 | 1.2102 | 1.2271 | 0.2102 | 0.2271 |
0.7 | 1.1638 | 1.1808 | 0.1638 | 0.1808 |
0.8 | 1.1124 | 1.1297 | 0.1124 | 0.1297 |
0.9 | 1.0572 | 1.0748 | 0.0572 | 0.0748 |
1.0 | 1.0000 | 1.0178 | 0 | 0.0178 |
−1.0 | 1.0142 | 1.0191 | 0.0142 | 0.0191 |
−0.9 | 1.0626 | 1.0788 | 0.0626 | 0.0788 |
−0.8 | 1.1093 | 1.1363 | 0.1093 | 0.1363 |
−0.7 | 1.1531 | 1.1897 | 0.1531 | 0.1897 |
−0.6 | 1.1928 | 1.2379 | 0.1928 | 0.2379 |
−0.5 | 1.2277 | 1.2799 | 0.2277 | 0.2799 |
−0.4 | 1.2570 | 1.3149 | 0.2570 | 0.3149 |
−0.3 | 1.2803 | 1.3426 | 0.2803 | 0.3426 |
−0.2 | 1.2972 | 1.3626 | 0.2972 | 0.3626 |
−0.1 | 1.3074 | 1.3747 | 0.3074 | 0.3747 |
0.0 | 1.3108 | 1.3788 | 0.3108 | 0.3788 |
0.1 | 1.3074 | 1.3747 | 0.3074 | 0.3747 |
0.2 | 1.2972 | 1.3626 | 0.2972 | 0.3626 |
0.3 | 1.2803 | 1.3426 | 0.2803 | 0.3426 |
0.4 | 1.2570 | 1.3149 | 0.2570 | 0.3149 |
0.5 | 1.2277 | 1.2799 | 0.2277 | 0.2799 |
0.6 | 1.1928 | 1.2379 | 0.1928 | 0.2379 |
0.7 | 1.1531 | 1.1897 | 0.1531 | 0.1897 |
0.8 | 1.1093 | 1.1363 | 0.1093 | 0.1363 |
0.9 | 1.0626 | 1.0788 | 0.0626 | 0.0788 |
1.0 | 1.0142 | 1.0191 | 0.0142 | 0.0191 |
−1.0 | 0 | 0 | 0 | 0 |
−0.9 | 0.0928 | 0.0925 | 0.0814 | 0.0961 |
−0.8 | 0.1740 | 0.1734 | 0.1547 | 0.1794 |
−0.7 | 0.2457 | 0.2449 | 0.2212 | 0.2523 |
−0.6 | 0.3100 | 0.3090 | 0.2822 | 0.3173 |
−0.5 | 0.3687 | 0.3676 | 0.3390 | 0.3764 |
−0.4 | 0.4235 | 0.4224 | 0.3928 | 0.4314 |
−0.3 | 0.4758 | 0.4747 | 0.4446 | 0.4838 |
−0.2 | 0.5266 | 0.5255 | 0.4953 | 0.5347 |
−0.1 | 0.5768 | 0.5757 | 0.5454 | 0.5849 |
0.0 | 0.6268 | 0.6257 | 0.5954 | 0.6349 |
0.1 | 0.6768 | 0.6757 | 0.6454 | 0.6849 |
0.2 | 0.7266 | 0.7255 | 0.6953 | 0.7347 |
0.3 | 0.7758 | 0.7747 | 0.7446 | 0.7838 |
0.4 | 0.8235 | 0.8224 | 0.7928 | 0.8314 |
0.5 | 0.8687 | 0.8676 | 0.8390 | 0.8764 |
0.6 | 0.9100 | 0.9090 | 0.8822 | 0.9173 |
0.7 | 0.9457 | 0.9449 | 0.9212 | 0.9523 |
0.8 | 0.9740 | 0.9734 | 0.9547 | 0.9794 |
0.9 | 0.9928 | 0.9925 | 0.9814 | 0.9961 |
1.0 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |
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Ellahi, R.; Zeeshan, A.; Hussain, F.; Abbas, T. Thermally Charged MHD Bi-Phase Flow Coatings with Non-Newtonian Nanofluid and Hafnium Particles along Slippery Walls. Coatings 2019, 9, 300. https://doi.org/10.3390/coatings9050300
Ellahi R, Zeeshan A, Hussain F, Abbas T. Thermally Charged MHD Bi-Phase Flow Coatings with Non-Newtonian Nanofluid and Hafnium Particles along Slippery Walls. Coatings. 2019; 9(5):300. https://doi.org/10.3390/coatings9050300
Chicago/Turabian StyleEllahi, Rahmat, Ahmed Zeeshan, Farooq Hussain, and Tehseen Abbas. 2019. "Thermally Charged MHD Bi-Phase Flow Coatings with Non-Newtonian Nanofluid and Hafnium Particles along Slippery Walls" Coatings 9, no. 5: 300. https://doi.org/10.3390/coatings9050300
APA StyleEllahi, R., Zeeshan, A., Hussain, F., & Abbas, T. (2019). Thermally Charged MHD Bi-Phase Flow Coatings with Non-Newtonian Nanofluid and Hafnium Particles along Slippery Walls. Coatings, 9(5), 300. https://doi.org/10.3390/coatings9050300