F. M. ABBASI， S. A. SHEHZAD， T. HAYAT， M. S. ALHUTHALI

1. Department of Mathematics， Quaid-i-Azam University， Islamabad， Pakistan

2. Department of Mathematics， Comsats Institute of Information Technology， Sahiwal， Pakistan

3. Nonlinear Analysis and Applied Mathematics （NAAM） Research Group， Department of Mathematics， Faculty of Science， King Abdulaziz University， Jeddah， Saudi Arabia

4. Department of Mathematics， COMSATS Institute of Information Technology Islamabad， Islamabad， Pakistan，E-mail: abbasisarkar@gmail.com

Mixed convection flow of jeffrey nanofluid with thermal radiation and double stratification*

F. M. ABBASI4， S. A. SHEHZAD2， T. HAYAT1，3， M. S. ALHUTHALI3

1. Department of Mathematics， Quaid-i-Azam University， Islamabad， Pakistan

2. Department of Mathematics， Comsats Institute of Information Technology， Sahiwal， Pakistan

3. Nonlinear Analysis and Applied Mathematics （NAAM） Research Group， Department of Mathematics， Faculty of Science， King Abdulaziz University， Jeddah， Saudi Arabia

4. Department of Mathematics， COMSATS Institute of Information Technology Islamabad， Islamabad， Pakistan，E-mail: abbasisarkar@gmail.com

Abstarct： This article addresses the two-dimensional laminar boundary layer flow of magnetohydrodynamic （MHD） Jeffrey nanofluid with mixed convection. Effects of thermal radiation， thermophoresis， Brownian motion and double stratifications are taken into account. Rosseland's approximation is utilized for the thermal radiation phenomenon. Convergent series solutions of velocity， temperature and nanoparticle concentration are developed. Graphs of dimensionless temperature and nanoparticle concentration are presented to investigate the influences of different emerging parameters. The values of skin-friction coefficient， local Nusselt and Sherwood numbers are computed and discussed for both Jeffrey and viscous fluids cases. We have observed that the temperature profile retarded for the larger values of Deborah number while an enhancement is noticed with the increasing values of ratio of relaxation to retardation times. Increasing values of thermal and nanoparticle concentration stratifications lead to a reduction in the temperature and nanoparticle concentration. The values of local Nusselt and Sherwood numbers are larger for the viscous fluid case when compared with Jeffrey fluid.

MHD flow， Jeffrey nanofluid， double stratification， thermal radiation， mixed convection

Introduction

The advancements in nanofluid technology increased largely during the past few years due to their higher thermal performances. Nanofluid is a suspendsion of very small size （1 nm-100 nm） metallic particles into a base liquid. Normally these fluids are made by using a mixture of solid particles into base fluids like water， engine oil， ethylene glycol （EG） etc.. The structure of nanoparticles consists of metal （Al， Ti， Fe，Au， Ag， etc.） carbide， nitride， metal oxide （CuO， TiO2，Al2O3etc.） and even nanoscale liquid droplets［1］. Such particles have shape of sphere， tubular， rod-like etc. Choi's investigation［2］shows that the implementation of the nanofluids is utilized in vast range of industrial manufacturing processes like textile， transportation，paper production， electronic equipment， energy production and many others. The basic characteristics of nanofluids are their thermal conductivities which are much higher than the other base fluids. The combination of such substances provides us a medium for heat transfer that behaves as a fluid but has the thermal conductivity of metal. In fact the addition of nanoparticles in the base liquids enhances the thermal conductivity due to which the heat transfer characteristics are increased substantially. Further， the magnetohydrodynamic （MHD） nanofluids are important in engineering and industrial processes. Especially， such fluids are involved in the optical modulators， optical grating， optical switches， tunable optical fiber filters，stretching of plastic sheets and metallurgy， polymer industry etc.. Many metallurgical processes involve the cooling of continuous strips or filaments by drawing them through a nanofluid. Such strips are sometimes stretched in the processes of drawing， thinning of copper wires and annealing. For such situations， thequality and desired characteristics of final product are obtained by drawing such strips in an electrically conducting fluid. The magnetic nanoparticles have key role in the construction of loudspeakers， magnetic cell separation， hyperthermia， drug delivery etc. The general applications of nanofluids include vehicle cooling，generating new types of fuel， reduction of fuel in electric power generation plant， cancer therapy， imaging and sensing etc. Some recent attempts on this topic can be seen in the references［3-10］.

The investigations on mixed convective transport in thermally or/and solute stratified fluids is a topic of tremendous interest during the past few years due to their occurrence in industrial and engineering applications. Examples of such applications are heat rejection from environment like lake， seas and rivers， thermal energy storage systems like solar ponds and heat transfer from thermal sources like condenser of the power plants［11］. Thermal or solute stratification of is a deposition of fluid layers which exists due to temperature or concentration variations or due to the presence of different fluids. It is very interesting and important to study the influences of thermal and solute stratifications of oxygen and hydrogen in the rivers and lakes. All the cultured species are directly affected by these effects. Further， the influences of thermal and solute stratifications are quite important for solar energy because the better stratification corresponds to higher energy performance. Ibrahim and Makinde［12］reported the boundary layer flow of viscous nanofluid past a vertical plate in the presence of thermal and solute stratifications. Hayat et al.［13］studied the impact of thermal stratification on mixed convection flow of Maxwell fluid with thermal radiation. Doubly stratified flow of MHD free convection micropolar fluid was addressed by Srinivasacharya and Upendar［14］. Simultaneous effects of thermal stratification and thermal radiation in mixed convection flow of thixotropic fluid induced by a stretching sheet was analyzed by Shehzad et al.［15］.

The emphasis of the present work is on the influence of double stratification in boundary layer flow of Jeffrey fluid with thermal radiation. Jeffrey fluid with nanoparticles is taken in the problem formulation. An incompressible laminar flow is considered in the presence of an applied magnetic field. The magnetic field is applied in the direction normal to the flow. Effects of thermophoresis and Brownian motion are also considered. The governing mathematical expressions are first converted into the dimensionless forms and then series solutions via homotopy analysis method （HAM）［16-20］are computed. Graphs are displayed for the dimensionless temperature and nanoparticle concentration profiles. Numerical values of physical quantities like skin-friction coefficient， local Nusselt and local Sherwood numbers are tabulated and discussed.

1. Problems development

We consider the MHD flow of Jeffrey nanofluid over a stretching sheet. The fluid flow and heat transfer are assumed to be incompressible and laminar. Magnetic field of strength 0B is applied normal to the flow field. The small magnetic Reynolds number is chosen. As a consequence the induced magnetic field is weaker in comparison to the applied magnetic field. Thus the induced magnetic field is not considered. Electric field is absent. All the physical properties are taken to be constant and independent of temperature and position. The effect of sheet axial conduction is neglected. The two-dimensional MHD boundary layer equations of an incompressible Jeffrey nanofluid are

The boundary conditions for the considered flow analysis are

where u and v are the velocity components in the x- and y-directions， ν the kinematic viscosity， λ1the ratio of relaxation to retardation times，2λ the retardation time，fρ the density of fluid， σ the Steffan-Boltzman constant， g the acceleration due to gravity，Tβ the thermal expansion coefficient，Cβ the concentration expansion coefficient， α the thermal diffusivity， τ=（ρc）p/（ρc）fthe ratio of nanoparticle heat capacity and the base fluid heat capacity，BD the Brownian diffusion coefficient，TD the thermophoretic diffusion coefficient， c is the stretching rate， a，b， d， e are dimensional constants and0T，0C are the reference temperature and reference concentration，respectively.

The radiative flux is accounted by employing the Rosseland assumption in the energy equation

in which σ*the Stefan-Boltzmann constant and k*the mean absorption coefficient. Further， the differences of temperature within the flow is assumed to be small such that4T may be expressed as a linear function of temperature. Expanding4T about T∞via Taylor's series and ignoring higher order terms， we have

By employing Eqs.（7） and （8）， Eq.（3） has the form

The Eqs.（2）-（6） and （9） can be reduced into the dimensionless form by introducing the following new variables

The above transformations satisfy Eq.（1） identically. The equations of linear momentum， energy and concentration in dimensionless form become

where β=λ2c is the Deborah number， M2=σ/ ρfcis the Hartman number， λ=Grx/the thermal buoyancy parameter with=gβ（T-T）x3/ T ∞the local Grashof number and=u（x）x/ν the w local Reynolds number， Pr=ν/α is the Prandtl number， Le=ν/DBis the Lewis number， Nb=is the Brownian motion parameter，is the thermophoresis parameter， Rd=（4σ*）/kk*the thermal radiation parameter， ST=a/b the thermal stratification parameter and SC=e/d the concentration stratification parameter. It is worth mentioning to point out that the analysis reduces to viscous fluid case when β=0=λ1.

The skin friction coefficient， the local Nusselt number and the local Sherwood number are

wherewτ is the shear stress along the stretching surface，wq is the surface heat flux andmq is the surface mass flux. The local skin-friction coefficient， local Nusselt and local Sherwood numbers in dimensionless forms are given below:

2. Homotopy analysis solutions

By choosing a set of base functions［16］

｛ηkexp（-nη），k≥0，n≥0｝

we can express （）fη， （）θη and （）φη in the following forms

The above initial guesses and auxiliary linear operators satisfy the properties

where Ci（i=1-7） are the arbitrary constants.

The zeroth order problems are defined as［17，18］:

Nθ［θ（η，q），f（η，q），φ（η，q ）］=

wheref?，θ? andφ? are the non-zero auxiliary parameters， ［0，1］q∈ is an embedding parameter andNf， Nθand Nφare the nonlinear operators. Putting q=0 and q=1 one has

When we increase the values of q from 0 to 1 then （，）fqη， （，）qθη and （，）qφη vary from0（）fη， θ0（η）， φ0（η） to f（η）， θ（η） and φ（η） By adopting Taylor series expansion， we have:

The convergence of above series highly depends upon the suitable values off?，θ? andφ?. Considering thatf?，θ? andφ? are selected properly such that Eqs.（32）-（34） converge at =1q and then we have

The general solutions can be written as follows:

3. Analysis and discussion

The computations of series solutions via homotopy analysis method involve the auxiliary parameters ?f， ?θand ?φwhich are used in adjusting and controlling the convergence rate of homotopic solutions. To select the appropriate values of such auxiliary parameters， we have drawn the -?curves at 15th-order of HAM deformations. From Fig.1 we observed that the suitable values off?，θ? andφ? are 1.10-≤ ?f≤-0.20， -1.00≤?θ≤-0.15， -1.00≤?φ≤-0.20. The series converges in the whole region of η when ?f=-0.6=?θ=?φ（see Table 1）.

Fig.1 ?-curves for the functions f（η）， θ（η） and φ（η） at 15thorder of approximations when λ1=0.3， β=0.2，M=0.7， λ=0.3=N ， Pr=1.0=Le， Nt=0.2= Nb， Rd=0.4 and ST=0.2=SC

Table 1 Convergence of homotopy solution for different order of approximations when λ1=0.3， β=0.2，λ=0.3=N ， Pr=1.0=Le， Nt=0.2=Nb，Rd=0.4， ST=0.2=SC　and ?f=-0.6=?θ= ?φ

Fig.2 Variations in the temperature distribution θ（η） vs η corresponding to different values of M when λ1=0.3，β=0.2， λ=0.3=N ， Pr=1.0=Le， Nt=0.2=Nb，Rd=0.4 and ST=0.2=SC

Fig.4 Variations in the temperature distribution θ（η） vs η corresponding to different values of λ1when β=0.2，M=0.7， λ=0.3=N ， Pr=1.0=Le， Nt=0.2= Nb， Rd=0.4 and ST=0.2=SC

Fig.3 Variations in the temperature distribution θ（η） vs η corresponding to different values of β when λ1=0.3，M=0.7， λ=0.3=N ， Pr=1.0=Le， Nt=0.2= Nb， Rd=0.4 and ST=0.2=SC

Fig.5 Variations in the temperature distribution θ（η） vs η corresponding to different values of λ when λ1=0.3，β=0.2， M=0.7， N=0.3， Pr=1.0=Le， Nt= 0.2=Nb， Rd=0.4 and ST=0.2=SC

Fig.6 Variations in the temperature distribution θ（η） vs η corresponding to different values of N when λ1=0.3，β=0.2， M=0.7， λ=0.3， Pr=1.0=Le， Nt= 0.2=Nb， Rd=0.4 and ST=0.2=SC

Impacts on the non-dimensional temperature distributionfunction （）θη fordifferentvaluesof Hartman number M Deborah number β， ratio of relaxation to retardation times1λ mixed convection parameters λ and N Prandtl number Pr thermophoresis parametertN thermal stratification parameterTS and thermal radiation parameter Rd are studied in the Figs.2-10. It is noted from Fig.2 that the temperature field is enhanced corresponding to the larger values of Hartman number M. Here an application of the transverse magnetic field normal to the flow direction creates a drag force known as Lorentz force. Such force caused a resistance in the fluid flow due to which the temperature and thermal boundary layer thickness are increased for the larger Hartman number. Figure 3 depicts that both the temperature and thermal boundary layer thickness are reduced with an increase in Deborah number. Physically， Deborah number β is directly proportional to the retardation time. Here larger Deborah number corresponds to higher retardation time. Such higher retardation time creates a reduction in the temperature and thermal boundary layer thickness. An increase in ratio of relaxation to retardation times leads to an enhancement inthe temperature profile and thermal boundary layer thickness （see Fig.4）. From Figs.5 and 6 we observed that the temperature and thermal boundary layer thickness are lower for higher values of mixed convection parameters λ and N. It is also noted that the variations in temperature due to λ are more pronounced in comparison to N even the values of λ are smaller than N.

Fig.7 Variations in the temperature distribution θ（η） vs η corresponding to different values of Pr when λ1=0.3，β=0.2， M=0.7， λ=0.3=N ， Le=1.0， Nt= 0.2=Nb， Rd=0.4 and ST=0.2=SC

Fig.8 Variations in the temperature distribution θ（η） vs η corresponding to different values of Ntwhen λ1=0.3，β=0.2， M=0.7， λ=0.3=N， Pr=1.0=Le，Nb=0.2， Rd=0.4 and ST=0.2=SC

Figure 7 depicts that an increase in Prandtl number leads to a decrease in the temperature and thermal boundary layer thickness. The suitable choice of Prandtl number is quite necessary in the industrial applications to control the heat transfer rate during the cooling process. From Fig.8， we have seen that the temperature and thermal boundary layer thickness are enhanced by increasing the values of thermophoresis parameter Nt. Physically the presence of nanoparticles enhanced the thermal conductivity of fluid. The fluid with stronger thermal conductivity has higher temperature and weaker thermal conductivity fluid corresponds to lower temperature. Here an increase in thermophoresis parameter leads to enhancement in the thermal conductivity of fluid due to which thicker thermal boundary layer exists. Figure 9 illustrates the variations in temperature field for the different values of thermal stratification parameterTS. Here it is analyzed that both the temperature and thermal boundary layer thickness are reduced with the increasing values of thermal stratification parameter. Due to an increase in thermal stratification parameter， the temperature difference between the sheet and ambient fluid is decreased that corresponds to a lower temperature and thinner thermal boundary layer thickness. Figure 10 elucidates that an increase in radiation parameter gives rise to the temperature and thermal boundary layer thickness. Due to the enhancement in the radiation parameter， more heat is emitted by the fluid that leads to higher temperature and thermal boundary layer thickness.

Fig.9 Variations in the temperature distribution θ（η） vs η corresponding to different values of STwhen λ1=0.3，β=0.2， M=0.7， λ=0.3=N ， Pr=1.0=Le，Nt=0.2=Nb， Rd=0.4 and SC=0.2

Fig.10 Variations in the temperature distribution θ（η） vs η corresponding to different values of Rd when λ1=0.3， β=0.2， M=0.7， λ=0.3=N ，Pr=1.0=Le， Nt=0.2=Nb　and ST=0.2=SC

Fig.11 Variations in the nanoparticle concentration distribution θ（η） vs η corresponding to different values of Le when λ1=0.3， β=0.2， M=0.7， λ=0.3=N ，Pr=1.0， Nt=0.2=Nb， Rd=0.4 and ST=0.2= SC

Fig.12 Variations in the nanoparticle concentration distribution θ（η） vs η corresponding to different values of Ntwhen λ1=0.3， β=0.2， M=0.7， λ=0.3=N ，Pr=1.0=Le， Nb=0.2， Rd=0.4 and ST=0.2= SC

Fig.13 Variations in the nanoparticle concentration distribution θ（η） vs η corresponding to different values of Nbwhen λ1=0.3， β=0.2， M=0.7， λ=0.3=N，Pr=1.0=Le， Nt=0.2， Rd=0.4 and ST=0.2= SC

Fig.14 Variations in the nanoparticle concentration distribution θ（η） vs η corresponding to different values of SCwhen λ1=0.3， β=0.2， M=0.7， λ=0.3=N ，Pr=1.0=Le， Nt=0.2=Nb， Rd=0.4 and ST= 0.2

Table 2 Numerical values of skin friction coefficient RCffor different values of λ1， M， β and λ when Pr=1.0=Le， Nt=0.2=Nb， Rd= 0.4 and ST=0.2=SC

Table 2 Numerical values of skin friction coefficient RCffor different values of λ1， M， β and λ when Pr=1.0=Le， Nt=0.2=Nb， Rd= 0.4 and ST=0.2=SC

λ1M β λ N -Re1x/2Cf 0.1 0.7 0.2 0.3 0.3 1.11097 0.4 0.7 0.2 0.3 0.3 0.97340 0.7 0.7 0.2 0.3 0.3 0.87539 0.3 0.0 0.2 0.3 0.3 0.79447 0.3 0.5 0.2 0.3 0.3 0.91104 0.3 1.0 0.2 0.3 0.3 1.20901 0.3 0.7 0.0 0.3 0.3 0.91586 0.3 0.7 0.3 0.3 0.3 1.05963 0.3 0.7 0.5 0.3 0.3 1.14651 0.3 0.7 0.2 0.0 0.3 1.17277 0.3 0.7 0.2 0.5 0.3 0.91683 0.3 0.7 0.2 1.0 0.3 0.69008

The effects of Lewis number Le， thermophoresis parametertN， Brownian motion parameterbN and nanoparticle concentration stratification parameterCS on the nanoparticle concentration field （）φη are investigated in the Figs.11-14. Figure 11 shows that the nanoparticle concentration and its associated boundary layer thickness are higher for smaller Lewis number and lower for larger Lewis number. Lewis number is dependent on the Brownian diffusion coefficient here. Larger Lewis number corresponds to weaker Brownian diffusion coefficient and smaller Lewis number implies stronger Brownian diffusion coefficient. Such change in Brownian diffusion coefficient is responsible for a reduction in the nanoparticle concentration. Increasing values of thermophoresis parameter enhance the nanoparticle concentration field （see Fig.12）.Here peak is attained when =0.8Nt and =1.0η， It is examined from Fig.13 that nanoparticle concentration field and its related boundary layer thickness are decreasing functions of Brownian motion parameter Nb， Figure 14 is presented to study the change in nanoparticle concentration profile for different values of concentration stratification parameterCS. Here we noted that the higher values ofCS show a lower nanoparticle concentration profile.

Table 3 Numerical values of local Nusselt number -（1+4/3Rd）θ＇（0）and local Sherwood number -φ＇（0） for different values of Rd， ST， SC， Nt， Nb， Le and Pr when M=0.7 and λ=0.3=N

Table 1 is computed to analyze the rate of convergence of homotopic solutions when1=0.3λ， =β 0.2， M=0.7， λ=0.3=N ， Pr=1.0=Le， Nt= 0.2=Nb， Rd=0.4， ST=0.2=SCand ?f=-0.6= ?θ=?φ. It is found that the values of f＇（0） converge from 10th-order of approximations while the values of （0）θ＇ and （0）φ＇ start to repeat from 24th-order of HAM deformations. Hence 24th-order of HAM computations gives us convergent series solutions. The values of skin-friction coefficient RCffor different values of1λ， M， β and λ are computed in Table 2 when N=0.3， Pr=1.0=Le， Nt=0.2=Nb， Rd=0.4 and ST=0.2=SC. From this Table， it is observed that the values of skin-friction coefficient are smaller for the higher values of 1λ and λ and larger for the higher values of M and β， Table 3 is constructed to investigate the values of local Nusselt number -（1+4/3Rd）θ＇（0）and local Sherwood number -φ＇（0） for different values of Rd， ST， SC， Nt， Nb， Le and Pr when λ1=0.3=λ=N ， β=0.2 and =0.7M. The values of local Nusselt and Sherwood numbers are quite opposite for TS and CS. Further， we also computed the values of local Nusselt and Sherwood numbers for viscous fluid case i.e.，β=0=λ1Here we observed that the numerical values of local Nusselt and Sherwood numbers are larger for viscous fluid case when we compared it with the non-Newtonian Jeffrey fluid.

4. Conclusions

We studied the two-dimensional laminar MHD flow of Jeffrey nanofluid with Brownian motion， thermophoresis and thermal radiation effects. Thermal and nanoparticle concentration stratifications effects are discussed. The main points of this research work are summarized below.

（1） To develop convergent HAM solutions， one has to compute 24th-order of deformations.

（2） Deborah number β causes a reduction in dimensionless temperature profile while an enhancement is noticed with the increasing values of ratio of relaxation to retardation times.

（3） Change in temperature profile for mixed convection parameter λ is more pronounced in comparison to N when the values of λ are smaller than N.

（4） Temperature and thermal boundary layer thickness are enhanced through larger thermophoresis parametertN.

（5） Temperature and nanoparticle concentration fields are larger for the smaller values ofTS andCS.

（6） Nanoparticle concentration profile is increased with an enhancement in thermophoresis parameter Ntbut is reducesd for larger Brownian motion parameter.

References

［1］CHEN H.， DING Y. Heat transfer and rheological behavior of nanofluids a review［J］. Advances in Transport Phenomena， 2009， 1: 135-177.

［2］CHOI S. U. S. Enhancing thermal conductivity of fluids with nanoparticles［C］. The Proceedings of the 1995 ASME International Mechanical Engineering Congress and Exposition. San Francisce， USA， 1995， 99-105.

［3］RASHIDI M. M.， BEG O. A. and ASADI M. et al. DTM-padé modeling of natural convective boundary layer flow of a nanofluid past a vertical surface［J］. Journal of Environmental Engineering， 2012， 4（1）: 13-24.

［4］TURKYILMAZOGLU M.， POP I. Heat and mass transfer of unsteady natural convection flow of some nanofluids past a vertical infinite flat plate with radiation effect［J］. International Journal of Heat and Mass Transfer， 2013，59（1）: 167-171.

［5］KHAN W. A.， MAKINDE O. D. MHD nanofluid bioconvection due to gyrotactic microorganisms over a convectively heat stretching sheet［J］. International Journal of Thermal Sciences， 2014， 81: 118-124.

［6］SHEIKHOLESLAMI M.， GORJI-BANDPY M. and GANJI D. D. MHD free convection in an eccentric semiannulus filled with nanofluid［J］. Journal of the Taiwan Institute of Chemical Engineers， 2014， 45（4）: 1204-1216.

［7］SHEIKHOLESLAMI M.， GORJI-BANDPY M. and GANJI D. D. Lattice Boltzmann method for MHD natural convection heat transfer using nanofluid［J］. Powder Technology， 2014， 254（2）: 82-93.

［8］IMTIAZ M.， HAYAT T. and HUSSAIN M. et al. Mixed convection flow of nanofluid with Newtonian heating［J］. European Physical Journal Plus， 2014， 129（5）: 1-11.

［9］HAYAT T.， ABBASI F. M. and AL-YAMI M. et al. Slip and Joule heating effects in mixed convection peristaltic transport of nanofluid with Soret and Dufour effects［J］. Journal of Molecular Liquids， 2014， 194（6）: 93-99.

［10］ LIN Y.， ZHENG L. and ZHANG X. Radiation effects on Marangoni convection flow and heat transfer in pseudoplastic non-Newtonian nanofluids with variable thermal conductivity［J］. International Journal of Heat and Mass Transfer， 2014， 77（4）: 708-716.

［11］ SRINIVASACHARYA D.， SURENDER O. Non-Darcy mixed convection in a doubly stratified porous medium with Soret-Dufour effects［J］. International Journal of Engineering Mathematics， 2014， 2014: Article ID 126218.

［12］ IBRAHIM W.， MAKINDE O. D. The effect of double stratification on boundary-layer flow and heat transfer of nanofluid over a vertical plate［J］. Computers and Fluids，2013， 86（7）: 433-441.

［13］ HAYAT T.， SHEHZAD S. A. and AL-SULAMI H. H. et al. Influence of thermal stratification on the radiative flow of Maxwell fluid［J］. Journal of the Brazilian Society of Mechanical Sciences and Engineering， 2013， 35（4）: 381-389.

［14］ SRINIVASACHARYA D.， UPENDAR M. Effect of double stratification on MHD free convection in a micropolar fluid［J］. Journal of the Egyptian Mathematical Society，2013， 21（3）: 370-378.

［15］ SHEHZAD S. A.， QASIM M. and ALSAEDI A. et al. Combined thermal stratified and thermal radiation effects in mixed-convection flow of a thixotropic fluid［J］. European Physical Journal Plus， 2013， 128（1）: 1-10.

［16］ LIAO S. J. Homotopy analysis method in nonlinear differential equations［M］. Heidelberg， Germany: Springer and Beijing， China: Higher Education Press， 2012.

［17］ ABBASBANDY S.， HASHEMI M. S. and HASHIM I. On convergence of homotopy analysis method and its application to fractional integro-differential equations［J］. Quaestiones Mathematicae， 2013， 36（1）: 93-105.

［18］ TURKYILMAZOGLU M. Solution of the Thomas-Fermi equation with a convergent approach［J］. Communications in Nonlinear Science and Numerical Simulation，2012， 17（11）: 4097-4103.

［19］ SHEHZAD S. A.， ALSAEDI A. and HAYAT T. et al. Three-dimensional flow of an Oldroyd-B fluid with variable thermal conductivity and heat generation/absorption［J］. Plos One， 2013， 8（11）: e78240.

［20］ ABBASBANDY S.， HAYAT T. and ALSAEDI A. et al. Numerical and analytical solutions for Falkner-Skan flow of MHD Oldroyd-B fluid［J］. International Journal of Numerical Methods for Heat and Fluid Flow， 2014，24（2）: 390-401.

（August 12， 2014， Revised May 7， 2016）

* Biography： F. M. ABBAS （1987-）， Male， Ph. D.，Assistant Professor