Analytical and Numerical Investigation of Fe3O4–Water Nanofluid Flow over a Moveable Plane in a Parallel Stream with High Suction
Abstract
1. Introduction
2. Modeling
3. Analytical Solutions via Singular Perturbation Technique
3.1. An Analytical Solution of Energy Equation
3.2. An Analytical Solution of the Blasuis Equation
4. Analytical Parametric Study
- Solutions in Equations (20) and (21) show that the temperatures profiles have exponential distributions.
- We notice that the solutions in Equations (20) and (21) do not contain the velocity ratio parameter or the slip factor , which indicates that, for high suction, the effect of these parameters on the temperature profiles and the local Nusselt number can be neglected compared to other existing parameters.
- Since we have , , and , , and the solution in Equation (21) results in a positive local Nusselt number.
- Additionally, since we have , , and , and . This means that, as the solid volume fraction increases the initial temperature of the wall layer, , decreases, while the thermal boundary layers thickness increases, which suggests that there are intersections points among curves and the temperature profiles decrease non-monotonically.
- Moreover, the solution in Equation (20) shows that, as the suction parameter increases or the Biot number decreases, the temperature profiles decrease monotonically.
- Additionally, the solution in (21) shows that as the suction parameter or the Biot number increases the wall temperature gradients (at ) and the local Nusselt number increase.
- The solution in Equation (20) shows that, as the suction parameter increases the wall temperature and the temperature profile decrease; therefore, the thermal boundary layers thickness decreases, while the Biot number has a neglected effect on the temperature layer’s thickness compared to other parameters.
- The solutions in Equations (27)–(29) do not contain the Biot number , which indicates that it has no effect on the fluid velocity and the Local skin friction coefficient.
- Since we have and , , and the solution in Equation (29) always results in a negative local skin friction coefficient for and a positive one for .
- Since we have and and for (), as the solid volume fraction parameter increases, the velocity profiles decrease (increases).
- The value of at which can be determined from Equation (27) and is given by
- Moreover, based on Equations (27)–(29), for () and (), as the suction parameter or the slip parameter increases, the velocity profiles decrease (increases) monotonically.
5. Numerical Results and Discussion
6. Conclusions
- The present singular perturbation technique results in a closed form asymptotic solution of the energy and Blasuis equations as a function of the physical parameters.
- The rapid calculation of the system solution (dynamic response) with acceptable accuracy demonstrates that the analytical solutions are effective for performing analytical parametric studies.
- An analytical parametric study is carried out to predict the impact of the system physical parameters on the temperature and velocity behaviors.
- The results of the numerical study confirms a high validation of the present analytical parametric study and their main results can be summarized as follows:
- Both the nanofluid velocity and temperature distributions are decelerated for growing the solid volume fraction and suction parameters.
- The raising in slip parameter causes an increment in the velocity profiles, and the raising in Biot number causes an increment in the temperature profiles.
- The local Nusselt number elevates along with boosting values of Biot number solid volume fraction and suction parameters.
Author Contributions
Funding
Conflicts of Interest
Abbreviations
Nomenclature | |
Bi | Biot number |
Cp | specific heat at constant pressure (J·kg−1·K−1) |
Cf | local skin-friction coefficient |
fw | suction parameter value |
dimensionless velocity | |
hf | convective heat transfer coefficient (W/m2 k) |
k | thermal conductivity (m2 s−1) |
N | velocity slip coefficient |
Nux | local Nusselt number |
Pr | Prandtl number, n/am |
Rew, Re∞ | Reynolds numbers |
T | temperature (K) |
u, v | velocity components along and axes (m/s) |
Uw, U∞ | the plate velocity and free stream velocity, respectively (m/s) |
x | coordinate in flow direction (m) |
y | coordinate perpendicular to flow direction (m) |
Vw | uniform transpiration velocity (m/s) |
Greek Symbols | |
α | thermal diffusivity (m2 s−1) |
β | coefficient of thermal expansion (1/K) |
γ | velocity ratio parameter |
η | similarity variable |
θ | dimensionless temperature |
solid volume fraction parameter | |
ψ | non-dimensional stream function |
δ | velocity slip parameter |
μ | dynamic viscosity (m2 s−1) |
ν | kinematic viscosity (m2 s−1) |
ρCp | heat capacity (J·kg−3·K−1) |
ρ | density (kg/ m3) |
Subscripts | |
f | fluid |
nf | ferrofluid |
s | nanoparticle |
w | condition at the wall |
∞ | condition at infinity |
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Chamkha, A.J.; Rashad, A.M.; EL-Zahar, E.R.; EL-Mky, H.A. Analytical and Numerical Investigation of Fe3O4–Water Nanofluid Flow over a Moveable Plane in a Parallel Stream with High Suction. Energies 2019, 12, 198. https://doi.org/10.3390/en12010198
Chamkha AJ, Rashad AM, EL-Zahar ER, EL-Mky HA. Analytical and Numerical Investigation of Fe3O4–Water Nanofluid Flow over a Moveable Plane in a Parallel Stream with High Suction. Energies. 2019; 12(1):198. https://doi.org/10.3390/en12010198
Chicago/Turabian StyleChamkha, A. J., A. M. Rashad, E. R. EL-Zahar, and Hamed A. EL-Mky. 2019. "Analytical and Numerical Investigation of Fe3O4–Water Nanofluid Flow over a Moveable Plane in a Parallel Stream with High Suction" Energies 12, no. 1: 198. https://doi.org/10.3390/en12010198
APA StyleChamkha, A. J., Rashad, A. M., EL-Zahar, E. R., & EL-Mky, H. A. (2019). Analytical and Numerical Investigation of Fe3O4–Water Nanofluid Flow over a Moveable Plane in a Parallel Stream with High Suction. Energies, 12(1), 198. https://doi.org/10.3390/en12010198