Dual Solutions and Stability Analysis of a Hybrid Nanofluid over a Stretching/Shrinking Sheet Executing MHD Flow
Abstract
:1. Introduction
2. Problem Formulation
3. Stability Analysis
4. Result and Discussion
Properties | Hybrid Nanofluid |
---|---|
Dynamic viscosity | |
Density | |
Thermal conductivity | where |
Heat capacity |
Fluids | k (W/m K) | ||
---|---|---|---|
Alumina () | 3970 | 765 | 40 |
Copper (Cu) | 8933 | 385 | 400 |
Water () | 997.1 | 4179 | 0.613 |
Waini | [31] | Present | Results | Waini | [31] | Present | Results | ||
---|---|---|---|---|---|---|---|---|---|
1st Soln | 2nd Soln | 1st Soln | 2nd Soln | 1st Soln | 2nd Soln | 1st Soln | 2nd Soln | ||
−1 | 0 | 2.194247 | −1.491281 | 2.19424658 | −1.49128119 | 7.073680 | 6.884548 | 7.0736798 | 6.88454793 |
−1 | 0.1 | -- | -- | 1.60888878 | −0.69818494 | -- | -- | 5.2238150 | 5.0737595 |
−3 | 0 | 1.521197 | −4.144746 | 1.52119229 | −4.14474572 | 7.497151 | 7.296176 | 7.4971508 | 7.2961762 |
−3 | 0.1 | -- | -- | 0.83697819 | −2.53135605 | -- | -- | 5.6316794 | 5.4925121 |
−5 | 0 | 0.844435 | −6.431507 | 0.84443506 | −6.43150738 | 7.858446 | 7.657801 | 7.8584457 | 7.6578012 |
−5 | 0.01 | -- | -- | 0.77597819 | −6.20435575 | -- | -- | 7.6375695 | 7.4426329 |
−9 | 0 | −0.517287 | −10.58983 | −0.51728551 | −10.5898303 | 8.473316 | 8.277676 | 8.4733163 | 8.27767616 |
−9 | 0.01 | -- | -- | −0.60021181 | −10.2417356 | -- | -- | 8.24709639 | 8.05819003 |
5. Conclusions
Abbreviation
Nomenclature | |||
a constant | nanoparticle volume fraction of the iron oxide. | ||
ambient temperature | M | magnetic parameter | |
differentiation with respect to | Porous parameter | ||
Thermal radiation | c | constant | |
effective density of hybrid nanofluid | Prandtl number | ||
effective density of nanofluid | skin friction coefficient | ||
effective dynamic viscosity of hybrid nanofluid | smallest eigen value | ||
effective dynamic viscosity of nanofluid | Stability transformed variable | ||
electrical conductivity | suction/injection velocity | ||
fluid fraction | T | Temperature | |
M | Hartmann/magnetic number | thermal conductivity of the hybrid nanofluid | |
heat capacitance of the hybrid nanofluid | thermal conductivity of the nanofluid | ||
heat capacitance of the nanofluid | t | time | |
hnf | Hybrid nanofluid | transformed variable | |
local Nusselt number | Unsteadiness parameter | ||
local Reynolds number | variable temperature at the sheet | ||
B | Magnetic field | u, v | velocity components |
nanofluid fraction | shrinking/stretching parameter | ||
mean absorption coefficient | Stefan–Boltzmann constant | ||
nanoparticle volume fraction of the copper | for suction parameter and for blowing parameter |
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Lund, L.A.; Omar, Z.; Khan, I.; Sherif, E.-S.M. Dual Solutions and Stability Analysis of a Hybrid Nanofluid over a Stretching/Shrinking Sheet Executing MHD Flow. Symmetry 2020, 12, 276. https://doi.org/10.3390/sym12020276
Lund LA, Omar Z, Khan I, Sherif E-SM. Dual Solutions and Stability Analysis of a Hybrid Nanofluid over a Stretching/Shrinking Sheet Executing MHD Flow. Symmetry. 2020; 12(2):276. https://doi.org/10.3390/sym12020276
Chicago/Turabian StyleLund, Liaquat Ali, Zurni Omar, Ilyas Khan, and El-Sayed M. Sherif. 2020. "Dual Solutions and Stability Analysis of a Hybrid Nanofluid over a Stretching/Shrinking Sheet Executing MHD Flow" Symmetry 12, no. 2: 276. https://doi.org/10.3390/sym12020276