ARTICLE

J. Cent. South Univ. (2019) 26: 1233-1249

DOI: https://doi.org/10.1007/s11771-019-4083-x

Heat transfer intensification in hydromagnetic and radiative 3D unsteady flow regimes: A comparative theoretical investigation for aluminum and γ-aluminum oxides nanoparticles

Naveed AHMED¹, ADNAN², Umar KHAN³, Syed Zulfiqar Ali ZAIDI⁴,Imran FAISAL⁵, Syed Tauseef MOHYUD-DIN¹

1. Department of Mathematics, Faculty of Sciences, HITEC University, Taxila Cantt, Pakistan;

2. Department of Mathematics, Mohi-ud-Din Islamic University, Nerian Sharif,Azad Jammu and Kashmir, Pakistan;

3. Department of Mathematics and Statistics, Hazara University, Mansehra, Pakistan;

4. Department of Mathematics, COMSATS University Islamabad, Abbottabad Campus, Pakistan;

5. Department of Mathematics, Taibah University, Medina, Saudi Arabia

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Abstract: This article investigates the colloidal study for water and ethylene glycol based nanofluids. The effects of Lorentz forces and thermal radiation are considered. The process of non-dimensionalities of governing equations is carried out successfully by means of similarity variables. Then, the resultant nonlinear nature of flow model is treated numerically via Runge-Kutta scheme. The characteristics of various pertinent flow parameters on the velocity, temperature, streamlines and isotherms are discussed graphically. It is inspected that the Lorentz forces favors the rotational velocity and rotational parameter opposes it. Intensification in the nanofluids temperature is observed for volumetric fraction and thermal radiation parameter and dominating trend is noted for γ-aluminum nanofluid. Furthermore, for higher rotational parameter, reverse flow is investigated. To provoke the validity of the present work, comparison between current and literature results is presented which shows an excellent agreement. It is examined that rotation favors the velocity of the fluid and more radiative fluid enhances the fluid temperature. Moreover, it is inspected that upturns in volumetric fraction improves the thermal and electrical conductivities.

Key words: conventional fluids; aluminum and γ-aluminum oxides; magnetic field; thermal radiation; Runge-Kutta scheme; shear stress; local rate of heat transfer

Cite this article as: Naveed AHMED, ADNAN, Umar KHAN, Syed Zulfiqar Ali ZAIDI, Imran FAISAL, Syed Tauseef MOHYUD-DIN. Heat transfer intensification in hydromagnetic and radiative 3D unsteady flow regimes: A comparative theoretical investigation for aluminum and γ-aluminum oxides nanoparticles [J]. Journal of Central South University, 2019, 26(5): 1233–1249. DOI: https://doi.org/10.1007/s11771-019-4083-x.

1 Introduction

Revelatory flow over a stretchable surface has attained much interest of the research community due to huge applications in mechanical as well as chemical industries. Other applications of such flows occur in thin plastic sheets, production of fiber and insulating material etc. Theoretically, the earlier work on the flow over a stretchable sheet was done by CRANE [1]. He investigated two- dimensional flow of viscous fluid over a stretchable surface. Inspired by the work of CRANE, WANG [2] contributed for three-dimensional flow due to stretchable surface which stretched in two perpendicular directions. Inspired by earlier innovative theories, numerous researchers focused on the analysis of flow regimes over stretching surfaces by considering various flow conditions and physical quantities like Lorentz forces, nonlinear thermal radiation, ohmic heating and chemical reaction etc. In this compliment, magnetohydrodynamic flow of non-Newtonian fluid over stretchable surface was reported by ANDERSSON [3]. Heat transfer phenomena over a continuously stretchable surface were explored by CARRAGHER et al [4].

A newly developed class of fluids called nanofluids attained huge attention of the researchers and industrial community. Nanofluids are the colloidal composition of the carrier fluids and nanoparticles of numerous metals and their oxides. The nanoparticles suspend in the base liquids consistently and stably. Nanofluids have distinct imperative characteristics that make them potentially useful in many technological and industrial areas regarding to heat transfer phenomena. The development of nanofluids provided a huge research area and gained much interest of the researchers because of multifarious range of applications comprised in pharmaceutical, heat exchanger, fuel cells, microelectronics, and domestic refrigerator etc. Firstly, CHOI [5] investigated effective thermal conductivity of the nano sized solid nanoparticles that enhanced the heat transfer rate significantly in the nanofluids. The development of nanofluids overcomes an important industrial and technological issue in which rapid heat transfer rate is required. SHEIKHOLESLAMI [6] examined the stimulations of magnetic field in an enclosure in the existence of porosity parameter. Boundary layer flow composed by H₂O and aluminum micro particles reported in Ref. [7]. Numerical investigation of nanofluid by encountering the influences of γ-nanoparticles was discussed in Ref. [8]. Squeezed flow of γ-aluminum by considering H₂O and C₂H₆O₂ between rotating plates was examined in Ref. [9].

Lately, mixed convection flow in triangular cavity and predictions on nanofluids were described in Refs. [10] and [11], respectively. Moreover, significant analysis for heat and mass transfer in cavities was discussed in Refs. [12–15]. Recently, SHEHZAD et al [16] extended boundary model for the third grade fluid by considering the stimulations of viscous dissipation. Furthermore, MHD 3D Jeffery model, MHD flow of Jeffery nanofluid and Maxwell fluid model were described in Refs. [17–19], respectively.

From the literature inspection, it is noted that there is no analysis on the intensifications in the velocity and thermal field in rotating frame by considering the aluminum and γ-aluminum nanoparticles. This study is carried out to fill this gap. The model is formulated successfully and handled by numerical scheme called Runge-Kutta method. The impacts of increasing flow parameters and volume fraction on the velocity and temperature distributions and streamlines and isotherms are presented. The comparison between current and existing data is also provided that the current results are accurate and reliable. Furthermore, Maxwell Garnett model, Brinkman and effective Prandtl models are used for effective thermal and physical characteristics of the nanofluids are used for thermal conductivity, dynamic viscosity and thermal conductivities of γ-nanofluids, respectively. Last section contains the key points of the study.

2 Description of problem

Considering 3D time-dependent rotating flow of electrically conducting water and ethylene glycol suspended by aluminum and γ-aluminum oxides nanoparticles between two parallel plates, the lower plate is situated at y=0 and upper plate is placed at a distance and nanofluids are squeezed with velocity The nanofluids and plates are rotating about positive y-axis in counter clockwise direction with angular component The lower plate of the channel is permeable and sucks the nanofluids with velocity –V₀(1–ct)^–1. Furthermore, lower plate is capable to stretching in x-direction and stretched with velocity U_w=ax/(1–ct). A magnetic field of intensity is imposed perpendicularly. The geometry of the flow model is shown in Figure 1.

The particular nanofluids model in the presence of applied magnetic field and thermal radiation is described by the following set of partial differential equations [21–23]:

                              (1)

Figure 1 Physical theme of problem

               (2)

   (3)

              (4)

                       (5)

The flow conditions at the boundaries of the channel are as follows:

at y=0                  (6)

and

at y=h(t)            (7)

In Eqs. (1)–(7), u, v and w are the components of the velocity along x, y, z directions; T is the temperature; μ_nf is effective dynamic viscosity; k_nf is effective thermal conductivity; ρ_nf is effective density; (ρc_p)_nf specific heat capacitance; σ^* is Stefan Boltzmann constant; k^* is mean absorption coefficient; a denotes the stretching rate of lower plate; B₀ is applied magnetic field; σ_nf is effective electrical conductivity; c shows the characteristic parameter and ct<1; p is the pressure and T_w is temperature at y=0. For particular flow model, the following effective models are used for dynamic viscosity, density, heat capacitance, thermal conductivity and electrical conductivity [1, 2]:

                     (8)

(For Al₂O₃–H₂O and Al₂O₃– C₂H₆O₂),

(For γAl₂O₃–H₂O),

(For γAl₂O₃–C₂H₆O₂) (9)

           (10)

(For Al₂O₃–H₂O and Al₂O₃–C₂H₆O₂),

 (11)

           (12)

Further, thermal and physical properties of the base liquids (water and ethylene glycol) and nanoparticles of aluminum (Al₂O₃) and γ-aluminum nanoparticles are shown in Table 1.

Table 1 Thermal and physical properties [1, 2]

In order to transform the particular flow model into nondimensional form, we have the following similarity transformations [3]:

,

   (13)

Substituting similarity variables, appropriate differentiation, effective models for nanofluids and thermal and physical properties of the base liquids and aluminum oxides nanoparticles in Eqs. (1)–(7), we have the following self-similar form of the nanofluid model.

On the basis of base liquids (water and ethylene) and aluminum and γ-aluminum nanoparticles, we have the following three models:

2.1 Al₂O₃–H₂O and Al₂O₃–C₂H₆O₂ model

       (14)

           (15)

 (16)

2.2 γ-Al₂O₃–H₂O model

      (17)

(18)

    (19)

2.3 γ-Al₂O₃–C₂H₆O₂ model

                           (20)

                            (21)

  (22)

Furthermore, conditions at the boundaries of the channel are as follows:

 at η=0                        (23)

and

at η=1                       (24)

The nondimensional physical parameters embedded in the flow models are as follows:

       (25)

The quantities of physical and practical interest like shear walls stress and local rate of heat transfer are defined in the following way:

At lower wall,

        (26)

At upper wall:

       (27)

With the help of similarity variables and effective models for nanofluids, we have the following mathematical form for skin friction and local rate of heat transfer:

At lower wall:

(For Al₂O₃–H₂O and Al₂O₃–C₂H₆O₂),

(For γ-Al₂O₃–H₂O),

(For γ-Al₂O₃–C₂H₂O₂),

(For Al₂O₃–H₂O and Al₂O₃–C₂H₆O₂),

(For γAl₂O₃–C₂H₂O₂),

(For γ-Al₂O₃–C₂H₂O₂)                    (28)

At upper plate:

(For Al₂O₃–H₂O and Al₂O₃–C₂H₆O₂),

(For γ-Al₂O₃–H₂O),

(For γAl₂O₃–C₂H₆O₂);

(For Al₂O₃–H₂O and Al₂O₃–C₂H₆O₂),

(For γ-Al₂O₃–H₂O),

(For γ-Al₂O₃–C₂H₆O₂)                   (29)

3 Solution of problem

For solution purpose of current flow model, we adopted Runge-Kutta numerical technique coupled with shooting techniques (for Refs. [4–6]). In order to initiate the Runge scheme, we made the following substitutions:

              (30)

First, we write the flow model in the following form:

                              (31)

              (32)

          (33)

By using substitutions, embedded in Eq. (30), we have the following system of first order ordinary differential equations:

H₁=H₂                                                    (34)

where

and

The corresponding initial conditions are:

                             (35)

Eqs. (34) and (35) form a system of the first order initial value problem. The solutions for the said system can be computed by means of Mathematica version 10.0.

In similar pattern, we can solve the flow models for γ-Al₂O₃–H₂O and γ-Al₂O₃–C₂H₆O₂ nanofluids.

4 Results and discussion

In this section, numerical results for the velocity fields, nanofluids temperature, wall skin friction and local rate of heat transfer are analyzed with the help of graphical aids and tables. These results are illustrated for various under consideration flow parameters. Moreover, for validation of the present study, a comparison table is provided which ensures the validity of the study.

4.1 Velocity field

This subsection contains the alterations in the velocity fields (F(η), F′(η) and angular velocity G(η)) due to varying suction parameter A (quotient of V₀ to ah), squeeze parameter a, the quotient of c (characteristics parameter) to a, magnetic parameter M and rotational parameter Ω. The numerical results are portrayed for Al₂O₃–H₂O, γ-Al₂O₃–H₂O and Al₂O₃–C₂H₆O₂, γ-Al₂O₃–C₂H₆O₂ nanofluids. It is imperative to mention that S>0 represents the movement of upper plate towards the lower once and S<0 indicates that the lower plate moves away from the upper plate.

Figure 2 highlights the effects of suction parameter A and squeeze parameter S on the velocity field F(η) for Al₂O₃–H₂O, γ-Al₂O₃–H₂O nanofluids. Figure 2(a) demonstrates the velocity field F(η) for different values of suction parameter A. It is noticed that the nanofluids velocity increases for higher suction parameter. In the presence of γ-aluminum oxide nanoparticles, the velocity increases quite rapidly as compared to aluminum oxide. In the vicinity of the upper and lower boundaries of the channel, almost similar changes in the velocity are investigated. Moreover, slightly reverse flow in the locality of the lower plate is observed due to stretching factor a. The effects of squeeze parameter are illustrated in Figure 2(b) over the domain of interest. As the squeeze parameter is the quotient of characteristic parameter (c) to the stretching rate of the lower sheet (a), the smaller squeeze parameter is due to slight stretching of the lower sheet and also for S>0 indicates that the upper plate accelerates towards the lower plate due to the fact that the fluids velocity starts increasing. On the other hand, S<0 shows that upper plate is fixed at some distance y=h and lower plate accelerates away from it. Due to away movement of the lower stretchable plate, drop in the velocity field F(η) is observed.

Figure 2 Influence of:(R_d=0.7, M=1.0, f=0.2, Ω=5)

Figures 3(a) and (b) illustrate the behavior of the velocity field F′(η) for varying suction and squeeze parameters, respectively. It is noticed that the velocity for Al₂O₃–H₂O, γ-Al₂O₃–H₂O nanofluids declines due to large suction. For both S>0 and S<0, the velocity field decreases and very rapid variations are observed in the region 0.2≤η≤0.8. Besides, the velocity field decreases quite slowly. On the other hand, opposite behavior of the squeeze parameter can be seen from Figure 3(b). The velocity field F′(n) increases when lower plate is fixed (which is placed at y=0) and upper plate placed at a variable distance moves towards the lower permeable sheet. The movement of lower stretching permeable sheet opposes the velocity field F′(n).

The effects of rotation of the plates on the velocity fields F(n) and F′(n) are depicted in Figures 4(a) and (b), respectively. The velocity profile F(n) shows dual behavior due to rotation parameter Ω. Near the lower plate (i.e., at η=0), the velocity of nanofluids decreases. The velocity field decreases rapidly for Al₂O₃–H₂O nanofluids and more fluid drags near the lower plate. The rotation of the plate alters the velocity field for γ-Al₂O₃–H₂O nanofluids inconsequentially. In the environs of η=0.5, both the velocity fields coincide and change the variations in the velocity field. Minimal variations in the velocity field F(η) are inspected near the upper part of the channel. The rotation of the system shows fascinating variations in the velocity field F′(η). For more rotating frame, considerable effects can be seen in the locality of upper and lower sheet. In this region, nanofluids velocity decreases rapidly for Al₂O₃–H₂O nanofluids. The effects are rotation on the velocity of γ-Al₂O₃–H₂O nanofluids being almost inconsequential. In the middle of the channel (i.e., η=0.5) the rotation parameter Ω favors the velocity component. For S>0, these effects are quite slow as compared to that of S<0.

The quotient of electromagnetic force to the viscous force which is known as Hartmann number is very imperative dimensionless physical quantity from industrial and technological point of view. In many industrial processes, flowing fluid may contain impurities which can affect the required production. In such scenario, considerable strength of magnetic field is applied due to the fact that motion of the fluid becomes slowdown and impurities remains at the bottom. Figures 5(a) and (b) highlight the influence of magnetic parameter M on the velocity field F(η) and the velocity field gradient F′(η) over the domain of interest. From Figure 5(a), it can be seen that for higher magnetic parameter M the velocity of Al₂O₃–H₂O and γ-Al₂O₃–H₂O nanofluids decreases. It is observed that in the presence of γ-aluminum oxide nanoparticles, the velocity of H₂O decelerates quite slowly as compared to that of aluminum oxide. On the other hand, alterations in the velocity gradient F′(η) are shown in Figure 5(b) for increasing magnetic parameter M. When lower stretching plate (S<0) moves away from the upper plate, the velocity component decreases rapidly as compared to that of S>0. Near the upper plate η=1 almost inconsequential effects of M on the velocity gradient component F′(η) are investigated. For γ-Al₂O₃–H₂O nanofluids almost negligible effects of magnetic field are observed throughout the channel.

Figures 6–8 demonstrate the behavior of transverse component of the velocity G(η) for varying magnetic parameter, rotation parameter and S, respectively. From Figures 6(a) and (b), it is obvious that the transverse velocity increases for strong magnetic parameter. For Al₂O₃–H₂O nanofluids, transverse velocity increases very rapidly as compared to that of γ-Al₂O₃–H₂O nanofluids. In the presence of γ-aluminum nanoparticles, the velocity field increases slowly. These slow variations are due to the effective dynamic viscosity of γ-aluminum oxide nanoparticles. Furthermore, it is investigated that when upper plate accelerates towards the lower stretchable plate, slight back flow produces near the lower plate (i.e., η=0). The back flow vanishes from lower to upper part of the channel. At the middle portion of the channel, these variations are very prominent and transverse velocity component gradually become slow down near both the plates. In Figure 6(b), it is shown that when lower stretching plate moves away from the upper part of the channel back flow near the lower plate vanishes and transverse velocity G(η) starts increasing rapidly as compared to that Figure 6(a).

Figure 3 Influence of:

Figure 4 Influence of:

Figure 5 Influence of:

Figure 6 Influence of:

Figure 7 Influence of Ω on G(η) for:

Figure 8 Influence of:

Figure 7 illustrates the behavior of transverse velocity G(η) for variable rotational parameter Ω over the domain of interest. As the rotational parameter is a quotient of ω₀ to the stretching of lower plate located at η=0, more stretching of the lower plate leads to smaller rotation of the plates. It is observed that for stronger rotational parameter the transverse velocity G(η) of Al₂O₃–H₂O and γ-Al₂O₃–H₂O nanofluids decreases very rapidly throughout the channel. In the middle portion of the channel, these variations are very rapid. Besides, decreasing behavior of the transverse velocity G(η) is gradually slow down. For S<0 (lower stretchable plate moves away from the upper plate), effects of Ω are rapid as compared to that of S>0 (upper plate accelerates towards lower plate). The parameter S shows very fascinating behavior for transverse component of the velocity G(η). These effects are depicted in Figures 8(a) and (b). It is observed that when lower plate is fixed at η=0 and upper plate accelerates towards it, the nanofluids velocity starts increasing. For Al₂O₃–H₂O, these effects are very prominent. On the other hand, when upper plate is fixed at η=1 and lower plate accelerates away from the upper plate (Figure 8(b)), transverse component decreases.

Figures 9–15 elucidate the impact of various under consideration flow parameters in the flow regimes of ethylene glycol in the presence of aluminum and γ-aluminum nanoparticles. It is observed that in ethylene glycol based nanofluids, velocity of the fluids is quite dominant as compared to that of Al₂O₃–H₂O and γ-Al₂O₃–H₂O nanofluids. This dominates behavior electrical and thermal conductivities of ethylene glycol and aluminum oxides nanoparticles. Moreover, effective thermal conductivity, density and thermal heat capacitance play major role.

4.2 Thermal field

This subsection contains the behavior of thermal field β(η) from η=0 to η=1. Both the cases for S>0 and S<0 are considered. The results are plotted for both aluminum and γ-aluminum nanoparticles over the finite domain. Prandtl number is fixed at 6.2 and 204 for water and ethylene glycol, respectively.

Figure 16 elucidates the temperature for increasing S. It is inspected that the movement of the upper plate towards lower plate favors the nanofluid temperature. In the presence of γ-Al₂O₃ nanoparticles, temperature increases slowly. In the portion 0.2≤η≤0.4, very rapid variations in the temperature filed β(η) are noted. In rest of the portion these variations gradually slow down. In the case of Al₂O₃–C₂H₆O₂ and γ-Al₂O₃–C₂H₆O₂ nanofluids self-similar thermal field increases quite slowly as compared to that of Al₂O₃–water and γ-Al₂O₃–water nanofluids. This difference is due to different values of Prandtl number and electrical conductivities of base fluids and nanoparticles.

Figures 17–20 elaborate the influence of volumetric fraction of the nanoparticles and radiation parameter R_d on the temperature distribution β(η). It can be seen that for higher volumetric fraction of the nanoparticles f, considerable variations are noted. For S<0, nanofluids temperature increases rapidly. Further, for γ-aluminum–H₂O, the temperature increases slowly. On the other hand, remarkable effects of thermal radiation parameter R_d on β(η) are observed for γ-aluminum–H₂O. For aluminum–H₂O nanofluids, temperature rises very slowly and near the upper part of the channel, these effects are almost inconsequential.

Figure 9 Influence of:

Figure 10 Influence of:

Figure 11 Influence of:

Figure 12 Influence of:

Figure 13 Influence of:

Figure 14 Influence of:

4.3 Thermal field

Figures 21 and 22 elaborate the impact of S on the streamlines for various under consideration flow parameters. From the streamlines pattern, it can be seen that there is no crisscross movement which shows that the flow is laminar. Further, from Figure 21(b), it is observed that the movement of the upper plate leads to back flow. Similarly,Figure 22 highlights the behavior of streamlines for varying rotational parameter Ω (for Ω=0.3 and Ω=2.0). It is noted that more back flow produces due to smaller Ω, and increasing Ω, the back flow region decreases. Figure 23 shows the variations in isotherms for both S>0 and S<0, respectively. Figures 24(a) and (b) show the influence of volumetric fraction f on thermal conductivity k_nf, specific heat capacitance (ρc_p)_nf and electrical conductivity σ_nf, respectively. The volumetric fraction f varies horizontally and the variations in the nanofluids properties are vertical. It is noted that aforementioned properties of nanofluids increase directly for higher volumetric fraction f. Furthermore, electrical conductivity σ_nf increases very slowly.

Figure 15 Influence of:

Figure 16 Influence of S for water based nanofluid (a) and ethylene glycol (b) (R_d=0.5, f=0.2, Ω=5, A=2)

Figure 17 Influence of f on β(η):

Figure 18 Influence of f on β(η) in ethylene glycol for:

Figure 19 Influence of R_d on β(η):

Figure 20 Influence of f on β(η) for ethylene glycol for:

4.4 Validity of study

Our flow model is wide-ranging model for nanofluids. For f=0, particular flow model is reduced into the model for carrier liquids. The comparison of present study for f=0 and M=0.5 is described in Tables 2 and 3. It is examined that the present results are in the best agreement with existing literature which provoked the reliability of the current study.

Figure 21 Streamlines pattern for:

Figure 22 Streamlines pattern for:

Figure 23 Isotherms pattern for:

5 Conclusions

A nonlinear time-dependent flow model comprising aluminum and γ-aluminum oxides nanoparticles by seeing the effects of Lorentz force and thermal radiation is presented. The specific model is treated numerically and concluded the following core findings:

1) It is examined that the velocity field F(η) increases when upper plate is moving towards the lower stretching plate and reverse behavior is noted that lower plate moves away from the upper plate.

2) Applied magnetic field opposes the velocity field F(η) and favors the rotational velocity G(η).

3) The rotational parameter Ω favors the fluid velocity F′(η) in the middle of the channel.

4) For increasing Ω, the velocity G(η) decreases and dominating behavior for γ-aluminum nanofluid is examined.

Figure 24 Influence of f on k_nf (a), (ρc_p)_nf (b) and σ_nf (c) for f=0.01, 0.04, 0.08, 0.12, 0.16, 0.20

Table 2 Comparison with existing literature for f=0, M=0.5

Table 3 Numerical values for F″(0) and F″(1)

5) The squeeze parameter, volumetric fraction of the nanoparticles and thermal radiation parameter affected the nanofluids temperature positively.

6) Streamlines show that for S<0, back flow produces and S>0 supports the forward flow.

7) For feasible values of f, thermal conductivity k_nf, specific heat capacitance and electrical conductivity σ_nf increase.

8) The present results for shear stresses at the upper and lower plates show an excellent agreement with existing literature under certain limitations.

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(Edited by YANG Hua)

中文导读

磁辐射3D非稳态流动体系的传热强化：Al和γ-Al₂O₃纳米粒子的对比研究

摘要：本文研究了水和乙二醇基纳米流体的胶体性质。考虑了洛伦兹力和热辐射的影响，通过相似变化，完成了控制方程的无量纲转化。然后，应用龙格-库塔法，对流动模型的非线性特性进行数值分析，对有关流速、温度、流线、等温线等流动参数的特性进行图解分析。结果表明，洛伦兹力有利于旋转速度，不利于其他旋转参数。体积分数和热辐射参数对Al和γ-Al₂O₃纳米流体的温度有正相关强化作用。此外，对于较高的旋转参数，研究其反向流动问题。与文献结果进行了比较，结果显示具有较高的吻合度。旋转有利于流体的流动速度，辐射流体越多，流体温度越高，而且体积分数的增加提高了热导率和电导率。

关键词：传统流体；铝和铝氧化物；磁场；热辐射；Runge-Kutta方案；剪切应力；局部热传递速率

Received date: 2018-10-16; Accepted date: 2018-12-12

Corresponding author: Umar KHAN, PhD, Assistant Professor; Tel: +92-332-8902728; E-mail: umar_jadoon4@yahoo.com; ORCID: 0000-0002-4518-2683