#
Analytical and Numerical Investigation of Fe_{3}O_{4}–Water Nanofluid Flow over a Moveable Plane in a Parallel Stream with High Suction

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## Abstract

**:**

_{3}O

_{4}–water nanoliquid flow over a porous moveable surface in a parallel free stream. The mechanisms of heat transfer are also modeled in the existence of Newtonian heating effect. The obtaining PDEs are transformed into a non-linear ODE system employing appropriate boundary conditions to diverse physical parameters. The governing ODE system is solved using a singular perturbation technique that results in an analytical asymptotic solution as a function of the physical parameters. The obtained solution allows us to carry out an analytical parametric study to investigate the impact of the physical parameters on the nonlinear attitude of the system. The precision of the proposed method is verified by comparisons between the numerical and analytical results. The results confirm that the proposed technique yields a good approximation to the solution as well as the solution calculation has no CPU time-consuming or round off error. Numerical solutions are computed and clarified in graphs for the model embedded parameters. Moreover, profiles of the skin friction coefficient and the heat transfer rate are also portrayed and deliberated. The data manifests that both solid volume fraction and slip impact significantly alter the flow profiles. Moreover, an upward trend in temperature is anticipated for enhancing Newtonian heating strength. Additionally, it was found that both the nanofluid velocity and temperature distributions are decelerated when the solid volume fraction and suction parameters increase. Furthermore, a rise in slip parameter causes an increment in velocity profiles, and a rise in Biot number causes an increment in the temperature profiles.

## 1. Introduction

## 2. Modeling

_{3}O

_{4})–water nanofluid passing through a moveable plane in parallel to a free stream of constant velocity U

_{w}in parallel with a constant free stream velocity U

_{∞}with high fluid suction imposed on the surface. The flow pattern and physical coordinate system is demonstrated in Figure 1. In this coordinate frame, the x-axis extends in parallel to the surface, while the y-axis extends upwards, normal to the surface. The temperature at the plane surface is deemed to have a constant value T

_{f}, which extends a heat transfer coefficient h

_{f}while the ambient temperature has a constant value T

_{∞}. The thermophysical properties of the nanofluid are given in Table 1. In addition, both the base fluid (i.e., water) and the nanoparticles are in thermal equilibrium, and no slip occurs between them. With the above assumptions, the simplifying governing equations of the problem are

_{nf}, μ

_{nf}, α

_{nf}, (ρC

_{p}), and (ρβ)

_{nf}are defined as (see Tiwari and Das [28])

_{w}and Re

_{∞}are the Reynolds numbers, ${\upsilon}_{f}$ is the kinematic coefficient of viscosity of base fluid, and α

_{f}is the thermal diffusivity of base fluid.

_{f}are proportional to ${x}^{-1/2}$, δ and Bi become independent of x and a true similarity is attained. It is manifested that the transpiration parameter f

_{w}= 0 (V

_{w}= 0) coincides with an impermeable surface, while f

_{w}< 0 (V

_{w}> 0) coincides to the status of fluid injection and f

_{w}> 0 (V

_{w}< 0) coincides to the status of the fluid suction or withdrawal (the current work). It is also manifested that velocity ratio parameters γ = 0 and γ = 1 coincides with a fixed plate in a moving fluid and with a moving plate in a quiescent fluid, respectively. The status 0 < γ < 1 is true when the plate and the fluid move in the same direction. If γ < 0, the free stream tends toward the positive x-direction, while the plate moves toward the negative x-direction. If γ > 1, the free stream is directed across the negative x-direction, while the plate moves across the positive x-direction. However, in this investigation, we inspect the status of γ ≤ 1, i.e., the direction of the free stream is specified (across the positive x-direction).

## 3. Analytical Solutions via Singular Perturbation Technique

#### 3.1. An Analytical Solution of Energy Equation

**Theorem**

**1.**

**Proof.**

#### 3.2. An Analytical Solution of the Blasuis Equation

**Theorem**

**2.**

**Proof.**

^{−6}and 3.9 × 10

^{−4}in approximating the temperature and velocity solutions, respectively. Figure 4 and Figure 5 show that the maximum relative error is within $0.0008\%$ and $0.045\%$ in approximating the Nusselt number and skin friction parameter, respectively. The results confirm that a good agreement between analytical and numerical solutions is achieved. Moreover, Figure 2, Figure 3, Figure 4 and Figure 5 show that the numerical data agree with the theoretical results (Theorems 1 and 2), which confirms the validity of the analytical approach and reveals that the method is sufficiently accurate for engineering applications.

## 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 $\gamma $ or the slip factor $\delta $, 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 ${k}_{s}>{k}_{f}>0$, ${(\rho {C}_{P})}_{s}>{(\rho {C}_{P})}_{f}>0$, and $0\le \varphi <0.5$, ${k}_{nf}\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}\sigma >0$, and the solution in Equation (21) results in a positive local Nusselt number.
- Additionally, since we have ${k}_{s}>{k}_{f}>0$, ${(\rho {C}_{P})}_{s}>{(\rho {C}_{P})}_{f}>0$, and $0\le \varphi <0.5$, ${k}_{nf}\propto \varphi ,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\sigma \propto \varphi \hspace{0.17em}$ and $\theta (0)\propto 1/\varphi \hspace{0.17em}$. This means that, as the solid volume fraction $\varphi $ increases the initial temperature of the wall layer, $\theta (0)$, decreases, while the thermal boundary layers thickness $O(\sigma /{f}_{w})$ increases, which suggests that there are intersections points among $\theta (t)$ curves and the temperature profiles decrease non-monotonically.
- Moreover, the solution in Equation (20) shows that, as the suction parameter ${f}_{w}$ increases or the Biot number $Bi$ decreases, the temperature profiles decrease monotonically.
- Additionally, the solution in (21) shows that as the suction parameter ${f}_{w}$ or the Biot number $Bi$ increases the wall temperature gradients (at $\eta =0$) and the local Nusselt number increase.
- The solution in Equation (20) shows that, as the suction parameter ${f}_{w}$ increases the wall temperature and the temperature profile decrease; therefore, the thermal boundary layers thickness decreases, while the Biot number $Bi$ 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 $Bi$, which indicates that it has no effect on the fluid velocity and the Local skin friction coefficient.
- Since we have ${\rho}_{s}>{\rho}_{f}>0$ and $0\le \varphi <0.5$, ${K}_{\varphi}>0$, and the solution in Equation (29) always results in a negative local skin friction coefficient for $\gamma <0.5$ and a positive one for $\gamma >0.5$.
- Since we have ${\rho}_{s}/{\rho}_{f}>7/2\hspace{0.17em}$ and $\hspace{0.17em}\varphi <\frac{1-2{\rho}_{s}/7{\rho}_{f}}{1-{\rho}_{s}/{\rho}_{f}}$ ${K}_{\varphi}\propto \varphi $ and for $\gamma >0.5$ ($\gamma <0.5$), as the solid volume fraction parameter $\varphi $ increases, the velocity profiles decrease (increases).
- The value of $\eta $ at which $\hspace{0.17em}{f}^{\prime}(\eta )=0.5$ can be determined from Equation (27) and is given by$${\eta}_{{f}^{\prime}=0.5}=\frac{1}{{k}_{\varphi}{f}_{w}}\mathrm{ln}\left(\frac{2{\left(1-\varphi \right)}^{5/2}}{{k}_{\varphi}{f}_{w}\mathsf{\delta}+{\left(1-\varphi \right)}^{5/2}}\right),$$
- Moreover, based on Equations (27)–(29), for $\gamma >0.5$ ($\gamma <0.5$) and ${\overline{f}}^{\u2033}(\eta )<0$ (${\overline{f}}^{\u2033}(\eta )>0$), as the suction parameter ${f}_{w}$ or the slip parameter $\delta $ increases, the velocity profiles decrease (increases) monotonically.

## 5. Numerical Results and Discussion

_{w}, slip factor δ, and Biot number Bi, with high values of suction parameter f

_{w}, on the behavior of nanofluid velocity and temperature components as well as the local skin-friction coefficient and the local Nusselt number. The results are shown in Figure 6, Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11. The present numerical study was performed for iron oxide–water nanofluid as a working fluid with various values of velocity ratio parameter γ in the range 0 ≤ γ ≤ 1. The corresponding thermo-physical properties [26] of the fluid and nanoparticles are shown in Table 1.

_{w}on the nanofluid velocity f’(η) and temperature profiles θ(η), respectively. The figure is limited to the status of the suction (lateral mass withdrawal over the plate surface out of the boundary layer regime). From these figures, it is manifested that velocity ratio parameter γ = 0, 0 < γ < 1, and γ = 1 coincides to a fixed plate in a movable fluid, a movable plate in a moving fluid, and a movable plate in a quiescent fluid, respectively. However, it is depicted that the imposition of a wall nanofluid suction (f

_{w}>> 0) tends to enhance the flow along the surface, which results in increasing the velocity profiles for γ < 0.5, while the opposite can be observed for γ > 0.5. In a similar pattern, it is manifested that an increase in nanoparticle volume fraction parameter $\varphi $ causes an enhancement in the nanofluid velocity for γ < 0.5, while the opposite occurs for γ > 0.5. Additionally, both the temperature profiles and thermal boundary layer elevate constantly with the augmenting volume fraction of the nanoparticles, while the reverse occurs with the suction parameter. This coincides with the physical pattern whereby, after the volume fraction of iron oxide boosts thermal conductivity, the thermal boundary layer thickness increases, as shown in Figure 7.

_{w}and $\varphi $ on the local skin friction coefficient ${C}_{f}{\left({\mathrm{Re}}_{w}+{\mathrm{Re}}_{\infty}\right)}^{1/2}$ and the local Nusselt number $N{u}_{x}{\left({\mathrm{Re}}_{w}+{\mathrm{Re}}_{\infty}\right)}^{-1/2}$, respectively, with various values of γ for the parallel moving plate. It is manifested that all values of the ${C}_{f}{\left({\mathrm{Re}}_{w}+{\mathrm{Re}}_{\infty}\right)}^{1/2}$ are positive as γ < 0.5 and negative when γ > 0.5, while γ = 0.5 attains ${C}_{f}{\left({\mathrm{Re}}_{w}+{\mathrm{Re}}_{\infty}\right)}^{1/2}$ = 0, since both the fluid and plate move with the same velocity. Conversely, the values of $N{u}_{x}{\left({\mathrm{Re}}_{w}+{\mathrm{Re}}_{\infty}\right)}^{-1/2}$ are positive for all γ. For γ < 0.5, development in f

_{w}causes a slight decline in the skin friction coefficient, while the reverse behavior can be seen for γ > 0.5. It was also noticed that the increment in ϕ has a tendency to diminish the ${C}_{f}{\left({\mathrm{Re}}_{w}+{\mathrm{Re}}_{\infty}\right)}^{1/2}$ as a result of the increment in the momentum thickness of the boundary layers for the status γ < 0.5, and the opposite impact is manifested for γ > 0.5. Moreover, it is evident in Figure 9a that a sufficient boosting of f

_{w}results in an increase in $N{u}_{x}{\left({\mathrm{Re}}_{w}+{\mathrm{Re}}_{\infty}\right)}^{-1/2}$ for all γ. This conduct is related to the remarkable reduction in the thermal boundary layers as f

_{w}boosts. However, as mentioned, the rise in volume fraction parameter $\varphi $ leads to an increase in the temperature profiles and thermal boundary layers, which results in an increase in $N{u}_{x}{\left({\mathrm{Re}}_{w}+{\mathrm{Re}}_{\infty}\right)}^{-1/2}$, as shown in Figure 9b. This is consistent with the physical manner in which the susceptibility of the thermal boundary layer thickness to $\varphi $ is concerned with the enhanced thermal conductivity of the nanofluid (see Table 1), which in turn enhances in thermal diffusivity and, consequently, following Equation (14), causes a significant evolution in the local Nusselt number.

## 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 |

C_{p} | specific heat at constant pressure (J·kg^{−1}·K^{−1}) |

C_{f} | local skin-friction coefficient |

f_{w} | suction parameter value |

${f}^{\prime}$ | dimensionless velocity |

h_{f} | convective heat transfer coefficient (W/m^{2} k) |

k | thermal conductivity (m^{2} s^{−1}) |

N | velocity slip coefficient |

Nu_{x} | local Nusselt number |

Pr | Prandtl number, n/a_{m} |

Re_{w}, Re_{∞} | Reynolds numbers |

T | temperature (K) |

u, v | velocity components along $x$ and $y$ axes (m/s) |

U_{w}, U_{∞} | the plate velocity and free stream velocity, respectively (m/s) |

x | coordinate in flow direction (m) |

y | coordinate perpendicular to flow direction (m) |

V_{w} | uniform transpiration velocity (m/s) |

Greek Symbols | |

α | thermal diffusivity (m^{2} s^{−1}) |

β | coefficient of thermal expansion (1/K) |

γ | velocity ratio parameter |

η | similarity variable |

θ | dimensionless temperature |

$\varphi $ | solid volume fraction parameter |

ψ | non-dimensional stream function |

δ | velocity slip parameter |

μ | dynamic viscosity (m^{2} s^{−1}) |

ν | kinematic viscosity (m^{2} s^{−1}) |

ρC_{p} | heat capacity (J·kg^{−3}·K^{−1}) |

ρ | density (kg/ m^{3}) |

Subscripts | |

f | fluid |

nf | ferrofluid |

s | nanoparticle |

w | condition at the wall |

∞ | condition at infinity |

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**Figure 2.**The effect of the physical parameters ${f}_{w}$, $\varphi $, $Bi$, and $\delta $ on the absolute error of the obtained analytical solution $\overline{\theta}\hspace{0.17em}(\hspace{0.17em}\eta \hspace{0.17em})$.

**Figure 3.**The effect of the physical parameters ${f}_{w}$, $\varphi $, $Bi$ and $\delta $ on the maximum absolute error of the obtained analytical solution ${\overline{f}}^{\prime}(\hspace{0.17em}\eta \hspace{0.17em})$.

**Figure 4.**The effect of the physical parameters ${f}_{w}$, $\varphi $, $Bi$, and $\delta $ on the maximum absolute error of the obtained analytical solution ${\overline{\theta}}^{\prime}(\hspace{0.17em}0\hspace{0.17em})$.

**Figure 5.**The effect of the physical parameters ${f}_{w}$, $\varphi $, $Bi$, and $\delta $ on the maximum absolute error of the obtained analytical solution ${\overline{f}}^{\u2033}(\hspace{0.17em}0\hspace{0.17em})$.

**Figure 6.**Velocity profiles for different values of (

**a**) suction parameter f

_{w}and (

**b**) solid volume fraction parameter $\varphi $ at different values of $\gamma $.

**Figure 7.**Temperature profiles for different values of (

**a**) suction parameter f

_{w}and (

**b**) soild volume fraction parameter $\varphi $.

**Figure 8.**Local skin friction coefficient for various values of (

**a**) suction parameter f

_{w}and (

**b**) nanoparticle volume fraction parameter $\varphi $.

**Figure 9.**Local Nusselt number for various values of (

**a**) suction parameter f

_{w}and (

**b**) nanoparticle volume fraction parameter $\varphi $.

**Figure 10.**(

**a**) Velocity profiles for different values of slip parameter δ at different values of $\gamma $ and (

**b**) temperature profiles for different values of Biot number Bi.

**Figure 11.**(

**a**) Local skin friction coefficient for various values of slip parameter δ and (

**b**) local Nusselt number for various values of Biot number Bi.

Property | Pure Water | (Fe_{3}O_{4}) |
---|---|---|

ρ (kg m^{−3}) | 997.1 | 5200 |

C_{p} (Jkg^{−1} K^{−1}) | 4179 | 670 |

k (W m^{−1} K^{−1}) | 0.613 | 6 |

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## Share and Cite

**MDPI and ACS Style**

Chamkha, A.J.; Rashad, A.M.; EL-Zahar, E.R.; EL-Mky, H.A.
Analytical and Numerical Investigation of Fe_{3}O_{4}–Water Nanofluid Flow over a Moveable Plane in a Parallel Stream with High Suction. *Energies* **2019**, *12*, 198.
https://doi.org/10.3390/en12010198

**AMA Style**

Chamkha AJ, Rashad AM, EL-Zahar ER, EL-Mky HA.
Analytical and Numerical Investigation of Fe_{3}O_{4}–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 Style**

Chamkha, A. J., A. M. Rashad, E. R. EL-Zahar, and Hamed A. EL-Mky.
2019. "Analytical and Numerical Investigation of Fe_{3}O_{4}–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